Research ArticleModelling and Optimizing an Open-Pit TruckScheduling Problem
Yonggang Chang12 Huizhi Ren1 and Shijie Wang1
1School of Mechanical Engineering Shenyang University of Technology Shenyang 110870 China2ChinaCoal Pingshuo Group Co Ltd Shuozhou 036006 China
Correspondence should be addressed to Huizhi Ren renhuizhi126com
Received 18 December 2014 Accepted 14 February 2015
Academic Editor Purushothaman Damodaran
Copyright copy 2015 Yonggang Chang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper addresses a special truck scheduling problem in the open-pit mine with different transport revenue consideration Amixed integer programmingmodel is formulated to define the problem clearly and a few valid inequalities are deduced to strengthenthe model Some properties and two upper bounds of the problem are proposed Based on these inequalities properties and upperbounds a heuristic solution approach with two improvement strategies is proposed to resolve the problem and the numericalexperiment demonstrates that the proposed solution approach is effective and efficient
1 Introduction
In an open-pit mine trucks are a kind of crucial equipmentwhich undertakes nearly all ore andwaste rock transportation[1] Typically the truck is loaded a full load ofmaterial (maybeore gangue or other waste rocks that cover ores and thusneed to be moved away) under an electric shovel transportsit to a dumping depot dumps its loaded material and thenreturns back to the shovel to carry out the next round Figure 1illustrates an open-pit photo and a simplified map for theroad network in an open-pit mine All of thematerial loadingpoints each of which is equipped with an electric shoveland dumping depots are relatively constant while trucks cantransport material from varied loading points and unload atdifferent depots [2] The efficient truck scheduling can notonly reduce the truck transportation cost but also increase theshovel utilization ratio and the mine productivity [1]
There are plenty of researches published on the operationsresearch in the open-pit mine but most of them focus on thepit design and production planning [3] such as the strategicultimate pit limit design problem [4ndash7] and tactical block-sequencing problem [8ndash11] Among researches on the tacticaland operational equipment-allocation in the open-pit minethe stochastic feature of equipment has been emphasizedBoth queuing theory and simulation were used as stochastic
methodologies tomodel and analyse the open-pit productionscheduling problem especially in the early published liter-atures For example Manla and Ramani [12] simulated thetruck haulage process which initialized the truck schedulingresearch Kappas and Yegulalp [13] used the queuing theorycombined with extensions of Markovian principles to ana-lyze the steady-state performance of a typical truck-shovelsystem and estimate its operation parameters Oraee and Asi[14] simulated the production process of trucks and shovels inSongun Copper Mine in Iran considering fuzzy parameterson each shovel Najor and Hagan [15] focused on stochasticbehavior in the truck-shovel system and provided a modeland analysis based on queuing theory Their analysis showedthat productivity can be overestimated by about 8 percentbecause of overlooking queuing
In some of the open-pit equipment routing and selectionmodels optimization methodologies that is integer pro-gramming and dynamic programming were used to deter-mine fleet size and allocation [3] For example White andOlson [16] applied networkmodels linear programming anddynamic programming methods to the truck schedulingproblem and proposed a feasible truck-dispatching systemThey divided the problem into three stages The first one wasto decide the shortest paths between all locations in theminewhich is a shortest path problem actually In the next stage a
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 745378 8 pageshttpdxdoiorg1011552015745378
2 Discrete Dynamics in Nature and Society
(a) An open-pit mine
L1
L2
L3
L4
L5L6
D1
D2
D3
(b) A sketch map of the open-pit road networkwhere 1198711 1198713 1198716 represent the loading pointsand11986311198632 and1198633 represent the dumping depots
Figure 1 An open-pit mine and a part of its road network
linear program was used to determine material flows alongthese paths Finally the dynamic program assigned trucks tothe optimized paths between shovels and dumpsTheir truck-dispatching system not only yields significant (10ndash20 percentin most cases) productivity improvements but also producedan effect on truck scheduling theory for its three-stage process[17] And Yang et al [18] adopted the three-stage process toresolve the truck dispatch problem
In this paper we consider the varied transportationrevenue for different material loading point and address thejoint path planning and tuck dispatching problem
The outline of this paper is as follows Section 2 describesthe considered open-pit truck scheduling problem withvaried truck transporting revenue in detail and provides itsinteger programming model In Section 3 a few validinequalities two properties of optimal solution and twoupper bounds for the problem are presented In Section 4 aheuristic approach with two improvement strategies is pro-posed to resolve the problem Section 5 tests the proposedupper bounds and solution approach Section 6 concludes thepaper
2 Problem Description andMathematical Model
21 Problem Description In the open pit there are usuallyseveral loading points and dumping depots (to be calleddumps later) as shown in Figure 1 At each loading pointthere is always a pile of loosened ores gangue or waste rocks(to be referred to as material collectively) since the load-ing point needs a loosening blasting before the excavationoperation starting Each loading point is assumed to be
equipped with one and only one electric shovel (without lossof generality it will be viewed as twodifferent loading points iftwo shovels are equipped) The electric shovel excavates andloads the loosened material onto the trucks at each loadingpoint and the truck unloads itself with its self-dischargingdevice at a dump Similar to the loading point we assumethat one dump can accommodate only one truck at the sametime and the unloading area of more than one truck loadingpositions will be viewed as several different dumps
It is worth noting that an electric shovel (or loading point)can load only one truck at the same time and a dump allowsonly one truck simultaneously There is often a park for thewaiting trucks near an operating electric shovel or dump It isactually the key task to reduce the truck waiting times in thetraditional truck scheduling problem
In our considered problem scenario a truck of materialfrom different loading point is given different transportrevenue and our optimization objective will be equal tothe total transport revenue completed in the schedulinghorizonThe revenue per truck loading (transporting a truckof material) of different loading point is always decided bythe management and the decision-making basis includes twoconsiderations the possible transport distance and the trans-port priority The former is easy to be understood since thelonger distance always causes more transport times and thusshould be given more revenue The transport priority comesfrom the mining sequence which has never been modeled asa kind of constraints in the traditional research [1 3] andorthe varied market demand The transport revenue per truckloading can be viewed as the product of the possible transportdistance and the transport priority
From the viewpoint of a scheduler who is in charge ofmaking truck scheduling decisions in a shift (8 hours) ourobjective is to maximize the total transport revenues basedon the given revenue value per truck loading The decisionsinclude the dump selection as well as the truck routingThe dump selection is virtually the same as the first-stagedecisions in literature [16 18] deciding the shortest pathbetween all locations in the mine
The constraints to be considered include the following(1) loading capacity constraint that is each shovel (load-
ing point) can load only one truck at the same time(2) unloading capacity constraint that is each dump can
unload only one truck at the same time(3) dump selection constraint that is a set of candidate
dumps is given for each loading point and thematerialfrom the loading point must be dumped at one of thecandidate dumps
22 The Mixed Integer Programming Model To define andformulate the problemexplicitly the followingnotations needto be introduced to represent all loading points dumpsmaterial to be loaded in the loading points trucks pathsparameters and so on
119878 set of available electric shovels (loading points) 119878 =1 2 1198731198781015840 1198781015840 = 119878 cup 0 0 denoting dummy shovels
Discrete Dynamics in Nature and Society 3
119879 set of available trucks119886119894 revenue per truck of material from electric shovel119894 119894 isin 119878119863119894 set of candidate dumps corresponding to material
(waste rock or coal) loaded at electric shovel 119894 119894 isin 119878119863 = cup
119894isin119878119863119894 including all dumps
119867 the horizon considered in the problem119898119894 max number of truck loadings an electric shovel 119894
can complete at most during the horizon 119894 isin 119878119870119894 119870119894= 1 2 119898
119894 set of possible trucks of rock
(coal) that electric shovel 119894 loads 119894 isin 119878(119896 119894) a truck of material loaded by shovel 119894 119894 isin 119878119896 isin 119870
119894 hereafter referred to as truck loading (119896 119894)
11987001198700= 0 (0 0) denoting dummy truck loading
1199010
119894 loading times per truck at electric shovel 119894 119894 isin 1198781199011
119895 unloading times per truck at depot 119895 119895 isin 119863
1198750
119895119894 times for an empty truck to move from depot 119895 to
shovel 119894 119895 isin 119863 119894 isin 1198781198751
119894119895 times for a full-loaded truck to move form shovel119894 to depot 119895 119895 isin 119863 119894 isin 119878119902119905
119894 times for truck 119905 to move from its initial position
to shovel 119894 119894 isin 119878 119905 isin 119879119872 very big plus number
To represent the dump selection and truck dispatchdecisions we introduce the following decision variables
11990911990511989610158401198941015840119896119894=
1 if truck 119905 transports (119896 119894)
immediatly after (1198961015840 1198941015840)
0 otherwise
for 119905 isin 119879 119894 1198941015840 isin 1198781015840 119896 isin 119870119894 1198961015840isin 1198701198941015840
119911119896119894=
1 if truck loading (119896 119894) is loaded
and transported in the horizon
0 otherwise
for 119894 isin 119878 119896 isin 119870119894
11990611989511989610158401198941015840119896119894
=
1 if material (119896 119894) is dumped immediatly
after (1198961015840 1198941015840) at despot 119895
0 otherwise
for isin 119863119894cap 1198631198941015840 119894 1198941015840isin 119878 119896 isin 119870
119894 1198961015840isin 1198701198941015840
119910119896119894119895=
1 if material (119896 119894) is dumped at despot 119895
0 otherwise
for119895 isin 119863119894119894 isin 119878 119896 isin 119870
119894
(1)
1198880
119896119894is complete time of loadingmaterial (119896 119894) for 119894 isin 119878
119896 isin 1198701015840
119894
1198881
119896119894is complete time of unloading material (119896 119894) for
119894 isin 119878 119896 isin 1198701015840119894
Based on the above notations the problem can be formu-lated as the following integer programmingmodel (similar tomodel in Tang et al [19] which involved a truck schedulingproblem in container terminals)
Max sum
119894isin119878
sum
119896isin119870119894
119886119894119911119896119894 (2)
st 1198880
119896+1119894minus 1198880
119896119894ge 1199010
119894 119894 isin 119878 119896 isin 119870
119894(3)
1198881
119896119894minus 1198881
11989610158401198941015840 ge 1199011
119895minus119872(1 minus 119906
11989511989610158401198941015840119896119894)
119894 1198941015840isin 119878 119896 isin 119870
119894
1198961015840isin 1198701198941015840 119895 isin 119863
119894cap 1198631198941015840
(4)
1198881
119896119894minus 1198880
119896119894ge sum
119895isin119863
119910119896119894119895(119875119894119895+ 1199011
119895) 119894 isin 119878 119896 isin 119870
119894 (5)
1198880
119896119894minus 1198881
11989610158401198941015840 ge sum
119895isin119863
11991011989610158401198941015840119895119875119895119894+ 1199010
119894
minus119872(1 minus sum
119905isin119879
11990911990511989610158401198941015840119896119894)
119894 1198941015840isin 119878 119896 isin 119870
119894 1198961015840isin 1198701198941015840
(6)
1198880
119896119894ge sum
119905isin119879
11990911990500119896119894
119902119905
119894+ 1199010
119894minus119872(1 minus sum
119905isin119879
11990911990500119896119894
)
119894 isin 119878 119896 isin 119870119894
(7)
1198881
119896119894le 119867 +119872(1 minus 119911
119896119894) 119894 isin 119878 119896 isin 119870
1015840
119894(8)
sum
119894isin1198781015840
sum
119896isin119870119894
11990911990511989610158401198941015840119896119894= 11991111989610158401198941015840 119905 isin 119879 119894
1015840isin 119878 119896
1015840isin 1198701198941015840 (9)
sum
1198941015840isin1198781015840
sum
1198961015840isin1198701198941015840
11990911990511989610158401198941015840119896119894= 119911119896119894 119905 isin 119879 119894 isin 119878 119896 isin 119870
119894 (10)
119909119905119896119894119896119894= 0 119905 isin 119879 119894 isin 119878 119896 isin 119870
119894 (11)
sum
119894isin1198781015840
sum
119896isin119870119894
11990911990511989611989400
= sum
119894isin1198781015840
sum
119896isin119870119894
11990911990500119896119894
= 1 119905 isin 119879 (12)
sum
1198941015840isin1198781015840
sum
1198961015840isin1198701015840
119894
11990611989511989610158401198941015840119896119894= sum
1198941015840isin1198781015840
sum
119896isin1198701015840
119894
11990611989511989611989411989610158401198941015840 = 119911119896119894
119905 isin 119879 119894 isin 119878 119896 isin 1198701015840
119894
(13)
119906119895119896119894119896119894= 0 119895 isin 119863
119894 119894 isin 119878 119896 isin 119870
1015840
119894(14)
sum
119894isin119878
sum
119896isin119870119894
11990611989511989611989400
= sum
119894isin119878
sum
119896isin119870119894
11990611989500119896119894
= 1 119895 isin 119863 (15)
sum
119895isin119863119894
119910119896119894119895= 119911119896119894 119894 isin 119878 119896 isin 119870
119894 (16)
4 Discrete Dynamics in Nature and Society
The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911
119896119894 that
is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon
119867 Constraints (9) and (10) guarantee that each valid truckloading (119911
119896119894= 1) is assigned to one and only one truck and
each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910
119896119894119895and 119911119896119894 that is each valid
truck loading has one and only one dumping depot
3 Valid Inequalities Property andUpper Bounds for the Problem
31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach
311 Electric Shovel Capacity Inequality Consider
sum
119896isin119870119894
1199111198961198941199010
119894le 119867 minusmin
119905isin119879
119902119905
119894minusmin119895isin119863119894
1199011
119894119895+ 1199011
119895 119894 isin 119878 (17)
Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min
119905isin119879119902119905
119894minus min
119895isin119863119894
1199011
119894119895is the practical time
limit for shovel 119894 where min119905isin119879119902119905
119894is the time for the nearest
empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min
119895isin119863119894
1199011
119894119895+ 1199011
119895 is the permitted least
retained time to transport and unload the last truck loadedby shovel 119894
L1
L2
L3
L4
D1
D2
D3
Figure 2 Illustration of the truck round trip
312 Unloading Capacity Inequality Consider
sum
119894isin119878
sum
119896isin119870119894
1199101198961198941198951199011
119895le 119867 minus min
119894isin1198941015840|119895isin1198631198941015840
min119905isin119879
119902119905
119894+ 1199010
119894+ 1199011
119894119895
119895 isin 119863
(18)
Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin
119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+1199011
119894119895 is the practi-
cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+
1199011
119894119895 is the earliest possible starting time of unloading the first
truck in depot 119895
313 Variables Relationship Inequality Consider
119911119896119894le 1199111198961015840119894 119896 le 119896
1015840 119894 isin 119878 (19)
Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands
32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties
Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894
moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863
119894 and unloads the material
(eg travelling from dump1198631to loading point 119871
1and to119863
3
in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)
It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot
Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890
1198951198941198951015840 = 119886119894(1198750
119895119894+1198751
1198941198951015840 +1199010
119894+1199011
119895) is called trip transport
value
Discrete Dynamics in Nature and Society 5
Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value
Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times
(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel
(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot
(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon
It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011
119894119895and1199010
119895119894) through
the path
Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to
sum
119905isin119879
sum
(1198951198941198951015840)isinΨ
11989711990511989511989411989510158401198901198951198941198951015840 (20)
where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has
spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips
The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound
Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750
119895119894+ 1198751
1198941198951015840 + 1199010
119894+ 1199011
119895) sdot
min1198850119894 1198851
1198951015840 where1198850119894 and119885
1
1198951015840 denote themaximum loading
capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively
In Property 2 1198850119894and 1198851
1198951015840 can be obtained through
formulae (17) and (18) respectively
33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem
331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum
119894isin1198781198861198941198850
119894
In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873
119879denote the number of available trucks then
119862119879= 119867 sdot 119873
119879denotes the total available truck times involved
in the problem Let 119903119894denote the optimal truck round trip
involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires
the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast
119894and 119864lowast
119894denote the required valuable truck times and trip
transport value corresponding to trip 119903119894 respectively
It is assumed that 1198941 1198942 119894
119873is the shovel sequence in
the descending order of 119864lowast119894 that is 119864lowast
1198941
ge 119864lowast
1198942
ge sdot sdot sdot ge 119864lowast
119894119873
and1198941 1198942 119894
119873 = 119878
Upper bound 2 can be described and formulated clearlybased on the above notations and assumption
332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value
119891 (119862119879) =
119864lowast
1198941
119862119879
119862119879le 1198850
1198941
119875lowast
1198941
1198861198941
1198850
1198941
+ 119864lowast
1198942
(119862119879minus 1198850
1198941
119875lowast
1198941
)
1198850
1198941
119875lowast
1198941
le 119862119879
le 1198850
1198941
119875lowast
1198941
+ 1198850
1198942
119875lowast
1198942
119873minus2
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873minus1
(119862119879minus
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873minus1
119875lowast
119894119873minus1
119873minus1
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873
(119862119879minus
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873
119875lowast
119894119873
(21)
The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues
4 Heuristic Solution Method to the Problem
41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here
Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875
0
119895119894+
1198751
1198941198951015840 + 1199010
119894+ 1199011
119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863
119894 119894 isin 119878 is called
expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip
Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps
Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in
6 Discrete Dynamics in Nature and Society
the scheduling and set 119878idle to include all trucks in the initialstate
Build a truck set 119878running and let 119878running = 0 in the initialstate
Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0
Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4
Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip
Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running
Go to Step A2
Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop
The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process
42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution
Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894
1 1198942 119894
119873in the
descending order of 119864lowast119894(see Section 33) Because the initial
truck position can be far away from the shovel with higher119864lowast
119894 the strategy should reassign more trucks to these shovels
for the objective pursuitSteps of Strategy 1 are listed as follows
Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899
Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894
119899+1 119894119899+2 119894
119873
Table 1 Parameters of the problem instances
Path Problem Trucks Shovels Dumps
1
1 73 6 32 71 6 33 74 6 34 69 6 3
2
5 84 6 46 82 6 47 85 6 48 85 6 4
Step B3 If 119905idle1199010119894 ge 120583119867119875lowast
119894lowast in the beginning reassign
lfloor119905idle119875lowast
119894lowast1205831199010
119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule
the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned
Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step
B2 otherwise terminate
Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps
Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained
Step C2 Let 1199031 1199032 119903
119898 denote the set of truck round trips
executed by truck 119905Set 119896 = 1
Step C3 Attempt to reassign truck round trip 119903119896to other
shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration
Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3
Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate
5 Computational Experiments
To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM
The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances
Discrete Dynamics in Nature and Society 7
Table 2 Experiment result for the small sized problems
Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp
1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1
Table 3 Experiment result and comparison for the practical problem size
Problem Objective value Relative imp UB GAPHeuristic Improvement
1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446
including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances
The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances
The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively
The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative
imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table
From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations
6 Conclusion
In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Discrete Dynamics in Nature and Society
(a) An open-pit mine
L1
L2
L3
L4
L5L6
D1
D2
D3
(b) A sketch map of the open-pit road networkwhere 1198711 1198713 1198716 represent the loading pointsand11986311198632 and1198633 represent the dumping depots
Figure 1 An open-pit mine and a part of its road network
linear program was used to determine material flows alongthese paths Finally the dynamic program assigned trucks tothe optimized paths between shovels and dumpsTheir truck-dispatching system not only yields significant (10ndash20 percentin most cases) productivity improvements but also producedan effect on truck scheduling theory for its three-stage process[17] And Yang et al [18] adopted the three-stage process toresolve the truck dispatch problem
In this paper we consider the varied transportationrevenue for different material loading point and address thejoint path planning and tuck dispatching problem
The outline of this paper is as follows Section 2 describesthe considered open-pit truck scheduling problem withvaried truck transporting revenue in detail and provides itsinteger programming model In Section 3 a few validinequalities two properties of optimal solution and twoupper bounds for the problem are presented In Section 4 aheuristic approach with two improvement strategies is pro-posed to resolve the problem Section 5 tests the proposedupper bounds and solution approach Section 6 concludes thepaper
2 Problem Description andMathematical Model
21 Problem Description In the open pit there are usuallyseveral loading points and dumping depots (to be calleddumps later) as shown in Figure 1 At each loading pointthere is always a pile of loosened ores gangue or waste rocks(to be referred to as material collectively) since the load-ing point needs a loosening blasting before the excavationoperation starting Each loading point is assumed to be
equipped with one and only one electric shovel (without lossof generality it will be viewed as twodifferent loading points iftwo shovels are equipped) The electric shovel excavates andloads the loosened material onto the trucks at each loadingpoint and the truck unloads itself with its self-dischargingdevice at a dump Similar to the loading point we assumethat one dump can accommodate only one truck at the sametime and the unloading area of more than one truck loadingpositions will be viewed as several different dumps
It is worth noting that an electric shovel (or loading point)can load only one truck at the same time and a dump allowsonly one truck simultaneously There is often a park for thewaiting trucks near an operating electric shovel or dump It isactually the key task to reduce the truck waiting times in thetraditional truck scheduling problem
In our considered problem scenario a truck of materialfrom different loading point is given different transportrevenue and our optimization objective will be equal tothe total transport revenue completed in the schedulinghorizonThe revenue per truck loading (transporting a truckof material) of different loading point is always decided bythe management and the decision-making basis includes twoconsiderations the possible transport distance and the trans-port priority The former is easy to be understood since thelonger distance always causes more transport times and thusshould be given more revenue The transport priority comesfrom the mining sequence which has never been modeled asa kind of constraints in the traditional research [1 3] andorthe varied market demand The transport revenue per truckloading can be viewed as the product of the possible transportdistance and the transport priority
From the viewpoint of a scheduler who is in charge ofmaking truck scheduling decisions in a shift (8 hours) ourobjective is to maximize the total transport revenues basedon the given revenue value per truck loading The decisionsinclude the dump selection as well as the truck routingThe dump selection is virtually the same as the first-stagedecisions in literature [16 18] deciding the shortest pathbetween all locations in the mine
The constraints to be considered include the following(1) loading capacity constraint that is each shovel (load-
ing point) can load only one truck at the same time(2) unloading capacity constraint that is each dump can
unload only one truck at the same time(3) dump selection constraint that is a set of candidate
dumps is given for each loading point and thematerialfrom the loading point must be dumped at one of thecandidate dumps
22 The Mixed Integer Programming Model To define andformulate the problemexplicitly the followingnotations needto be introduced to represent all loading points dumpsmaterial to be loaded in the loading points trucks pathsparameters and so on
119878 set of available electric shovels (loading points) 119878 =1 2 1198731198781015840 1198781015840 = 119878 cup 0 0 denoting dummy shovels
Discrete Dynamics in Nature and Society 3
119879 set of available trucks119886119894 revenue per truck of material from electric shovel119894 119894 isin 119878119863119894 set of candidate dumps corresponding to material
(waste rock or coal) loaded at electric shovel 119894 119894 isin 119878119863 = cup
119894isin119878119863119894 including all dumps
119867 the horizon considered in the problem119898119894 max number of truck loadings an electric shovel 119894
can complete at most during the horizon 119894 isin 119878119870119894 119870119894= 1 2 119898
119894 set of possible trucks of rock
(coal) that electric shovel 119894 loads 119894 isin 119878(119896 119894) a truck of material loaded by shovel 119894 119894 isin 119878119896 isin 119870
119894 hereafter referred to as truck loading (119896 119894)
11987001198700= 0 (0 0) denoting dummy truck loading
1199010
119894 loading times per truck at electric shovel 119894 119894 isin 1198781199011
119895 unloading times per truck at depot 119895 119895 isin 119863
1198750
119895119894 times for an empty truck to move from depot 119895 to
shovel 119894 119895 isin 119863 119894 isin 1198781198751
119894119895 times for a full-loaded truck to move form shovel119894 to depot 119895 119895 isin 119863 119894 isin 119878119902119905
119894 times for truck 119905 to move from its initial position
to shovel 119894 119894 isin 119878 119905 isin 119879119872 very big plus number
To represent the dump selection and truck dispatchdecisions we introduce the following decision variables
11990911990511989610158401198941015840119896119894=
1 if truck 119905 transports (119896 119894)
immediatly after (1198961015840 1198941015840)
0 otherwise
for 119905 isin 119879 119894 1198941015840 isin 1198781015840 119896 isin 119870119894 1198961015840isin 1198701198941015840
119911119896119894=
1 if truck loading (119896 119894) is loaded
and transported in the horizon
0 otherwise
for 119894 isin 119878 119896 isin 119870119894
11990611989511989610158401198941015840119896119894
=
1 if material (119896 119894) is dumped immediatly
after (1198961015840 1198941015840) at despot 119895
0 otherwise
for isin 119863119894cap 1198631198941015840 119894 1198941015840isin 119878 119896 isin 119870
119894 1198961015840isin 1198701198941015840
119910119896119894119895=
1 if material (119896 119894) is dumped at despot 119895
0 otherwise
for119895 isin 119863119894119894 isin 119878 119896 isin 119870
119894
(1)
1198880
119896119894is complete time of loadingmaterial (119896 119894) for 119894 isin 119878
119896 isin 1198701015840
119894
1198881
119896119894is complete time of unloading material (119896 119894) for
119894 isin 119878 119896 isin 1198701015840119894
Based on the above notations the problem can be formu-lated as the following integer programmingmodel (similar tomodel in Tang et al [19] which involved a truck schedulingproblem in container terminals)
Max sum
119894isin119878
sum
119896isin119870119894
119886119894119911119896119894 (2)
st 1198880
119896+1119894minus 1198880
119896119894ge 1199010
119894 119894 isin 119878 119896 isin 119870
119894(3)
1198881
119896119894minus 1198881
11989610158401198941015840 ge 1199011
119895minus119872(1 minus 119906
11989511989610158401198941015840119896119894)
119894 1198941015840isin 119878 119896 isin 119870
119894
1198961015840isin 1198701198941015840 119895 isin 119863
119894cap 1198631198941015840
(4)
1198881
119896119894minus 1198880
119896119894ge sum
119895isin119863
119910119896119894119895(119875119894119895+ 1199011
119895) 119894 isin 119878 119896 isin 119870
119894 (5)
1198880
119896119894minus 1198881
11989610158401198941015840 ge sum
119895isin119863
11991011989610158401198941015840119895119875119895119894+ 1199010
119894
minus119872(1 minus sum
119905isin119879
11990911990511989610158401198941015840119896119894)
119894 1198941015840isin 119878 119896 isin 119870
119894 1198961015840isin 1198701198941015840
(6)
1198880
119896119894ge sum
119905isin119879
11990911990500119896119894
119902119905
119894+ 1199010
119894minus119872(1 minus sum
119905isin119879
11990911990500119896119894
)
119894 isin 119878 119896 isin 119870119894
(7)
1198881
119896119894le 119867 +119872(1 minus 119911
119896119894) 119894 isin 119878 119896 isin 119870
1015840
119894(8)
sum
119894isin1198781015840
sum
119896isin119870119894
11990911990511989610158401198941015840119896119894= 11991111989610158401198941015840 119905 isin 119879 119894
1015840isin 119878 119896
1015840isin 1198701198941015840 (9)
sum
1198941015840isin1198781015840
sum
1198961015840isin1198701198941015840
11990911990511989610158401198941015840119896119894= 119911119896119894 119905 isin 119879 119894 isin 119878 119896 isin 119870
119894 (10)
119909119905119896119894119896119894= 0 119905 isin 119879 119894 isin 119878 119896 isin 119870
119894 (11)
sum
119894isin1198781015840
sum
119896isin119870119894
11990911990511989611989400
= sum
119894isin1198781015840
sum
119896isin119870119894
11990911990500119896119894
= 1 119905 isin 119879 (12)
sum
1198941015840isin1198781015840
sum
1198961015840isin1198701015840
119894
11990611989511989610158401198941015840119896119894= sum
1198941015840isin1198781015840
sum
119896isin1198701015840
119894
11990611989511989611989411989610158401198941015840 = 119911119896119894
119905 isin 119879 119894 isin 119878 119896 isin 1198701015840
119894
(13)
119906119895119896119894119896119894= 0 119895 isin 119863
119894 119894 isin 119878 119896 isin 119870
1015840
119894(14)
sum
119894isin119878
sum
119896isin119870119894
11990611989511989611989400
= sum
119894isin119878
sum
119896isin119870119894
11990611989500119896119894
= 1 119895 isin 119863 (15)
sum
119895isin119863119894
119910119896119894119895= 119911119896119894 119894 isin 119878 119896 isin 119870
119894 (16)
4 Discrete Dynamics in Nature and Society
The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911
119896119894 that
is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon
119867 Constraints (9) and (10) guarantee that each valid truckloading (119911
119896119894= 1) is assigned to one and only one truck and
each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910
119896119894119895and 119911119896119894 that is each valid
truck loading has one and only one dumping depot
3 Valid Inequalities Property andUpper Bounds for the Problem
31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach
311 Electric Shovel Capacity Inequality Consider
sum
119896isin119870119894
1199111198961198941199010
119894le 119867 minusmin
119905isin119879
119902119905
119894minusmin119895isin119863119894
1199011
119894119895+ 1199011
119895 119894 isin 119878 (17)
Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min
119905isin119879119902119905
119894minus min
119895isin119863119894
1199011
119894119895is the practical time
limit for shovel 119894 where min119905isin119879119902119905
119894is the time for the nearest
empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min
119895isin119863119894
1199011
119894119895+ 1199011
119895 is the permitted least
retained time to transport and unload the last truck loadedby shovel 119894
L1
L2
L3
L4
D1
D2
D3
Figure 2 Illustration of the truck round trip
312 Unloading Capacity Inequality Consider
sum
119894isin119878
sum
119896isin119870119894
1199101198961198941198951199011
119895le 119867 minus min
119894isin1198941015840|119895isin1198631198941015840
min119905isin119879
119902119905
119894+ 1199010
119894+ 1199011
119894119895
119895 isin 119863
(18)
Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin
119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+1199011
119894119895 is the practi-
cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+
1199011
119894119895 is the earliest possible starting time of unloading the first
truck in depot 119895
313 Variables Relationship Inequality Consider
119911119896119894le 1199111198961015840119894 119896 le 119896
1015840 119894 isin 119878 (19)
Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands
32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties
Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894
moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863
119894 and unloads the material
(eg travelling from dump1198631to loading point 119871
1and to119863
3
in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)
It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot
Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890
1198951198941198951015840 = 119886119894(1198750
119895119894+1198751
1198941198951015840 +1199010
119894+1199011
119895) is called trip transport
value
Discrete Dynamics in Nature and Society 5
Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value
Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times
(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel
(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot
(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon
It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011
119894119895and1199010
119895119894) through
the path
Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to
sum
119905isin119879
sum
(1198951198941198951015840)isinΨ
11989711990511989511989411989510158401198901198951198941198951015840 (20)
where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has
spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips
The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound
Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750
119895119894+ 1198751
1198941198951015840 + 1199010
119894+ 1199011
119895) sdot
min1198850119894 1198851
1198951015840 where1198850119894 and119885
1
1198951015840 denote themaximum loading
capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively
In Property 2 1198850119894and 1198851
1198951015840 can be obtained through
formulae (17) and (18) respectively
33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem
331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum
119894isin1198781198861198941198850
119894
In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873
119879denote the number of available trucks then
119862119879= 119867 sdot 119873
119879denotes the total available truck times involved
in the problem Let 119903119894denote the optimal truck round trip
involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires
the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast
119894and 119864lowast
119894denote the required valuable truck times and trip
transport value corresponding to trip 119903119894 respectively
It is assumed that 1198941 1198942 119894
119873is the shovel sequence in
the descending order of 119864lowast119894 that is 119864lowast
1198941
ge 119864lowast
1198942
ge sdot sdot sdot ge 119864lowast
119894119873
and1198941 1198942 119894
119873 = 119878
Upper bound 2 can be described and formulated clearlybased on the above notations and assumption
332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value
119891 (119862119879) =
119864lowast
1198941
119862119879
119862119879le 1198850
1198941
119875lowast
1198941
1198861198941
1198850
1198941
+ 119864lowast
1198942
(119862119879minus 1198850
1198941
119875lowast
1198941
)
1198850
1198941
119875lowast
1198941
le 119862119879
le 1198850
1198941
119875lowast
1198941
+ 1198850
1198942
119875lowast
1198942
119873minus2
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873minus1
(119862119879minus
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873minus1
119875lowast
119894119873minus1
119873minus1
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873
(119862119879minus
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873
119875lowast
119894119873
(21)
The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues
4 Heuristic Solution Method to the Problem
41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here
Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875
0
119895119894+
1198751
1198941198951015840 + 1199010
119894+ 1199011
119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863
119894 119894 isin 119878 is called
expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip
Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps
Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in
6 Discrete Dynamics in Nature and Society
the scheduling and set 119878idle to include all trucks in the initialstate
Build a truck set 119878running and let 119878running = 0 in the initialstate
Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0
Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4
Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip
Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running
Go to Step A2
Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop
The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process
42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution
Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894
1 1198942 119894
119873in the
descending order of 119864lowast119894(see Section 33) Because the initial
truck position can be far away from the shovel with higher119864lowast
119894 the strategy should reassign more trucks to these shovels
for the objective pursuitSteps of Strategy 1 are listed as follows
Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899
Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894
119899+1 119894119899+2 119894
119873
Table 1 Parameters of the problem instances
Path Problem Trucks Shovels Dumps
1
1 73 6 32 71 6 33 74 6 34 69 6 3
2
5 84 6 46 82 6 47 85 6 48 85 6 4
Step B3 If 119905idle1199010119894 ge 120583119867119875lowast
119894lowast in the beginning reassign
lfloor119905idle119875lowast
119894lowast1205831199010
119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule
the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned
Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step
B2 otherwise terminate
Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps
Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained
Step C2 Let 1199031 1199032 119903
119898 denote the set of truck round trips
executed by truck 119905Set 119896 = 1
Step C3 Attempt to reassign truck round trip 119903119896to other
shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration
Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3
Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate
5 Computational Experiments
To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM
The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances
Discrete Dynamics in Nature and Society 7
Table 2 Experiment result for the small sized problems
Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp
1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1
Table 3 Experiment result and comparison for the practical problem size
Problem Objective value Relative imp UB GAPHeuristic Improvement
1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446
including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances
The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances
The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively
The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative
imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table
From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations
6 Conclusion
In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 3
119879 set of available trucks119886119894 revenue per truck of material from electric shovel119894 119894 isin 119878119863119894 set of candidate dumps corresponding to material
(waste rock or coal) loaded at electric shovel 119894 119894 isin 119878119863 = cup
119894isin119878119863119894 including all dumps
119867 the horizon considered in the problem119898119894 max number of truck loadings an electric shovel 119894
can complete at most during the horizon 119894 isin 119878119870119894 119870119894= 1 2 119898
119894 set of possible trucks of rock
(coal) that electric shovel 119894 loads 119894 isin 119878(119896 119894) a truck of material loaded by shovel 119894 119894 isin 119878119896 isin 119870
119894 hereafter referred to as truck loading (119896 119894)
11987001198700= 0 (0 0) denoting dummy truck loading
1199010
119894 loading times per truck at electric shovel 119894 119894 isin 1198781199011
119895 unloading times per truck at depot 119895 119895 isin 119863
1198750
119895119894 times for an empty truck to move from depot 119895 to
shovel 119894 119895 isin 119863 119894 isin 1198781198751
119894119895 times for a full-loaded truck to move form shovel119894 to depot 119895 119895 isin 119863 119894 isin 119878119902119905
119894 times for truck 119905 to move from its initial position
to shovel 119894 119894 isin 119878 119905 isin 119879119872 very big plus number
To represent the dump selection and truck dispatchdecisions we introduce the following decision variables
11990911990511989610158401198941015840119896119894=
1 if truck 119905 transports (119896 119894)
immediatly after (1198961015840 1198941015840)
0 otherwise
for 119905 isin 119879 119894 1198941015840 isin 1198781015840 119896 isin 119870119894 1198961015840isin 1198701198941015840
119911119896119894=
1 if truck loading (119896 119894) is loaded
and transported in the horizon
0 otherwise
for 119894 isin 119878 119896 isin 119870119894
11990611989511989610158401198941015840119896119894
=
1 if material (119896 119894) is dumped immediatly
after (1198961015840 1198941015840) at despot 119895
0 otherwise
for isin 119863119894cap 1198631198941015840 119894 1198941015840isin 119878 119896 isin 119870
119894 1198961015840isin 1198701198941015840
119910119896119894119895=
1 if material (119896 119894) is dumped at despot 119895
0 otherwise
for119895 isin 119863119894119894 isin 119878 119896 isin 119870
119894
(1)
1198880
119896119894is complete time of loadingmaterial (119896 119894) for 119894 isin 119878
119896 isin 1198701015840
119894
1198881
119896119894is complete time of unloading material (119896 119894) for
119894 isin 119878 119896 isin 1198701015840119894
Based on the above notations the problem can be formu-lated as the following integer programmingmodel (similar tomodel in Tang et al [19] which involved a truck schedulingproblem in container terminals)
Max sum
119894isin119878
sum
119896isin119870119894
119886119894119911119896119894 (2)
st 1198880
119896+1119894minus 1198880
119896119894ge 1199010
119894 119894 isin 119878 119896 isin 119870
119894(3)
1198881
119896119894minus 1198881
11989610158401198941015840 ge 1199011
119895minus119872(1 minus 119906
11989511989610158401198941015840119896119894)
119894 1198941015840isin 119878 119896 isin 119870
119894
1198961015840isin 1198701198941015840 119895 isin 119863
119894cap 1198631198941015840
(4)
1198881
119896119894minus 1198880
119896119894ge sum
119895isin119863
119910119896119894119895(119875119894119895+ 1199011
119895) 119894 isin 119878 119896 isin 119870
119894 (5)
1198880
119896119894minus 1198881
11989610158401198941015840 ge sum
119895isin119863
11991011989610158401198941015840119895119875119895119894+ 1199010
119894
minus119872(1 minus sum
119905isin119879
11990911990511989610158401198941015840119896119894)
119894 1198941015840isin 119878 119896 isin 119870
119894 1198961015840isin 1198701198941015840
(6)
1198880
119896119894ge sum
119905isin119879
11990911990500119896119894
119902119905
119894+ 1199010
119894minus119872(1 minus sum
119905isin119879
11990911990500119896119894
)
119894 isin 119878 119896 isin 119870119894
(7)
1198881
119896119894le 119867 +119872(1 minus 119911
119896119894) 119894 isin 119878 119896 isin 119870
1015840
119894(8)
sum
119894isin1198781015840
sum
119896isin119870119894
11990911990511989610158401198941015840119896119894= 11991111989610158401198941015840 119905 isin 119879 119894
1015840isin 119878 119896
1015840isin 1198701198941015840 (9)
sum
1198941015840isin1198781015840
sum
1198961015840isin1198701198941015840
11990911990511989610158401198941015840119896119894= 119911119896119894 119905 isin 119879 119894 isin 119878 119896 isin 119870
119894 (10)
119909119905119896119894119896119894= 0 119905 isin 119879 119894 isin 119878 119896 isin 119870
119894 (11)
sum
119894isin1198781015840
sum
119896isin119870119894
11990911990511989611989400
= sum
119894isin1198781015840
sum
119896isin119870119894
11990911990500119896119894
= 1 119905 isin 119879 (12)
sum
1198941015840isin1198781015840
sum
1198961015840isin1198701015840
119894
11990611989511989610158401198941015840119896119894= sum
1198941015840isin1198781015840
sum
119896isin1198701015840
119894
11990611989511989611989411989610158401198941015840 = 119911119896119894
119905 isin 119879 119894 isin 119878 119896 isin 1198701015840
119894
(13)
119906119895119896119894119896119894= 0 119895 isin 119863
119894 119894 isin 119878 119896 isin 119870
1015840
119894(14)
sum
119894isin119878
sum
119896isin119870119894
11990611989511989611989400
= sum
119894isin119878
sum
119896isin119870119894
11990611989500119896119894
= 1 119895 isin 119863 (15)
sum
119895isin119863119894
119910119896119894119895= 119911119896119894 119894 isin 119878 119896 isin 119870
119894 (16)
4 Discrete Dynamics in Nature and Society
The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911
119896119894 that
is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon
119867 Constraints (9) and (10) guarantee that each valid truckloading (119911
119896119894= 1) is assigned to one and only one truck and
each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910
119896119894119895and 119911119896119894 that is each valid
truck loading has one and only one dumping depot
3 Valid Inequalities Property andUpper Bounds for the Problem
31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach
311 Electric Shovel Capacity Inequality Consider
sum
119896isin119870119894
1199111198961198941199010
119894le 119867 minusmin
119905isin119879
119902119905
119894minusmin119895isin119863119894
1199011
119894119895+ 1199011
119895 119894 isin 119878 (17)
Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min
119905isin119879119902119905
119894minus min
119895isin119863119894
1199011
119894119895is the practical time
limit for shovel 119894 where min119905isin119879119902119905
119894is the time for the nearest
empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min
119895isin119863119894
1199011
119894119895+ 1199011
119895 is the permitted least
retained time to transport and unload the last truck loadedby shovel 119894
L1
L2
L3
L4
D1
D2
D3
Figure 2 Illustration of the truck round trip
312 Unloading Capacity Inequality Consider
sum
119894isin119878
sum
119896isin119870119894
1199101198961198941198951199011
119895le 119867 minus min
119894isin1198941015840|119895isin1198631198941015840
min119905isin119879
119902119905
119894+ 1199010
119894+ 1199011
119894119895
119895 isin 119863
(18)
Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin
119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+1199011
119894119895 is the practi-
cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+
1199011
119894119895 is the earliest possible starting time of unloading the first
truck in depot 119895
313 Variables Relationship Inequality Consider
119911119896119894le 1199111198961015840119894 119896 le 119896
1015840 119894 isin 119878 (19)
Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands
32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties
Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894
moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863
119894 and unloads the material
(eg travelling from dump1198631to loading point 119871
1and to119863
3
in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)
It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot
Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890
1198951198941198951015840 = 119886119894(1198750
119895119894+1198751
1198941198951015840 +1199010
119894+1199011
119895) is called trip transport
value
Discrete Dynamics in Nature and Society 5
Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value
Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times
(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel
(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot
(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon
It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011
119894119895and1199010
119895119894) through
the path
Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to
sum
119905isin119879
sum
(1198951198941198951015840)isinΨ
11989711990511989511989411989510158401198901198951198941198951015840 (20)
where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has
spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips
The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound
Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750
119895119894+ 1198751
1198941198951015840 + 1199010
119894+ 1199011
119895) sdot
min1198850119894 1198851
1198951015840 where1198850119894 and119885
1
1198951015840 denote themaximum loading
capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively
In Property 2 1198850119894and 1198851
1198951015840 can be obtained through
formulae (17) and (18) respectively
33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem
331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum
119894isin1198781198861198941198850
119894
In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873
119879denote the number of available trucks then
119862119879= 119867 sdot 119873
119879denotes the total available truck times involved
in the problem Let 119903119894denote the optimal truck round trip
involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires
the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast
119894and 119864lowast
119894denote the required valuable truck times and trip
transport value corresponding to trip 119903119894 respectively
It is assumed that 1198941 1198942 119894
119873is the shovel sequence in
the descending order of 119864lowast119894 that is 119864lowast
1198941
ge 119864lowast
1198942
ge sdot sdot sdot ge 119864lowast
119894119873
and1198941 1198942 119894
119873 = 119878
Upper bound 2 can be described and formulated clearlybased on the above notations and assumption
332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value
119891 (119862119879) =
119864lowast
1198941
119862119879
119862119879le 1198850
1198941
119875lowast
1198941
1198861198941
1198850
1198941
+ 119864lowast
1198942
(119862119879minus 1198850
1198941
119875lowast
1198941
)
1198850
1198941
119875lowast
1198941
le 119862119879
le 1198850
1198941
119875lowast
1198941
+ 1198850
1198942
119875lowast
1198942
119873minus2
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873minus1
(119862119879minus
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873minus1
119875lowast
119894119873minus1
119873minus1
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873
(119862119879minus
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873
119875lowast
119894119873
(21)
The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues
4 Heuristic Solution Method to the Problem
41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here
Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875
0
119895119894+
1198751
1198941198951015840 + 1199010
119894+ 1199011
119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863
119894 119894 isin 119878 is called
expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip
Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps
Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in
6 Discrete Dynamics in Nature and Society
the scheduling and set 119878idle to include all trucks in the initialstate
Build a truck set 119878running and let 119878running = 0 in the initialstate
Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0
Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4
Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip
Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running
Go to Step A2
Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop
The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process
42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution
Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894
1 1198942 119894
119873in the
descending order of 119864lowast119894(see Section 33) Because the initial
truck position can be far away from the shovel with higher119864lowast
119894 the strategy should reassign more trucks to these shovels
for the objective pursuitSteps of Strategy 1 are listed as follows
Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899
Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894
119899+1 119894119899+2 119894
119873
Table 1 Parameters of the problem instances
Path Problem Trucks Shovels Dumps
1
1 73 6 32 71 6 33 74 6 34 69 6 3
2
5 84 6 46 82 6 47 85 6 48 85 6 4
Step B3 If 119905idle1199010119894 ge 120583119867119875lowast
119894lowast in the beginning reassign
lfloor119905idle119875lowast
119894lowast1205831199010
119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule
the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned
Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step
B2 otherwise terminate
Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps
Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained
Step C2 Let 1199031 1199032 119903
119898 denote the set of truck round trips
executed by truck 119905Set 119896 = 1
Step C3 Attempt to reassign truck round trip 119903119896to other
shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration
Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3
Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate
5 Computational Experiments
To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM
The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances
Discrete Dynamics in Nature and Society 7
Table 2 Experiment result for the small sized problems
Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp
1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1
Table 3 Experiment result and comparison for the practical problem size
Problem Objective value Relative imp UB GAPHeuristic Improvement
1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446
including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances
The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances
The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively
The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative
imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table
From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations
6 Conclusion
In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
4 Discrete Dynamics in Nature and Society
The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911
119896119894 that
is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon
119867 Constraints (9) and (10) guarantee that each valid truckloading (119911
119896119894= 1) is assigned to one and only one truck and
each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910
119896119894119895and 119911119896119894 that is each valid
truck loading has one and only one dumping depot
3 Valid Inequalities Property andUpper Bounds for the Problem
31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach
311 Electric Shovel Capacity Inequality Consider
sum
119896isin119870119894
1199111198961198941199010
119894le 119867 minusmin
119905isin119879
119902119905
119894minusmin119895isin119863119894
1199011
119894119895+ 1199011
119895 119894 isin 119878 (17)
Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min
119905isin119879119902119905
119894minus min
119895isin119863119894
1199011
119894119895is the practical time
limit for shovel 119894 where min119905isin119879119902119905
119894is the time for the nearest
empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min
119895isin119863119894
1199011
119894119895+ 1199011
119895 is the permitted least
retained time to transport and unload the last truck loadedby shovel 119894
L1
L2
L3
L4
D1
D2
D3
Figure 2 Illustration of the truck round trip
312 Unloading Capacity Inequality Consider
sum
119894isin119878
sum
119896isin119870119894
1199101198961198941198951199011
119895le 119867 minus min
119894isin1198941015840|119895isin1198631198941015840
min119905isin119879
119902119905
119894+ 1199010
119894+ 1199011
119894119895
119895 isin 119863
(18)
Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin
119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+1199011
119894119895 is the practi-
cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905
119894+1199010
119894+
1199011
119894119895 is the earliest possible starting time of unloading the first
truck in depot 119895
313 Variables Relationship Inequality Consider
119911119896119894le 1199111198961015840119894 119896 le 119896
1015840 119894 isin 119878 (19)
Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands
32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties
Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894
moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863
119894 and unloads the material
(eg travelling from dump1198631to loading point 119871
1and to119863
3
in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)
It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot
Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890
1198951198941198951015840 = 119886119894(1198750
119895119894+1198751
1198941198951015840 +1199010
119894+1199011
119895) is called trip transport
value
Discrete Dynamics in Nature and Society 5
Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value
Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times
(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel
(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot
(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon
It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011
119894119895and1199010
119895119894) through
the path
Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to
sum
119905isin119879
sum
(1198951198941198951015840)isinΨ
11989711990511989511989411989510158401198901198951198941198951015840 (20)
where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has
spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips
The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound
Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750
119895119894+ 1198751
1198941198951015840 + 1199010
119894+ 1199011
119895) sdot
min1198850119894 1198851
1198951015840 where1198850119894 and119885
1
1198951015840 denote themaximum loading
capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively
In Property 2 1198850119894and 1198851
1198951015840 can be obtained through
formulae (17) and (18) respectively
33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem
331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum
119894isin1198781198861198941198850
119894
In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873
119879denote the number of available trucks then
119862119879= 119867 sdot 119873
119879denotes the total available truck times involved
in the problem Let 119903119894denote the optimal truck round trip
involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires
the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast
119894and 119864lowast
119894denote the required valuable truck times and trip
transport value corresponding to trip 119903119894 respectively
It is assumed that 1198941 1198942 119894
119873is the shovel sequence in
the descending order of 119864lowast119894 that is 119864lowast
1198941
ge 119864lowast
1198942
ge sdot sdot sdot ge 119864lowast
119894119873
and1198941 1198942 119894
119873 = 119878
Upper bound 2 can be described and formulated clearlybased on the above notations and assumption
332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value
119891 (119862119879) =
119864lowast
1198941
119862119879
119862119879le 1198850
1198941
119875lowast
1198941
1198861198941
1198850
1198941
+ 119864lowast
1198942
(119862119879minus 1198850
1198941
119875lowast
1198941
)
1198850
1198941
119875lowast
1198941
le 119862119879
le 1198850
1198941
119875lowast
1198941
+ 1198850
1198942
119875lowast
1198942
119873minus2
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873minus1
(119862119879minus
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873minus1
119875lowast
119894119873minus1
119873minus1
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873
(119862119879minus
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873
119875lowast
119894119873
(21)
The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues
4 Heuristic Solution Method to the Problem
41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here
Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875
0
119895119894+
1198751
1198941198951015840 + 1199010
119894+ 1199011
119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863
119894 119894 isin 119878 is called
expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip
Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps
Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in
6 Discrete Dynamics in Nature and Society
the scheduling and set 119878idle to include all trucks in the initialstate
Build a truck set 119878running and let 119878running = 0 in the initialstate
Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0
Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4
Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip
Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running
Go to Step A2
Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop
The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process
42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution
Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894
1 1198942 119894
119873in the
descending order of 119864lowast119894(see Section 33) Because the initial
truck position can be far away from the shovel with higher119864lowast
119894 the strategy should reassign more trucks to these shovels
for the objective pursuitSteps of Strategy 1 are listed as follows
Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899
Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894
119899+1 119894119899+2 119894
119873
Table 1 Parameters of the problem instances
Path Problem Trucks Shovels Dumps
1
1 73 6 32 71 6 33 74 6 34 69 6 3
2
5 84 6 46 82 6 47 85 6 48 85 6 4
Step B3 If 119905idle1199010119894 ge 120583119867119875lowast
119894lowast in the beginning reassign
lfloor119905idle119875lowast
119894lowast1205831199010
119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule
the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned
Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step
B2 otherwise terminate
Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps
Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained
Step C2 Let 1199031 1199032 119903
119898 denote the set of truck round trips
executed by truck 119905Set 119896 = 1
Step C3 Attempt to reassign truck round trip 119903119896to other
shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration
Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3
Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate
5 Computational Experiments
To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM
The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances
Discrete Dynamics in Nature and Society 7
Table 2 Experiment result for the small sized problems
Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp
1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1
Table 3 Experiment result and comparison for the practical problem size
Problem Objective value Relative imp UB GAPHeuristic Improvement
1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446
including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances
The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances
The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively
The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative
imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table
From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations
6 Conclusion
In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 5
Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value
Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times
(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel
(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot
(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon
It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011
119894119895and1199010
119895119894) through
the path
Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to
sum
119905isin119879
sum
(1198951198941198951015840)isinΨ
11989711990511989511989411989510158401198901198951198941198951015840 (20)
where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has
spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips
The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound
Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750
119895119894+ 1198751
1198941198951015840 + 1199010
119894+ 1199011
119895) sdot
min1198850119894 1198851
1198951015840 where1198850119894 and119885
1
1198951015840 denote themaximum loading
capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively
In Property 2 1198850119894and 1198851
1198951015840 can be obtained through
formulae (17) and (18) respectively
33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem
331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum
119894isin1198781198861198941198850
119894
In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873
119879denote the number of available trucks then
119862119879= 119867 sdot 119873
119879denotes the total available truck times involved
in the problem Let 119903119894denote the optimal truck round trip
involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires
the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast
119894and 119864lowast
119894denote the required valuable truck times and trip
transport value corresponding to trip 119903119894 respectively
It is assumed that 1198941 1198942 119894
119873is the shovel sequence in
the descending order of 119864lowast119894 that is 119864lowast
1198941
ge 119864lowast
1198942
ge sdot sdot sdot ge 119864lowast
119894119873
and1198941 1198942 119894
119873 = 119878
Upper bound 2 can be described and formulated clearlybased on the above notations and assumption
332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value
119891 (119862119879) =
119864lowast
1198941
119862119879
119862119879le 1198850
1198941
119875lowast
1198941
1198861198941
1198850
1198941
+ 119864lowast
1198942
(119862119879minus 1198850
1198941
119875lowast
1198941
)
1198850
1198941
119875lowast
1198941
le 119862119879
le 1198850
1198941
119875lowast
1198941
+ 1198850
1198942
119875lowast
1198942
119873minus2
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873minus1
(119862119879minus
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus2
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873minus1
119875lowast
119894119873minus1
119873minus1
sum
119899=1
119886119894119899
1198850
119894119899
+ 119864lowast
119894119873
(119862119879minus
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
)
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
le 119862119879le
119873minus1
sum
119899=1
1198850
119894119899
119875lowast
119894119899
+ 1198850
119894119873
119875lowast
119894119873
(21)
The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues
4 Heuristic Solution Method to the Problem
41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here
Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875
0
119895119894+
1198751
1198941198951015840 + 1199010
119894+ 1199011
119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863
119894 119894 isin 119878 is called
expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip
Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps
Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in
6 Discrete Dynamics in Nature and Society
the scheduling and set 119878idle to include all trucks in the initialstate
Build a truck set 119878running and let 119878running = 0 in the initialstate
Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0
Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4
Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip
Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running
Go to Step A2
Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop
The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process
42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution
Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894
1 1198942 119894
119873in the
descending order of 119864lowast119894(see Section 33) Because the initial
truck position can be far away from the shovel with higher119864lowast
119894 the strategy should reassign more trucks to these shovels
for the objective pursuitSteps of Strategy 1 are listed as follows
Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899
Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894
119899+1 119894119899+2 119894
119873
Table 1 Parameters of the problem instances
Path Problem Trucks Shovels Dumps
1
1 73 6 32 71 6 33 74 6 34 69 6 3
2
5 84 6 46 82 6 47 85 6 48 85 6 4
Step B3 If 119905idle1199010119894 ge 120583119867119875lowast
119894lowast in the beginning reassign
lfloor119905idle119875lowast
119894lowast1205831199010
119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule
the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned
Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step
B2 otherwise terminate
Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps
Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained
Step C2 Let 1199031 1199032 119903
119898 denote the set of truck round trips
executed by truck 119905Set 119896 = 1
Step C3 Attempt to reassign truck round trip 119903119896to other
shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration
Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3
Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate
5 Computational Experiments
To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM
The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances
Discrete Dynamics in Nature and Society 7
Table 2 Experiment result for the small sized problems
Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp
1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1
Table 3 Experiment result and comparison for the practical problem size
Problem Objective value Relative imp UB GAPHeuristic Improvement
1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446
including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances
The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances
The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively
The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative
imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table
From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations
6 Conclusion
In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Discrete Dynamics in Nature and Society
the scheduling and set 119878idle to include all trucks in the initialstate
Build a truck set 119878running and let 119878running = 0 in the initialstate
Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0
Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4
Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip
Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running
Go to Step A2
Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop
The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process
42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution
Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894
1 1198942 119894
119873in the
descending order of 119864lowast119894(see Section 33) Because the initial
truck position can be far away from the shovel with higher119864lowast
119894 the strategy should reassign more trucks to these shovels
for the objective pursuitSteps of Strategy 1 are listed as follows
Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899
Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894
119899+1 119894119899+2 119894
119873
Table 1 Parameters of the problem instances
Path Problem Trucks Shovels Dumps
1
1 73 6 32 71 6 33 74 6 34 69 6 3
2
5 84 6 46 82 6 47 85 6 48 85 6 4
Step B3 If 119905idle1199010119894 ge 120583119867119875lowast
119894lowast in the beginning reassign
lfloor119905idle119875lowast
119894lowast1205831199010
119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule
the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned
Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step
B2 otherwise terminate
Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps
Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained
Step C2 Let 1199031 1199032 119903
119898 denote the set of truck round trips
executed by truck 119905Set 119896 = 1
Step C3 Attempt to reassign truck round trip 119903119896to other
shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration
Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3
Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate
5 Computational Experiments
To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM
The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances
Discrete Dynamics in Nature and Society 7
Table 2 Experiment result for the small sized problems
Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp
1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1
Table 3 Experiment result and comparison for the practical problem size
Problem Objective value Relative imp UB GAPHeuristic Improvement
1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446
including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances
The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances
The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively
The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative
imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table
From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations
6 Conclusion
In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 7
Table 2 Experiment result for the small sized problems
Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp
1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1
Table 3 Experiment result and comparison for the practical problem size
Problem Objective value Relative imp UB GAPHeuristic Improvement
1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446
including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances
The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances
The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively
The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative
imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table
From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations
6 Conclusion
In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Discrete Dynamics in Nature and Society
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)
References
[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002
[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007
[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010
[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965
[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989
[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002
[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006
[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985
[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988
[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001
[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006
[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971
[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991
[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004
[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006
[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992
[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003
[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006
[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of