multiproduct aggregate production planning with fuzzy ......common in aggregate production planning....

302 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 3, MAY 2003

Multiproduct Aggregate Production Planning WithFuzzy Demands and Fuzzy Capacities

Richard Y. K. Fung, Jiafu Tang, and Dingwei Wang

Abstract—Given the uncertain market demands and capacitiesin production environment, this paper discusses some practical ap-proaches to modeling multiproduct aggregate production planningproblems with fuzzy demands, fuzzy capacities, and financial con-straints. By formulating the fuzzy demand, fuzzy equation, andfuzzy capacities, a fuzzy production-inventory balance equationfor single period and a dynamic balance equation are formulatedas fuzzy/soft equations and they represent the possibility levels ofmeeting the market demands. Using this formulation and interpre-tation, a fuzzy multiproduct aggregate production planning modelis developed, and its solutions using parametric programming, bestbalance and interactive techniques are introduced to cater to dif-ferent scenarios under various decision making preferences. Usingthe proposed models and techniques, first, the decision maker canselect a preferred production plan with a common satisfaction levelor different combinations of preferred possibility level and satis-faction levels, according to the market demands and available pro-duction capacities, and second, the obtained structure of the op-timal solution can help decision maker in aggregate productionplanning. The decision maker can also make a preferred and rea-sonable production plan corresponding to one’s most concernedcriteria. Hence, decision makers not only can come up with a rea-sonable aggregate production plan with minimum efforts, but alsohave more choices of making a preferred aggregate plan based onhis most concerned criteria. These models can effectively enhancethe capability of an aggregate plan to give feasible family disag-gregation plans under different scenarios with fuzzy demands andcapacities. Simulation and the results of analysis on the proposedtechniques are also given in detail in this paper.

Index Terms—Aggregate production planning, fuzzy demands,fuzzy equation, mathematical programming, interactive proce-dure.

I. INTRODUCTION

A GGREGATE PRODUCTION PLANNING (APP) [1] isconcerned with the determination of production, inven-

tory, and workforce levels required to meet aggregate market de-mands at the level of product type. Product families with similarproduct costs and demand seasonality are grouped into a producttype. APP is an important tactical level planning activity in a

Manuscript received September 12, 2000; revised April 30, 2003. This paperwas supported by a Strategic Research Grant (SRG) from City University ofHong Kong, Project Number 7001227, the National Natural Science Founda-tion, NSFC No. 70002009 of P.R.C., the Excellent Youth Teacher Programof Ministry of Education of China, and Liaoning Provincial Natural ScienceFoundation, No. 20022019. This paper was recommended by Associate EditorW. Gruver.

R. Y. K. Fung is with the Department of Manufacturing Engineering & En-gineering Management, City University of Hong Kong, Kow loon, Hong Kong,P.R.C. (e-mail: [email protected]).

J. Tang and D. Wang are with the Department of Systems Engineering, Schoolof Information Science and Engineering, Northeastern University, Shenyang110004, P.R.C.

Digital Object Identifier 10.1109/TSMCA.2003.817032

production management system. Other forms of family disag-gregation plans, such as master production schedule, capacityplan, and material requirements plan all depend on APP in a hi-erarchical way [2]. Holt [3] suggested an optimal staffing, pro-duction, and inventory policy for a single manufacturer underthe assumption of a given sales forecast, and they proposed acontinuous time-varied model (HMMS) [3]. HMMS model isone of the best known classical models for tackling aggregatedproduction and inventory planning problems. It aims at mini-mizing the total cost of regular payroll, overtime and layoffs,inventory, stock-outs, and machine set up. The linear decisionrules for production level and workforce level are derived to dealwith problems in a single product category. Since then, much at-tention has been directed toward aggregate production planning,and different models and approaches have been developed. Theapproaches to APP vary from mathematically optimal proce-dures [3] and [4], simulation and search methods [5] to heuristicdecision rules [6]. Among the optimal models, linear program-ming has received the widest acceptance. These models can becategorized into deterministic optimization models [3], [7], sto-chastic programming models [7], [8], and fuzzy optimizationmodels [6], [9]–[11].

Bergstrom and Smith [7] generalized the HMMS approachto a multiproduct formulation, which was further extended byHausman [8] into a stochastic programming model to caterto the randomness in product demand. Bitran and Yanassee[12] considered the problem of determining production plansover a number of time periods under stochastic demands withknown distribution functions. A distribution bound for theproblem could be obtained by a deterministic approximation tothe original stochastic problem. These stochastic programmingmodels and methods are usually based on the concept ofrandomness and probability, and they are limited to tacklinguncertainties with some probability distributions. The decisionsachieved by stochastic programming can only take the formof a distribution function, which can do little to help decisionmaking in practical scenarios.

Some other forms of uncertainties, such as fuzzy demands,capacities with tolerance, imprecise process times, etc., arecommon in aggregate production planning. These types ofuncertainties cannot be fully described by frequency-basedprobability distributions. Therefore, there is a need to formulateAPP using fuzzy set theory [13], [14] and fuzzy optimizationmethods [6], [9], [10], [15]–[18].

Rinks [6] tackled aggregate production planning problemsusing fuzzy logic and fuzzy linguistics, and developed aproduction and a workforce algorithm using a series of ap-proximately 40 relational assignment rules. Lee [9] discussed

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FUNG et al.: MULTIPRODUCT AGGREGATE PRODUCTION PLANNING WITH FUZZY 303

fuzzy aggregate production planning problems for singleproduct types, under fuzzy objective, fuzzy workforce levelsand fuzzy demands in different time periods. The workforcelevel and product demands were expressed in fuzzy numberswith tolerance. In this paper [9], a linear programming modelwith fuzzy objective and fuzzy constraints was developed,and fuzzy solutions under different levels could be attainedusing parametric programming technology. Wang and Fang[11] developed a genetic-based approach for linear APP withfuzzy objectives and resources. Tang [10], [19] developeda fuzzy approach for formulating integrated production andinventory problems with triangular fuzzy demands. It aims tominimize the total costs of quadratic production cost and linearinventory holding cost. A fuzzy solution is developed by whichthe solutions corresponding to different satisfaction levels areobtained.

These fuzzy optimization models and approaches mainlyconsider a single product type [3], [6], [9] and they tend to over-look the financial constraint in a practical business environment[2], [3], [6], [7], [9]–[11], [19]. In general, a production plan isnot only constrained by production capacities (e.g., machinesand workforce), but also by financial resources, i.e., capitallevel, and all these constraints should be considered in APP. Atpresent, there are no appropriate formulation and interpretationapproaches to production-inventory balance equation under afuzzy environment suitable for APP.

In practical production planning systems, the aggregatemarket demands of a product in each period are not alwaysclear, but uncertain in nature. These demands can be expressedas

1) random number with probability distribution;2) fuzzy triangular number;3) interval number, while the production capacities can

be represented as fuzzy numbers with tolerance.

Taking into account these factors and the financial constraintsin the production environments, this paper proposes a practicalapproach to modeling multiproduct aggregate production plan-ning problems with fuzzy demands and fuzzy capacities. Withthe use of fuzzy equation and fuzzy addition, the fuzzy pro-duction-inventory balance equation for single period and thedynamic balance equation are formulated as soft equations interms of degrees of truth, which can be interpreted as possi-bility levels for meeting the market demands. With this formu-lation and interpretation, a fuzzy multiproduct aggregate pro-duction planning (FMAPP) model is developed. This model cantake the form of a parametric programming model, a best bal-ance model or an interactive model, to cater to different sce-narios and decision maker’s preference. A parametric program-ming-based fuzzy solution and an interactive procedure are pro-posed to solve the problem. With the proposed models and so-lution, decision makers can make a reasonable APP econom-ically, and more readily under his preferred criteria. This ap-proach effectively allows an aggregate plan to give a feasiblefamily disaggregation plan under different circumstances, espe-cially under fuzzy demands and fuzzy capacities.

The remainder of the paper is organized as follows: Nota-tion and formulation of the problem will be presented in de-

tail in Section II. Section III discusses the formulation of fuzzydemands, fuzzy capacities, fuzzy production-inventory balanceequation, and the development of the FMAPP model. The para-metric programming-based fuzzy solution and the interactiveprocedures for the model are discussed in Sections IV and V,respectively. An illustrative example and the results of simula-tion are presented in Section VI, and the conclusion is given inSection VII.

II. NOTATION AND FORMULATION

Assume that a company manufacturestypes of productsto meet the market demand over a planning horizon. APPis to determine the production, workforce, inventory, and back-order levels for product in period at minimum total cost. Amultiproduct aggregate production planning (MAPP) problemunder financial constraints with the aim to minimize the totalcost can be formulated by crisp mathematics intoa MAPP model as follows:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

wheredecision variable, inventory level of productat theend of period ;decision variable, production level of productin pe-riod ;decision variable, assigned workforce level for product

in period ,decision variable, back-order level of productin pe-riod ;demands for productin period ,unit production costs including materials and overheadfor product in period ,unit workforce costs for productin period ;unit inventory costs for productin each period;unit back-order costs for productin each period;unit production capacity (man-hours) for product;available workforce level in period,available financial resources in period.production capacity allowance percentage;the capacity of the warehouse;the initial inventory level for product;the initial back-order level for product.


In the MAPP model, (1) represents the objective of mini-mizing the total cost of production, inventory and back-order,(2) is the production-inventory balance equation, (3) and (4)denote the constraints of production capacity and financial re-sources in each period, (5) gives the warehouse capacity con-straint, (6) reflects the relation between production level andproduction capacity, (7) indicates that at least one of the back-order and inventory level in a period is zero, and (8) implies theinitial and nonnegative constraints of the decision variables.

III. FUZZY MULTIPRODUCT AGGREGATE

PRODUCTION PLANNING

A. Formulating the Fuzzy Multiproduct Aggregate ProductionPlanning Model

In real-world scenarios, it is difficult to determine product de-mands in advance in each period precisely, especially overa long planning horizon. These demands are not a certainty, butan estimate or imprecise/interval number. It is assumed in thispaper that the demand is a triangular fuzzy number, denotedby . The workforce level in each periodis assumed to take the form of a fuzzy number, denoted by,where the available capacity level is, and the largest accept-able tolerance is . Hence, MAPP problems can be formulatedinto a fuzzy mathematical model.

Owing to the fuzziness of demand, the production-inventorybalance (2) is not a crisp, but a fuzzy equation, namely, a softequation in this paper. It implies that the equation is met in termsof a degree of truth.

As a result, an MAPP problem with fuzzy demands andfuzzy production capacities can be formulated into a fuzzymultiproduct aggregate production planning (FMAPP) modelas follows:

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

where denotes a soft equation. It implies that the production-inventory balance equation is achieved in terms of a degree oftruth. The formulations and interpretation of fuzzy demands,fuzzy capacity and fuzzy production-inventory balance equationare explained in the following subsections.

B. Fuzzy Demands and Their Addition

Let the fuzzy demands be a triangular fuzzy number, i.e.,, where , and are the

most possible, most pessimistic and most optimistic values offuzzy demands, respectively. The membership functionexpresses the possibility measurement of fuzzy demands ac-cording to the decision maker’s view. For example, it may be

with highest possibility, and with thelowest possibility, and between and and between

and with a possibility level between zero and one.Hence the membership function can be given as

Let and be two triangular fuzzy numbers, the additionof two fuzzy numbers remains a triangular fuzzy number, whichcan be denoted as . Similarly,this concept can be extended to the addition of any number oftriangular fuzzy numbers.

Let be the total demands of productfromthe period 1 to a certain period. According to the extensionprinciple [14], the membership function is defined as

(17)

where and are the maximum and minimum operators re-spectively.

represents the possibility measurement of total fuzzydemands of productfrom period 1 to period according to thedecision maker.

Obviously, the total demands can also be de-scribed by a triangular fuzzy number, i.e.,

.Triangular fuzzy demands and total demands can be illus-

trated as shown in Fig. 1(a) and (b), respectively.

C. Fuzzy Equations and Their Addition

Production-inventory balance (10) in the FMAPP model isnot a crisp equation, but a fuzzy one. It implies that the equationis met in terms of a degree of truth, and it can be expressed asa membership function depending on the fuzzy demands

. Therefore

(18)

With this formulation, the production-inventory balance (10)indicates that the sum of production level in this period andthe difference between inventory level and back-order level inthe previous period minus the difference between inventory andback-order level in this period should not differ from the de-mand too much.The larger the difference, the smaller the de-gree of truth that the equation holds, and the less likely thatmarket demands can be met. In this sense, reflects thepossibility level at which the market demands are met. For ex-ample, if , the dif-


Fig. 1. Formulation of fuzzy demands and fuzzy production-inventory balance equation.

ference is zero, and the production-inventory balance equationis strictly satisfied, and the market demand is most possiblymet, i.e., the possibility level equals one. On the contrary, if

, the difference isgreater than the tolerance, and the production inventory balanceequation holds with a degree of truth being zero, and the marketdemand is unlikely met. Of course, when the right-hand of theequation becomes , the market demand will be likelymet, however it will bring higher inventory levels and holdingcosts, thus, it is not always practical, and therefore the equationhas a pessimistic degree of truth in this case. This formulationis illustrated as shown in Fig. 1, and the interpretation coincideswith the crisp situation when the demand is crisp, and theproduction-inventory balance equation holds absolutely over theentire planning horizon.

With this interpretation, given the decision maker’s requestthat the APP is to meet the market demands for productin aperiod with possibility level , the soft equation can be ex-pressed as follows:

(19)

as illustrated in Fig. 1.The addition of two soft equations is also a soft equation and

its degree of truth is defined as the logic addition of the degreesof truth for each member of soft equations according to the ex-tension principle [14]. It can be explained in the following ex-ample.

Let and be two soft equations, then theiraddition is denoted as in which the degree oftruth satisfies

(20)

according to max-min theory. Similarly, this operation can beextended to the addition of any number of soft equations.

By applying the addition of soft equations to (10), we canhave

(21)

more general, for

(22)

D. Formulation of the Fuzzy Production-Inventory BalanceEquations

With the formulation of fuzzy addition and fuzzy equation,the fuzzy production-inventory balance equation in a singleperiod can be interpreted as the possibility level of the APPmeeting the fuzzy demands. It can be formulated as in (19).Under the same types of fuzzy demands, the production-in-ventory balance equation in a single period is equivalent tothe dynamic balance equation for productfrom period 1 tocertain period , i.e., for , (19)is equivalent to

(23)

which can be illustrated as shown in Fig. 1(d).Therefore, the production-inventory balance equation under

a fuzzy environment can be formulated as in (19) or (23).

E. Formulation of the Fuzzy Production Capacity

Assuming the production capacity in period is a fuzzynumber with tolerance, its membership function re-


flects the decision maker’s level of satisfaction with the con-sumption of production capacity, and is defined as follows:

(24)

For example, if the consumption of production capacity is lowerthan the available capacity , decision maker is of course fullysatisfied. As the consumption of capacity increases, decisionmaker’s satisfaction level decreases, and the decision maker isbecoming dissatisfied when the consumption reaches the toler-ance point .

For a given threshold level, the decision maker’s satisfac-tion level with the consumption of production capacity can beformulated as , i.e.,

(25)

In general, the fuzzy demand may be extended to the casesof general L-R type and trapezoidal numbers. In those cases, theAPP can be formulated into the above model only if the fuzzydemands in each period are of the same type.

IV. PARAMETRIC PROGRAMMING MODEL

FMAPP model is a model of fuzzy nonlinear programmingwith fuzzy inequality and equality constraints. Unlike a crispmodel, the optimal solution here is not crisp but a fuzzy set “de-cision” which has different meanings depending on its defini-tion and formulation. APP is an important upstream level de-cision activity, and the decision is the source of family disag-gregation plan at downstream levels. Therefore, the sole op-timal solution to APP cannot guarantee that the feasible dis-aggregation plan will always be achieved. The diversity of thesolutions and multiple solutions to APP can offer more oppor-tunities to achieve the feasible disaggregation plan particularlyunder a fuzzy environment. From this point of view, the solutionof the model is “optimal” subject to some preferred possibilitylevels for meeting the market demands and satisfaction levelswith consumption of production capacities. In this case, a fuzzysolution [8], [14], [18], [20]–[22] can be obtained using para-metric programming techniques.

A preferred level is given to denote decisionmaker’s request for possibility level at which the APP meetsthe market demands, the soft equation can be defuzzified suchthat the cumulative demand of product typefrom period 1 to afuture period should be met with a possibility level higher than. At the same time, another threshold level can

be given to express the decision maker’s satisfaction level withthe consumption of production capacity which should not beless than . With this understanding, the FMAPP model can betransformed into the following equivalent parametric aggregateproduction planning (PAPP) model

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

where . The production-inventory balance equationin the original fuzzy optimization model FMAPP is expressedin terms of inequality constraints (27) and (28) as per the deci-sion maker’s request for possibility level at which APP can meetmarket demands.and are the preferred parameters, which re-flect the constraints of the possibility level of meeting the marketdemands and satisfaction level with the consumption of produc-tion capacity, respectively.

For given preferred satisfaction levelsand , the PAPPmodel becomes a parametric nonlinear programming model. Let

be an optimal solution to the para-metric nonlinear programming model, the following propertieson the solution to the PAPP model can be easily proved.

Property 1: In the case of increases with thecommon value of and .

Property 2: Given a specified increases with the.Property 3: Given a specified increase with the.For given preferred satisfaction levelsand , the optimal so-

lution is a fuzzy solution to the orig-inal fuzzy optimization problem FMAPP. That is to say, it’s anoptimal solution in a fuzzy sense, because it depends on the de-cision maker’s preference on the possibility leveland the sat-isfaction level . In particular, the optimal solution is crisp when

. Therefore, the fuzzy solution to FMAPP reflects thedecision maker’s preference and subjectivity, and it can increasethe chances and ability of constructing an APP to guarantee thefeasibility of a family disaggregation plan, especially under afuzzy environment.

Of course, in the PAPP model, the possibility levelandthe satisfaction level may have common or different values.Under some practical situations, the decision maker may expecta higher or lower possibility level of meeting the market de-mands for a certain product type in a certain period, hence dif-ferent possibility levels can be applied. Similarly, different sat-isfaction levels may also be selected in different periods tosuit practical situation and the decision maker’s preference. Insummary, the above model is flexible and can reflect the deci-sion maker’s preference in a fuzzy environment.


V. BESTBALANCE MODEL AND INTERACTIVE PROCEDURES

As mentioned in the previous section, the total cost increasesproportionally with the possibility level and the satisfactionlevel . In this case, decision maker can make a reasonableplan in light of his preferred possibility level and satisfactionlevel. However, in some cases, decision maker not only con-cerns about total cost, but also prefers to make a balance amongtotal cost, possibility level of meeting market demands and sat-isfaction level on the consumption of production capacities.

Let denote the decision maker’s required possi-bility level of meeting market demands from period 1 to a spe-cific period and denote the decision maker’s sat-isfaction levels with production capacity consumption in period

and the total cost in planning horizon respectively. There-fore, the APP problem concerns with not total cost alone anymore, but a best balance among the total cost, the possibility ofmeeting market demands and the satisfaction with consumptionof production capacity, and it can be expressed in a balanced ag-gregate production planning (BAPP) model as follows:

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

where the original production-inventory balance equation isgiven in (36), is the costs incurred in periodand have been described in (17) and (24) respectively,while is defined as

(44)

where is an aspiration level of the total cost pre-ferred by decision maker, and is an acceptable tolerance,say 5%–15% of .

The BAPP model can be transformed into the following non-linear programming model:

(45)

(46)

(47)

(27), (28), (30), (31), (32), (33), (34) (48)

and it can be solved by traditional nonlinear optimization tech-niques. The optimal solution is a crisp and

unique solution. However, this optimal solution may not be de-sirable or acceptable to decision maker in practice, especiallyunder a fuzzy environment. On the contrary, multiple solutionsunder different criteria may be preferred. The decision maker’smost important criteria may include minimizing the total cost,maximizing the possibility level of meeting market demands oncertain products in a specified period, and maximizing the sat-isfaction level with consumption of production capacities overspecified periods. To cater for this situation, an interactive pro-cedure for APP is proposed to help the decision maker make apreferred plan from a pool of optimal solutions under differentcriteria. The interactive procedure can be described as follows:

A. Interactive Procedure for APP

Step 1) Initialization: Given an initialized threshold re-flecting decision maker’s overall acceptable satis-faction level with total cost, consumption of pro-duction capacities, and possibility level required formeeting market demands.

Step 2) Input the decision maker’s most concerned criteriaindexes (CI) set

where represents the criterion for total cost.

represent the criteria for meeting market demandsof product in period ;and represent thecriteria for the consumption of production capacitiesin period .

Step 3) Construct an interactive aggregate production plan-ning (IAPP) model as follows:

(49)

(50)

(51)

(52)

(53)

(54)

(30), (31), (32), (33), (34) (55)

where denotes the possibility level ofmeeting the market demands on productin periodand behaves in the same way as in (18), andis an aggregated satisfaction level with the criteriaof concerns, and are theindicators of criteria index, and only one of themequals 1 at a time.

Step 4) Set initial criteria index and optimal solutionpool to be empty.


TABLE IBASIC DATA AND MODEL PARAMETERS OF THESIMPLE EXAMPLE

Step 5) Reset indicators of criteria indexesas follows:

Step 6) Solve the IAPP model by traditional constrainedoptimization techniques, and reset the optimalsolution pool , where

is the optimal solution with thethcriterion.

Step 7) if , go to Step 8; else, , go toStep 5.

Step 8) Input decision maker’s most preferred criteria anddisplay the corresponding optimal solutions, stop.

Of course, in this interactive procedure, multiple indicatorsfor criteria indices can equal 1 simultaneously, and in this

case the indicator vector represents a combination of multiplecriteria at a given time. The values of indicatorcan be furtherextended to represent the weights of the criteria when it assumesa value in [0, 1], and the multiple criteria can be also consideredsimultaneously.

With this interactive procedure, an optimal solution pool [23]consisting of a set of potential optimal solutions with var-

ious criteria will be obtained [24], from which decision makercan interactively establish a production plan to suit particularcircumstances. The interactive procedure and the optimal so-lution pool reflect decision maker’s preference, and can offerdecision maker more choices to facilitate the establishment ofdownstream family disaggregation plans under a fuzzy environ-ment.

VI. I LLUSTRATIVE EXAMPLE AND ANALYSIS

To illustrate the models proposed in this paper and assess theeffect of the possibility level and the satisfaction level onthe total cost and production plan in the parametric program-ming model, a simple simulation example using optimizationtool box of software Matlab 5.0 is presented in this section. Thebasic datum of the example and model parameters are shown inTable I. For this simple example, two types of products denotedby product 1 and product 2, are produced in periods 1–4 to meetthe customer demands, of which the value is an approximate

one and it can be formulated as a triangular fuzzy number. Forexample, the demands of product 1 in period 1 is 85 units mostpossibly, at least 80 units (pessimistically), and not more than90 (optimistically). The unit production costs of product maybe varied with product types and periods, for example product1 has identical unit costs in different periods and ithas different unit costs for product 2. Other input dataand parameters, e.g., workforce costs, capacity, initial inventorylevel, inventory costs, back-order costs capacity coefficients canbe explained in similar way.

The simulation is conducted in three parts. The first part aimsto analyze the effect of the parameters on the aggregate plan inthe parametric programming model. The second and last partsfocus on performance of the best balance model (BAPP) and theinteractive procedure respectively. The simulation results andanalysis are presented as follows.

A. Simulation of the Parametric Programming Model and theResults

1) Effect of Common Values of and on the TotalCost: To analyze the effect of the common values on the totalcost, common level and are scaled by discretesizing [0,1] in equal steps of 0.05. Table II presents the production,workforce, inventory and back-order levels for product 1 and2 in each period and the corresponding total costs when thevalue of both and varies in steps from 0.60 to 1.0. Theeffect of the common values on the total cost are depicted inFig. 2. It can be seen from Table II that the total cost increaseswith and linearly as shown in Fig. 2. The simulation resultssuggest the optimal production policy that the inventory levelsof product 1 in each period (except when ) andproduct 2 in the final period are zero, and the back-order ineach period is absolutely not permitted, and the workforce levelis proportional to the production level. These results coincidewith the optimal production policies as shown in Table III.

2) Effect of Level on the Total Cost Given Specified Valuesof : In this part of the simulation, two specified values of thepossibility level are used, i.e., and , and variesfrom 0.0 to 1.0 discretely in equal steps of 0.10. The resultsshown in Fig. 4 show that with both of the specified values of,


TABLE IIAGGREGATEPRODUCTION PLAN WITH COMMON VALUES OF� AND BY PARAMETRIC PROGRAMMING

the total cost increases linearly with, i.e.,and . As

indicated in the formula, however, there are no great changes inthe total cost for different levels of for a specified . Hence,

has little effect on the total cost for a specified. That meansfor a given possibility level of meeting the market demands, thesatisfaction with the consumption of capacities has little effecton the total cost.

Results of the simulation also conclude that for any specified, the inventory costs increase with, while both the production

cost and workforce cost maintain at certain levels except for theworkforce cost when . The average inventory cost ofincreases proportionally to. It implies that both the productioncost and the workforce cost are only dependent on the possi-bility level of meeting the market demands. In fact, in order tomaintain the possibility levels of meeting the market demand,the tight constraints of production capacity in each period canonly bring more inventory levels and warehouse holding cost,while the production and workforce costs do not decrease as aFig. 2. Total cost varies with common values of theta and gamma.


TABLE IIIOPTIMAL STRUCTURE OFAGGREGATEPRODUCTIONPLAN AND TOTAL COST

Fig. 3. Total cost varies with theta and gamma simultaneously.

whole. This result reveals a hint that the relaxation of toleranceof the production capacity in each period may decrease the totalcosts.

The simulation also implies that asincreases, the produc-tion level of product 2 increases in the first two periods anddecreases in the last two periods; and the production levels forproduct 1 are almost unchanged withexcept for the last pe-riod, and it is produced in the optimal way that the inventory andbackorder levels in each period remain zero. The optimal pro-duction policies shown in Table III help explain this situation.

It can be concluded that the results of the simulation not onlyreflect the effect of on total cost given a specified, but alsoimply that has substantial impacts on the total cost and hencethe production plan, while has relatively little effect.

3) Effect of on the Total Cost Given Specified Values of: In this part of the simulation, two specified values of, i.e.,

0.7 and 1.0 are selected, andvaries from 0.0 to 1.0 discretelyin equal step size of 0.10. The results are shown in Fig. 5.

It can be seen that for any given, the total cost increaseslinearly with , i.e., ,and . The productionlevel of product 2 increases rapidly in the first two periods anddecreases in the last period with, whereas the production level

Fig. 4. Total cost varies with gamma given specified theta.

Fig. 5. Total cost varies with theta given specified gamma.

for product 1 in each period satisfies that the inventory level ineach period is almost zero for any value of.

As shown in Fig. 5, there is significant difference in the totalcost between neighboring levels ofwith the same values of

, while there are little difference between neighboring levelsof with the same . It can be also concluded thathas greatimpacts on the total cost and APP, whilehas relatively littleeffect.

4) Combined Effects of and on the Total Cost and Pro-duction Plan: In this part of the simulation, 100 sets of valuesof and are used, where 20 sets are with common values,40 sets are with either or specified, and the remaining arerandom numbers in [0 1]. The resulting production, workforce,inventory and back-order levels, as well as the total cost behavelinearly as shown in Table III. The total costs with varyingand

change as shown in Fig. 3.It can be seen from the optimal structure depicted in Table III

that the production level for product 1 decreases slightly in thefirst three periods, and then increases slightly withand rapidlywith in the last period. On the other hand, the production levelfor product 2 increases significantly in the first two periods andthen decreases rapidly withand slightly with respectively


TABLE IVAGGREGATEPRODUCTION PLAN BY WAY OF INTERACTIVE PROCEDURE

in the last two periods. The workforce level varies proportion-ally to the production level. The inventory level for product 1 isslightly affected by both and , and mostly around zero overthe entire planning period, nevertheless the inventory level forproduct 2 increase rapidly with bothand in the planningperiods except in the last one. The back-order level for eachproduct over the entire planning period remains zero owing tothe high back-order penalty imposed in this example. The op-timal production policies coincide with the results in the afore-mentioned three parts of the simulation. As shown in Table IIIand Fig. 3, the total cost increases significantly with the level

and only slightly with . So far, it can be concluded that the

parameter plays a more influential role than in the model,hence the possibility level has more significant impacts on theproduction plan.

B. Simulation Results on the Best Balance Model

The best balance model is simulated using the Minimax func-tion in the optimization tool box of the software Matlab 5.0.The resulting best balance degree, as well as the production,workforce, inventory and back-order levels and the total cost areshown in the first row in Table IV. It can be seen that the best bal-ance among them is at a satisfaction level of 0.5455 with a costof 170 954 pounds. That means when a higher possibility level


of meeting market demand or a satisfaction level higher than0.5455 is requested, the total cost will exceed 170 954 pounds,giving a less satisfaction level than 0.5455 with total cost, andvice versa.

C. Simulation Results on the Interactive Procedure

Given an initial overall acceptable satisfaction level with thetotal cost,consumption of capacities and possibility level

, the interactive procedure is implemented using the con-strained optimization function in Matlab 5.0, the preferred so-lutions under different criteria are obtained as shown in Table IV.

Following the best balance criteria, the criteria number 0 in-dicates the aggregate plan under criteria of minimizing the totalcosts. The lows with criteria numbered 1–4 correspond to the so-lutions under criteria of maximizing possibility level of meetingmarket demands of product 1 in periods 1–4, respectively. Sim-ilarly the lows with criteria number 5-8 correspond to the solu-tions under criteria of maximizing possibility level of meetingmarket demands of product 2 in periods 1–4, respectively. Whilethe criteria numbered 9–12 correspond to the ones of maxi-mizing satisfaction of consumption of production capacity inperiods 1–4, respectively. It can be seen from the Table IV thatgiven an acceptable level , the optimum satisfactionlevel of the total cost can reach 0.6297 with a total cost of 170406 pounds. While the possibility levelsof meeting the marketdemands of products 1 and 2 in each period can reach 1.0, andthe corresponding satisfaction levelswith consumption of ca-pacities can reach 1.0 too.

It can be seen from the total cost column in Table IV thatthe minimum total cost is 170 406 pounds achieved at criterianumber 0, and the maximum total cost is 171 575 pounds ob-tained at criteria number 5 and 12, and there is a differenceamong other criteria, particularly the differences is distinctiveamong different types of criteria. For example, there is differ-ences of 387 pounds between the minimum value of total costsunder criteria of possibility of meeting market demands and ofsatisfaction of consumption of production capacities. Moreover,there is distinctive differences in solution under different dif-ferent types of criteria. Both the possibility levels under criterianumbered and satisfaction levels under criteria numberedtype are achieved as 100%, which coincides the actual sce-nario.

D. Implication for Decision Procedure

In summary, on the one hand, decision maker can select a pre-ferred production plan with a common satisfaction level, e.g.,according to the market demands and available production ca-pacities, if are chosen, the third row in Table IIshows the corresponding APP. Of course, one may choose dif-ferent combination of and based on the information and im-plication given in Figs. 2–5. The structure of the optimal solu-tion presented in Table III can help decision maker in aggregateproduction planning. On the other hand, one can choose pre-ferred production plan corresponding to one’s most concernedcriteria, e.g., if one concerns most with best balance among thetotal cost, meeting market demands and consumption of capaci-ties, then the first row in Table IV gives the corresponding plan.Similarly, if the total cost is of major concern, then the second

row in Table IV offers the feasible plan. Of course, if the criteriaof meeting market demands for product 1 in period 4 is of mostconcern, then the sixth row in Table IV suggest afeasible aggregate plan.

VII. CONCLUSION

Taking into account the fuzzy demands and fuzzy capacitiesin each period, this paper proposes a fuzzy approach to formu-lating MAPP problems under financial constraints. The fuzzyproduction-inventory balance equation is formulated as a softequation with a degree of truth, and it can be interpreted asthe possibility level of meeting the market demands. With thisformulation and interpretation, a FMAPP model is developed,and it is transformed into a parametric programming model anda best balance model to cater for different scenarios and de-cision maker’s preferences. A parametric programming basedfuzzy solution and an interactive aggregate production plan-ning procedure to solve the problem are also discussed. Withthe proposed models and approaches, on the one hand, first, thedecision maker can select a preferred production plan with acommon satisfaction level or different combinations of possi-bility and satisfaction levels, according to the market demandsand available production capacities, secondly, the structure ofthe optimal solution presented in Table III can help decisionmaker in APP. On the other hand, the decision maker can alsomake a preferred and reasonable production plan correspondingto one’s most important criteria. The approach could not guar-antee a global optimal solution, however the solution is nearoptimal and it can effectively enhance the capability that anaggregate plan can have feasible family disaggregation plansunder different circumstances, especially under the environmentof fuzzy demands and fuzzy capacities.

ACKNOWLEDGMENT

REFERENCES

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Richard Y. K. Fung received the B.Sc. (Hons.)and M.Phil. degrees from University of Aston,Birmingham, U.K., and the Ph.D. degree fromLoughborough University, Loughborough, U.K.

He is the Founder and Convenor of the govern-ment-funded Forum for Environmental Supply ChainManagement (FESCM), and the Forum for IndustrialLearning Organizations (FILO). In addition to under-taking various academic and administrative functionswith the City University of Hong Kong, Hong Kong,he is also the Principal Investigator of a number of

research projects funded by the University and the Hong Kong SAR Govern-ment. He has published over 100 research and scientific papers, around 50 ofwhich appear in international referred journals. He is the Director of the En-terprise Knowledge Integration and Transfer Laboratory (E-KIT Lab.), and theGroup Coordinator of Design and Manufacturing Informatics Activity Groupin the Department of MEEM. His current research interests include ERP, fuzzysystems and applications, product design and management, logistics and supplychain management, enterprise knowledge integration, product life-cycle man-agement.

Jiafu Tang received the B.Sc. degree in mathematicsfrom Hunan Normal University, Changsha, China, in1989, and the M.Sc. degree and Ph.D. both in controltheory and systems engineering from NortheasternUniversity, Shenyang, R.O.C, in 1992 and 1999 re-spectively.

He is currently Professor in the Research Instituteof System Engineering, School of InformationScience and Engineering, Northeastern University(NEU), in Shenyang, R.O.C. He was ResearchAssistant and Senior Research Associate at City

University of Hong Kong, in 1998 and 2000, respectively, and at the HongKong Polytechnic University as Research Fellow in 2002. He authored twoChinese books in Scientific House of China, Beijing, and China MachinePress, Beijing, in 2000 and 2001, and he has published more than 40 papers ininternational and local journals. He is currently interested in fuzzy optimizationtheory and its applications, supply chain planning and logistics management,quality design in new product development, quality function deployment.

Dingwei Wang received the B.E. degree in au-tomatic instrument from Northeastern University(NEU), Shenyang, R.O.C, the M.S. degree insystems engineering from Huazhong University ofScience and Technology, Wuhan, P.R.C., and thePh.D. degree in control theory and applications fromNEU.

He is currently Professor and Chair with the Re-search Institute of Systems Engineering, School ofInformation Science and Engineering, NEU. He hasbeen a Post-Doctoral Fellow at the North Carolina

State University, Raleigh. Since 2001, he has been a Panel Member of the Eval-uation Committee of Management Science Department of National Natural Sci-ence of Foundation of China (NSFC). He is currently interested in modeling andoptimization, soft computing, genetic algorithms, agile manufacturing, produc-tion planning and scheduling.

Dr. Wang is a member of The New York Academy of Sciences and a memberof The Economic and Management Systems Committee of The Chinese Associ-ation Automation. He is now member of Editorial board of International Journalof Computer and Operations Research, Editorial board of International Journalof Fuzzy Optimization and Decision Making.

multiproduct aggregate production planning with fuzzy ......common in aggregate production planning....

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