procurement scheduling for complex projects with fuzzy activity durations and lead times

Computers & Industrial Engineering xxx (2014) xxx–xxx

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Computers & Industrial Engineering

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Procurement scheduling for complex projects with fuzzy activitydurations and lead times

0360-8352/$ - see front matter � 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cie.2013.12.009

⇑ Corresponding author at: Indian Institute of Management, Prabandh Nagar, OffSitapur Road, Lucknow, Uttar Pradesh 226013, India. Tel.: +91 9158405056.

E-mail addresses: [email protected], [email protected] (V. Dixit),[email protected] (R.K. Srivastava), [email protected] (A. Chaudhuri).

1 Tel.: +91 522 2736613.2 Tel.: +91 522 2736663.

Please cite this article in press as: Dixit, V., et al. Procurement scheduling for complex projects with fuzzy activity durations and lead times. CompIndustrial Engineering (2014), http://dx.doi.org/10.1016/j.cie.2013.12.009

Vijaya Dixit a,b,⇑, Rajiv K. Srivastava b,1, Atanu Chaudhuri b,2

a Institute of Management Technology, 35 km Milestone, Katol Road, Nagpur, Maharashtra 441502, Indiab Indian Institute of Management, Prabandh Nagar, Off Sitapur Road, Lucknow, Uttar Pradesh 226013, India

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:Procurement scheduleComplex productsUncertaintyFuzzy integer programming

a b s t r a c t

Material procurement is a vital activity for manufacturing complex products like ships and aircrafts withlong manufacturing cycle times. The project activities are interlinked through numerous precedence andsuccession rules. Given these interdependencies, it is difficult to ascertain the durations of the activitiesprecisely at planning stage and creates uncertainty in the requirement dates of the items. The lead timesof items are also not known accurately and result in uncertainty in the availability dates. Consequently,inventory holding and shortage costs incurred cannot be crisply defined. The objective of this paper is todevelop a procurement scheduling model considering the above uncertainties.

We have developed a method to calculate fuzzy holding and shortage costs and used those as fuzzy costcoefficients in the procurement scheduling model. It minimizes the sum of the above costs under budgetconstraints and generates optimal ordering schedule. It is applied for procurement scheduling of a realship building project. Two types of sensitivity analyses were performed: first to understand the effectof variation of degree of uncertainty on total cost and on stage budget requirements and second to studythe effect of changes in allocated stage budget parameters on total cost. The results indicate that totalcost can be reduced significantly if stage wise budgets are determined considering the uncertaintiesrather than allocating budget upfront and treating them as constraints. The sensitivity analyses per-formed, helps in identifying the most sensitive stage of the project and determine the ranges in whichstage-wise budgets can be varied.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Material procurement is a vital activity particularly for manu-facturing complex products like ships, aircrafts with long cycletimes. Manufacturing of such products is usually managed as aproject, with interim products defined for each stage of the project.The production processes for such products are predominantlyassembly based with complex activities interlinked through prece-dence rules (Koenig, Taylor, MacDonald, Lamb, & Dougherty, 1997).Given these interdependencies, it is difficult to ascertain the dura-tions of the activities precisely at planning stage and this createsuncertainty in the requirement dates/time unit of the items.Timely material availability plays a critical role for on-time andbudgeted completion of such projects (Manavazhi & Adhikari,2002). However, most of the materials required for these activities

are in modular or sub-assembly form and are procured from differ-ent suppliers with lead times that are not known accurately andthis result in uncertainty in the availability dates/time unit ofitems. The combined effect of requirement and supplier lead timeuncertainties leads to unavailability of some items required at pro-duction site, subsequently delaying the project, and can also resultin high inventory carrying cost for those items which are not re-quired immediately. Moreover, cash inflows for such projects takeplace in stages marking the start of key events and need to be opti-mally utilized to ensure availability of right material at the righttime. All these complexities make procurement scheduling of com-plex projects significantly challenging for practicing managers par-ticularly for budget constraint projects.

A significant gap exists in the approaches being practiced byproduction and procurement departments. Production schedulingis based on modern Product-oriented Work Breakdown Structure(PWBS), where each interim product, considered as the end prod-uct of a stage, is manufactured by collaborative efforts of crossfunctional teams. But, procurement scheduling still continues tofollow a traditional functional approach, which does not take intoaccount the relationship between inter-departmental dependen-cies of work schedules. This leads to further mismatch between

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2 V. Dixit et al. / Computers & Industrial Engineering xxx (2014) xxx–xxx

the planned requirement and actual availability of items. Materialcosts in complex projects account for 60–70% of total costs. Price-reduction is extensively used as a competitive tool to grab orders(Kini, 1999; Yeo & Ning, 2002). Yeo and Ning (2002) and Sun andLiu (2008), emphasize procurement and logistics as a major areaof constraint and an opportunity which can be exploited to signif-icantly improve the overall performance of the project. This can beachieved if some of the currently available management paradigmsare applied to manage project procurement while considering theuncertainties/impreciseness prevalent during planning stage.

In this paper, we focus on synchronizing materials procurementwith production planning while addressing the associated uncer-tainties and budget constraints. The activity durations and leadtimes of items are uncertain/imprecise and are represented as fuz-zy numbers. Due to these uncertainties inventory holding andshortage costs incurred cannot be crisply defined for a particularorder time unit. We develop a method to calculate fuzzy holdingand shortage costs and use those as fuzzy cost coefficients in theprocurement scheduling model which minimizes their sum underbudget constraints and generates optimal ordering schedule.

The motivation of this research came from visiting a leadingshipyard where we found that the long shipbuilding projects in-volved in manufacturing complex end products face similar chal-lenges as described above. A ship is a self-sustaining unit,equipped with multiple support systems to make it functionallyindependent in far seas. It is divided into blocks as PWBS of theproduct. Each block is unique and material requirement of eachblock is different. For example, the most basic material ‘‘steel’’ var-ies across the blocks in terms of thickness. The requirements of theoutfitting materials, modules, sub-assemblies vary as the blockerection advances. These emphasize the need for alignment ofmaterial procurement schedule with the production sequence. Inthis work, we develop a procurement scheduling model for a ship-building project named ‘‘Anchor Handling Tug Supply Vessel’’using the data obtained from this project.

Our approach helps to align material procurement schedulingwith PWBS based production scheduling followed while manufac-turing complex products. The results of the proposed model indi-cate that fuzzy inventory holding and shortage costs can bereduced significantly if stage wise budgets are determined consid-ering the uncertainties in activity durations and lead times of itemsrather than allocating the budget upfront and treating them as con-straints. We also perform two types of sensitivity analyses: first tounderstand the effect of variation of degree of uncertainty on totalcost and on budget requirement of stages and the second to studythe effect of changes in allocated stage budget parameters on totalcost. These sensitivity analyses can help practicing managers toidentify the most sensitive stage of the project and determinethe ranges in which stage-wise budgets can be varied.

2. Related literature and motivation for research

Time buffers can be preferred over safety stocks when the var-iation is in lead-times of input items, as opposed to demand fluctu-ations (Whybark & Williams, 1976). Our work is also set in thesame context where quantity requirements of items are knownbut activity durations and lead times are not known precisely.Thus, while designing a procurement scheduling model, time buf-fers need to be decided, based on the trade-off between inventoryholding and shortage costs.

Yeo and Ning (2002), describe that complex products manufac-tured in an Engineer Procure Construct (EPC) setting, are made upof a large number of interconnected subsystems and components,requiring considerable human efforts and financial commitment.The EPC activities are time-phased according to specified prece-

Please cite this article in press as: Dixit, V., et al. Procurement scheduling for coIndustrial Engineering (2014), http://dx.doi.org/10.1016/j.cie.2013.12.009

dence rules, resource requirements and constraints. Due to thesecomplexities, accurate prediction of activities’ durations is very dif-ficult. Materials are the foundation of constructed facilities andtheir costs represent a major portion of total costs in EPC projects.Procurement is a connecting function between engineering andconstruction. It is highly dependent on external companies andbuyer’s control is very limited, especially in outsourcing and pur-chasing long lead-time equipments where no buffer inventory ismaintained by suppliers. These circumstances make lead time anuncertain parameter.

In MRP system lead time is deterministic; if all the orders ofitems are released at the order date (difference of requirement dateand lead time), then total cost will automatically be minimal. But,if uncertainty or impreciseness exists, the arrival of item is uncer-tain. In this case, for each specific value of order date used in theMRP method, some shortage cost or inventory holding cost exists(Dolgui & Ould-Louly, 2002). Laufer, Denker, and Shenhar (1996)opine that systematic and integrative planning should be madeaccording to prevalent uncertainty. Timing of decisions and theirdegree of completeness should be adjusted according to the stabil-ity and preciseness of the information on hand. Thus, attentionshould be paid to the quality and completeness of the informationat an early stage. So that managers can employ essential redundan-cies to formulate an integrative project plan with maximum reli-ability and adaptability. Our work addresses such impreciseness/uncertainty in activity durations and lead times of items by theuse of fuzzy numbers (discussed in details in Section 3).

The traditional approach of procurement scheduling used byengineering firms is based on a constant safety lead time appliedto the whole duration of the construction phase. Caron and Mar-chet (1998) point out two main shortcomings of this approach:firstly it does not take into account the different stages of the con-struction process and considers every planning period in the sameway; secondly it does not take explicit account of the uncertaintyfactors that influence the delivery and the construction processes,causing delivery delays and variability in productivity. They haveprovided a relationship between the ‘‘construction progress’’, the‘‘required availability’’ and the ‘‘actual availability’’ curves. A sto-chastic model evaluates the safety stock and the correspondingsafety lead time taking into the variability in delivery dates andthe rate of progress of construction. However, it is difficult to applysuch a model, particularly in case of one of a kind make to orderprojects (like shipbuilding) where historical data is hardly avail-able for stochastic modeling. The concepts of construction progressand availability curves are theoretically conceivable but are diffi-cult to implement quantitatively.

Yeo and Ning (2006) provided an improved buffer managementincorporated within the critical supply chain management frame-work. Their focus is on dynamically managing the feeding buffersinserted between the two important sets of control dates namely,the Promised Delivery (PD) dates by the equipment vendors andthe Required-On-Site (ROS) dates according to the master construc-tion schedule. However, their research is entirely qualitative in nat-ure and strives to generate insights from broad industrial practices.Further, the above proposed models (Caron & Marchet, 1998; Yeo &Ning, 2006) address the procurement scheduling problem at anaggregate planning level and do not drill down to the execution le-vel to finally decide the ordering dates for every individual item.Our work overcomes this gap and obtains an optimal order datefor each item, based on the uncertainty in its requirement dateand lead time. It also performs quantitative sensitivity analyses,for degree of uncertainty and stage budget requirements, to deriveexecutable insights for material procurement decisions.

In large projects, where several dependent items are ordered,delay in one order is sufficient to delay the whole project with pen-alty. A model developed by Ronen and Trietsch (1988), Ronen and

mplex projects with fuzzy activity durations and lead times. Computers &


V. Dixit et al. / Computers & Industrial Engineering xxx (2014) xxx–xxx 3

Trietsch (1993) includes the effect of additional inventory holdingcost of items which are already present but cannot be used becauseof the non-availability of their predecessor item. The model appliesdifferential calculus to derive a formula similar to newsboy prob-lem, to calculate optimal ordering time of each item with probabi-listic lead time, which results in the minimum expected weightedtotal of holding cost and lateness penalty. But they have not con-sidered the impreciseness in requirement times of items arisingdue to uncertainty in activity durations and do not address pro-curement scheduling under stage budget constraints.

Manavazhi and Adhikari (2002) targeted a research on highwayprojects in Nepal. It aimed at ascertaining the occurrence of mate-rial and equipment procurement delays and then, to perform anassessment of the causes of the delays and the magnitude of theirimpact on project costs. Yildirim (2004) studied the piecewise pro-curement of a large-scale project. The author modeled a dynamicprocurement game where he investigates the effects of the physi-cal progress of the project on parties’ relationships. Elazouni andGab-Allah (2004) have carried out extensive research on finance-based scheduling of construction projects. An integer-program-ming based scheduling model provides financially feasibleschedules that balance the financing requirements of activities atany period with the cash available during that same period. This ap-proach minimizes the total project duration while addressing thefixed credit limits. Further, Elazouni and Metwally (2005) utilizedgenetic algorithm technique to devise finance-based schedules thatmaximize project profit through minimizing financing costs andindirect costs. Later, Elazouni and Metwally (2007) expanded theabove finance-based scheduling model for relaxed credit limits sce-nario. Sun and Liu (2008) underpin the importance of material pro-curement planning (MPP) and recognize it as a capital-intensivedecision that often accounts for a large portion of the total operatingcosts. They have formulated a fuzzy minimum-risk MPP problemwhich incorporates uncertainty in material’s price, product’s de-mand and the uncertainty of changing raw material to finished prod-uct. However, above models lack in considering the effect ofuncertainty arising due to simultaneous impreciseness in activitydurations and lead times of items.

James (1989) emphasized the importance of alignment of PWBSin design as well as in scheduling equipment procurement. Hedeveloped a decision support system for shipyards based on leadtime and backward scheduling calculations. This approach, thoughstill widely used, fails to analyze the planned order dates from thepoint of view of uncertainty and budget constraints. Cakravastiaand Diawati (1999) proposed a system dynamic model to facilitateassessment of logistic operating performance of shipbuildingindustry in Indonesia. The model developed is mainly concernedwith the integration of money flow together with material andinformation flow. Dwivedi and Crisp (2003) carried out a detailedstudy of current trends in material management in the shipbuild-ing industry. They observe the development of computer softwarewith the advancement of automation in material handling. Yue,Wang, and Zhang (2008) analyzed the modern shipbuilding logis-tics in Chinese context and put forward a shipbuilding logisticsmanagement system based on MRP-II and JIT.

Procurement planning literature has also witnessed contribu-tions in the selection of procurement methods and identificationof appropriate criteria for the same. Many authors have applied dif-ferent methodologies like Multi-Attribute Utility Technology(MAUT) analysis (Love, Skitmore, & Earl, 1998), Analytic HierarchyProcess (AHP) and Parker’s judging alternative technique (Alhazmi& McCaffer, 2000), MAUT and AHP (Cheung, Lam, Leung, & Wan,2001), Case-based reasoning (Luu, Thomas Ng, & Chen, 2003). Thisstream of research identifies ‘‘uncertainty’’ as an important dimen-sion for aligning procurement scheduling with production plan-ning of complex manufactured products.


The above review shows that procurement scheduling literatureof complex projects lacks developments based on line of thinkingas conceptualized by Ronen and Trietsch (1988), Ronen and Tri-etsch (1993) and Caron and Marchet (1998). The models discussedabove do not provide guidance on how to determine the optimalordering schedule of items at execution level while consideringuncertainties in activity durations and lead times along with bud-get constraints in one holistic model. Our work is an attempt tofurther initiate research on the same lines and to understand theeffect of variation of degree of uncertainty on the budget require-ment of stages through sensitivity analyses, which has not yet beenwell explored in the literature of complex projects.

3. Modeling under uncertainty

Two common modeling approaches for uncertainty are stochas-tic and fuzzy mathematical programming. In stochastic program-ming, uncertainty is modeled by discrete or continuousprobability functions which are defined based on historical evi-dence and application of statistical techniques (Demirli & Yimer,2008).

When reliable past data is unavailable or when sound measure-ment is difficult to perform, probabilistic modeling will be inappro-priate. In this situation, qualitative data described by linguisticexpressions derived from experience, intuition or subjective man-agerial judgment is used (Dubois & Prade, 1994; Petrovic, 2001).Fuzzy set theory, proposed by Zadeh (1965), considers uncertain/imprecise parameters as fuzzy numbers with satisfaction level ofa range defined by membership function. It provides a method tolinguistically represent knowledge. Bellman and Zadeh (1970) pre-sented applications of fuzzy theory to the various decision makingprocesses in a fuzzy environment. In the context of one of a kindcomplex projects like ship, there are hardly any historical dataavailable. Each shipbuilding project is unique with respect to mate-rial requirements and the way the activities of the project are exe-cuted. Managers have to rely on their subjective judgments andpast experience to evaluate relevant parameters. They find it easierto provide information in a range; with optimistic, most likely andpessimistic values. Therefore, in this research, we have taken L–RTriangular Fuzzy Numbers (TFN) to represent the activity durationsand lead times of items.

Triangular membership functions are simple to use. They can beplotted by limited information available in the form of modal(most likely) value of the linguistic set with its upper and lowerbounds. The distribution of the degree of membership betweenthese boundaries can then be considered as linear. Managers findit easy to give information in this form with modal value beingreferred to as most likely value and bounds as optimistic and pes-simistic values. If other forms of membership functions are to beused, we need more auxiliary information at the selected interme-diate points within these bounds. However, getting such informa-tion is not possible in practical setting, as managers hardly haveany data for the intermediate points. L–R Triangular Fuzzy Num-bers have been extensively used in literature for modeling impre-cise information mainly due to their computational efficiency.Pedrycz (1994) provides mathematical justification for the use ofTFN based on balance entropy and error free reconstruction prop-ositions. Utilizing same concepts, Herrera and Martinez (2000)stipulate use of TFN as one of the necessary and sufficient condi-tions for defining linguistic term set to avoid any loss of informa-tion during the transformation process from linguistic 2-tuple tonumeric values and vice versa. Dubois, Foulloy, Mauris, and Prade(2004) have shown that the triangular possibility distribution is alegitimate transformation of the uniform probability distributionwith the same support, and it is an upper bound of all the




possibility transforms associated with all the bounded symmetricuni-modal probability distributions with the same support. Fur-ther, performing computations like fuzzy addition and subtractionon triangular fuzzy numbers yield the same type of membershipfor the resulting fuzzy number. Ease of computations and ease ofeliciting the data from managers using TFN further motivate theiruse (Lai & Hwang, 1992; Liang, 2006; Liang, 2008).

A Fuzzy number ~P is a fuzzy set defined on the set of real num-bers R, characterized by means of a membership function l~PðxÞsatisfying following properties:

1. l~PðxÞ is piecewise continuous.2. l~PðxÞ is a convex fuzzy subset.3. l~PðxÞ is the normality of a fuzzy subset, implying that for at

least one element x0, called core (modal/ most likely) value ofthe fuzzy number, the membership function value must be 1i.e. l~Pðx0Þ ¼ 1.

A L–R Triangular Fuzzy Number is the one whose membershipfunction has following form:

l~PðxÞ ¼LððPm � xÞ=aÞ if x 6 Pm

Rððx� PmÞ=bÞ if x P Pm

�

where Pm is the core/modal/most likely value, a, b > 0 are left andright spreads respectively of the fuzzy number. L and R are contin-uous, non-increasing functions defined in the interval [0,1) asL(x) = max(0,(1 � x)) and R(x) = max(0,(1 � x)). Values of L and Rfunctions decrease from 1 to 0, and fulfill the conditionsL(0) = R(0) = 1. Thus, a L–R TFN can be represented as ðPm;a;bÞL—R.Further, addition and subtraction of L–R TFN ~P1 and ~P2 are definedas follows (Dubois & Prade, 1978).

~P1 � ~P2 ¼ ðPm1 þ Pm

2 ;a1 þ a2;b1 þ b2Þ

~P1 � ~P2 ¼ ðPm1 þ Pm

2 ;a1 þ b2;b1 þ a2Þ

The c-cut set of a fuzzy number ~P is defined as~Pc ¼ fx 2 R : l~PðxÞ � cg. For a L–R TFN ~P as defined above, the c-cut set is given by

~Pc ¼ ½Pm � ð1� cÞa; Pm þ ð1� cÞb�:

Dubois and Prade (1987) defined an interval-valued expectationof fuzzy numbers, bounded by the expectations calculated from itsupper and lower distribution functions. Goetschel and Voxman(1986) introduced possibilistic interpretation of ordering andintroduced following method for ranking fuzzy numbers, withthe motivation to give less importance to the lower levels of fuzzynumbers. If ~P1 and ~P2 are two TFNs, with c-cut sets.

~Pc1 ¼ ½P

m1 � ð1� cÞa1; P

m1 þ ð1� cÞb1�

~Pc2 ¼ ½P

m2 � ð1� cÞa2; Pm

2 þ ð1� cÞb2�

then;

~P1 6~P2 ()

Z 1

0fPm

1 � ð1� cÞa1 þ Pm1 þ ð1� cÞb1g

6

Z 1

0fPm

2 � ð1� cÞa2 þ Pm2 þ ð1� cÞb2g

Based on this principle, Carlsson and Fukker (2001) introducedthe notion of lower possibilistic and upper possibilistic mean val-ues and defined interval-valued possibilistic mean and crisp possi-bilistic mean value of a continuous possibility distribution. For atriangular fuzzy number ~P


Lower possibilistic mean value M�ð~PÞ ¼Z 1

0c�fPm � ð1� cÞagdc=

Z 1

0cdc

¼ Pm � a=3

Upper possibilistic mean value M�ð~PÞ¼Z 1

0c�fPmþð1�cÞbgdc=

Z 1

0cdc

¼ Pmþb=3

Crisp possibilistic mean value Mð~PÞ ¼ ðM�ð~PÞ þM�ð~PÞÞ=2¼ Pm þ ðb� aÞ=6

Since ranking method given by Goetschel and Voxman (1986)directly matches the managers’ general perceptions of optimistic,pessimistic and most likely values and Carlsson and Fukker(2001) used the same principle. We adopt method by Carlssonand Fukker (2001) to calculate crisp possibilistic mean value ofTFN. In this paper, we term Pm as most likely value, Po = Pm � aas the optimistic value and Pp = (Pm + b) as the pessimistic valueof fuzzy number, as stated by mangers. We represent, L–R TFN as~P ¼ Po; Pm; Pp as shown in Fig. 1. Based on these notations crisp pos-sibilistic mean value of ~P is Mð~PÞ ¼ Pm þ ðb� aÞ=6 ¼ ð4 � PmþðPm � aÞ þ ðPm þ bÞÞ=6 ¼ ðPo þ 4 � Pm þ PpÞ=6. In the subsequentsection, we will use this method to convert fuzzy number into itsequivalent crisp value.

4. Model formulation

Let,

m

i

plex projects with fuzzy

be the index for item i
t be the index for time unit j be the index for stage of the project PA(k) be the set of predecessors of activity k k be the index for activity in project
network
h be the index for activity in set PA(k) Oi be the crisp ordering time unit of item i hi be the holding cost per unit earliness
per unit price of item i
ci be the shortage cost per unit lateness
per unit price of item i
Hit be the equivalent crisp holding cost of
item i if it is ordered in time unit t
Cit be the equivalent crisp shortage cost of
Tit be the equivalent crisp total cost of item
i if it is ordered in time unit t
Pi be the price of item i Sj be the set of time units for jth stage Bj be the budget for stage j TB be the total budget available for the
project

ð~dk ¼ ðdok; d

mk ; d

pkÞ
be the fuzzy duration of activity k
~Ri ¼ ðRoi ;R

mi ;R

pi Þ
be the fuzzy requirement time period of
item i
~Li ¼ ðLo
i ; Lmi ; L

pi Þ
be the fuzzy lead time of item i
~Oi ¼ ðOoi ;O

mi ;O

pi Þ
be the fuzzy ordering time period of
item i
~Ai ¼ ðAo
i ;Ami ;A

pi Þ
be the fuzzy arrival time period of item i
~dLit ¼ ðdoLit; d

mLit; d

pLitÞ
be the fuzzy earliness in the arrival of
~dRit ¼ ðdo
Rit ; dmRit ; d

pRitÞ
be the fuzzy lateness in the arrival of
activity durations and lead times. Computers &


Fig. 1.


Please cite this article in pIndustrial Engineering (201

~Hit ¼ ðHo
it ;Hmit ;H

pitÞ
be the fuzzy holding cost of item i if it is
ordered in time unit t
~Cit ¼ ðCo
it ;Cmit ;C

pitÞ
be the fuzzy shortage cost of item i if it
is ordered in time unit t
~ESk ¼ ðESo
k; ESmk ; ESp

kÞ
be fuzzy early start of activity k
�
be the symbol for fuzzy addition asdefined below
~P1 � ~P2 ¼ ððPo1 þ Po

2Þ; ðPm1 þ Pm

2 Þ; ðPp1 þ Pp

2ÞÞ
� be the symbol for fuzzy subtraction as
defined below
~P1 � ~P2 ¼ ððPo
1 � Pp2Þ; ðP

m1 � Pm

2 Þ; ðPp1 � Po

2ÞÞ

Assumptions:

1. We model uncertain/imprecise activity durations and leadtimes of items as L–R Triangular Fuzzy Numbers. As explainedand justified in Section 3.

2. We use Carlsson and Fukker (2001) as the method to arrive atthe equivalent crisp possibilistic mean of fuzzy costs. Asexplained and justified in Section 3.

3. We assume that holding cost coefficient remains constantthroughout the time periods.

4. Total holding cost varies linearly with lateness (in terms of timeunits).

5. We assume that shortage cost coefficient remains constantthroughout the time periods.

6. Total shortage cost varies linearly with lateness (in terms oftime units).

7. However, assumptions 3,4,5,6 do not change the methodologyand even the case of varying holding/shortage cost coefficient,linear/non-linear cost functions can easily be incorporated.Because holding/shortage costs corresponding to each discreettime unit are calculated separately.

4.1. Fuzzy costs

The fuzzy requirement time of an item is given by the early starttime of the activity for which it is required. Chen and Huang (2007)proposed a FPERT (Fuzzy PERT) method where the duration of eachactivity in the project network is characterized as a L–R TFN. The for-ward pass yields the fuzzy early-start as ~ESk ¼maxh2PAðkÞð ~ESh � ~dhÞ.Thus, fuzzy requirement time ~Ri of an item i required in activity kis equal to fuzzy earliest start ~ESk.

Suppose order of item i is placed on a crisp time unit Oi = t. Thenits fuzzy arrival time is given by~Ai ¼ Oi � ~Li

If fuzzy requirement time is greater than fuzzy arrival time,~Ri > ~Ai i:e: ~Ri > ðOi � ~LiÞ, then fuzzy earliness in arrival of the itemis given by:~dLit ¼ ~Ri � ðOi � ~LiÞ

~dLit ¼ ð~Ri � ~LiÞ � Oi

Fuzzy number representation.

ress as: Dixit, V., et al. Procurement scheduling for co4), http://dx.doi.org/10.1016/j.cie.2013.12.009

Let us term ð~Ri � ~LiÞ ¼ ~Oi as fuzzy order time. Then,

~dLit ¼ ~Oi � Oi

Therefore, ~dLit is defined as:

~dLit ¼maxf0; ð~Oi � OiÞg

Since, ~dLit corresponding to each Oi is a fuzzy number, the resul-tant inventory holding cost ~Hit ¼ hi � ~dLit � Pi will also be a fuzzynumber with optimistic, most likely and pessimistic values~Hit ¼ ðHito;Hitm;HitpÞ and its membership function defined as:

lðHitÞ ¼ðHit � Ho

itÞ=ðHmit � Ho

itÞ Hoit � Hit � Hm

it

ðHpit � HitÞ=ðHp

it � Hmit Þ Hm

it � Hit � Hpit

0 otherwise

8><>:

Using Carlsson and Fukker (2001) the equivalent crisp possibi-listic mean holding cost is given by:

Hit ¼ ðHoit þ 4 � Hm

it þ HpitÞ=6

If fuzzy requirement time is lesser than fuzzy arrival time, then~Ri < ~Ai i:e: ~Ri < ðOi � ~LiÞ Thus, fuzzy lateness in arrival of the itemcan be expressed as:

~dRit ¼ ðOi � ~LiÞ � ~Ri

~dRit ¼ Oi � ð~Ri � ~LiÞ

~dRit ¼ Oi � ~Oi

~dRit ¼maxf0; ðOi � ~OiÞg

Since, ~dRit corresponding to each Oi is a fuzzy number, the resultantinventory shortage cost ~Cit ¼ ci � ~dRit � Pi will also be a fuzzy numberwith optimistic, most likely and pessimistic values

~Cit ¼ ðCoit; Cm

it ; CpitÞ and its membership function can be defined

as:

lðCitÞ ¼ðCit � Co

itÞ=ðCmit � Co

itÞ Coit � Cit � Cm

it

ðCpit � CitÞ=ðCp

it � Cmit Þ Cm

it � Cit � Cpit

0 otherwise

8><>:

Using Carlsson and Fukker (2001) the equivalent crisp possibi-listic mean shortage cost is given by

Cit ¼ ðCoit þ 4 � Cm

it þ CpitÞ=6

The total cost corresponding to time unit t for item i is given bysum of holding and shortage costs.

Tit ¼ ðHoit þ 4 � Hm

it þ HpitÞ=6þ ðCo

it þ 4 � Cmit þ Cp

itÞ=6

As an illustration, consider an item with fuzzy requirement time~Ri ¼ ð7:5;10;10:5Þ and lead time ~Li ¼ ð1:5;3;3:5Þ then fuzzy ordertime period is given by ~Oi ¼ ~Ri � ~Li ¼ ð7:5;10;10:5Þ � ð1:5;3;3:5Þ¼ ð4;7;9Þ (Refer Fig. 2). Holding costs and shortage costs calcula-tions corresponding to each possible crisp Oi are explained below.

If Oi lies on the left side of fuzzy number range withmembership value equal to 0 i.e. crisp Oi is strictly less than fuzzy

Fig. 2. Fuzzy order date representation.



Fig. 3.1. Crisp equivalent costs curves.

Fig. 3.2. Total cost most likely curve.


~Oi ¼ ð~Ri � ~LiÞ. Then for any scenario of requirement time or leadtime the arrival of item will always be early and we will incur onlyholding cost. For example, for crisp Oi = 2.

~dLiðt¼2Þ ¼ ðð4� 2Þ; ð7� 2Þ; ð9� 2ÞÞ

hi � ~dLiðt¼2Þ � Pi ¼ 0:5 � ð2;5;7Þ � Pi

~Riðt¼2Þ ¼ ð1;2:5;3:5Þ � Pi

~Riðt¼2Þ ¼ ð0;0;0Þ

If Oi lies on right side of fuzzy number range with membershipvalue equal to 0 i.e. crisp Oi is strictly greater than fuzzy ~Oi. Thenfor any scenario of requirement or lead time, the arrival of itemwill always be late and we will incur only shortage cost. For exam-ple, let us consider crisp Oi = 12.

~dRiðt¼12Þ ¼ ðð12� 9Þ; ð12� 7Þ; ð12� 4ÞÞ

ci � ~dRiðt¼12Þ � Pi ¼ 0:6 � ð3;5;8Þ � Pi

~Ciðt¼12Þ ¼ ð1:8;3;4:8Þ � Pi

~Hiðt¼12Þ ¼ ð0;0;0Þ

If Oi lies within the range of fuzzy number with membershipvalue greater than 0 then both holding and shortage cost will be in-curred. For example, let us consider crisp Oi = 6. For most likely andpessimistic scenarios the item will be ordered early and holdingcost will be incurred. However, for optimistic scenario the itemwill be ordered late and shortage cost will be incurred.

~dLiðt¼6Þ ¼ ð0; ð7� 6Þ; ð9� 6ÞÞ

hi � ~dLiðt¼6Þ � Pi ¼ 0:5 � ð0;1;3Þ � Pi

~Hiðt¼6Þ ¼ ð0;0:5;1:5Þ � Pi

~dRiðt¼6Þ ¼ ð0;0; ð6� 4ÞÞ � Pi

Ci � ~dRiðt¼6Þ � Pi ¼ 0:6 � ð0;0;2Þ � Pi

~Ciðt¼6Þ ¼ ð0;0;1:2Þ � Pi

If Oi = 8 then for pessimistic scenario, the item will be procuredearly and holding cost will be incurred. However, for optimisticand most likely scenarios the item will be late and shortage costwill be incurred.~dLiðt¼8Þ ¼ ð0;0; ð9� 8ÞÞ

hi � ~dLiðt¼8Þ � Pi ¼ 0:5 � ð0;0;1Þ � Pi

~Hiðt¼8Þ ¼ ð0;0;0:5Þ � Pi

~dRiðt¼8Þ ¼ ð0; ð8� 7Þ; ð8� 4ÞÞ

Ci � ~dRiðt¼8Þ � Pi ¼ 0:6 � ð0;1;4Þ � Pi

~Ciðt¼8Þ ¼ ð0;0:6;2:4Þ � Pi

The crisp equivalents of holding, shortage and total costs for theitem for all possible crisp order time units are plotted in a graph asshown in Fig. 3.1 below. The total cost curve is convex with itsminimum value falling in the range of fuzzy order time period.


Ideally, the minimum value would have been zero in case require-ment and lead times were known precisely as shown in Fig. 3.2 formost likely value. However, due to impreciseness in information, apenalty equal to the minimum value is incurred, as shown inFig. 3.1. This sensitizes managers to take steps to reduce variabilityof activity durations at their end and lead times of items at sup-plier’s end. If all items are ordered on time units correspondingto their respective minima without considering any budget con-straint their sum will give the minimum value of penalty incurredin the form of holding and shortage costs due to uncertainty. Thissets a lower bound for the objective function of the model. Withthis understanding, we proceed to develop the model below.

4.2. Procurement scheduling model

We define decision variables xit as,

xit ¼1; if Oi ¼ t i:e: item i is ordered on time unit t

0; otherwise

�

oj the excess cash availability variable of stage j.Our objective given by (1) is to place orders for the items such

that sum of fuzzy inventory holding and shortage costs is mini-mized. This is constrained by budget allocated to each stage bytop management or contract terms for cash inflows. If the bud-get allocated to a particular stage is more than its requirementthe excess cash available can be transferred to subsequent stages.Excess cash availability variables, of previous stage oj�1 and ofthe given stage oj connect cash flows of stages. Thus, budget con-straint of stage j, given by inequality (2), has parameter of stagebudget availability Bj, excess cash availability variables of previous



Table 1Budget allocated.

Stage Budget allocated in million USD

Budget Stage 1 B1 1.9234338Budget Stage 2 B2 0.6117169Budget Stage 3 B3 1.9502339Budget Stage 4 B4 2.67137132Total budget TB 7.15675592

Table 2Budget required.

Stage Budget required in million USD

Budget Stage 1 B1 1.4258531Budget Stage 2 B2 3.685984786Budget Stage 3 B3 2.035389392Budget Stage 4 B4 0.00952864135


stage oj-1 and of the given stage oj. Further, the total expenditure ofthe project should not exceed the total budget available, given byinequality (3). Also, each item should be ordered only once, givenby equality (4).

~Z1 ¼ MinimizeX

i

Xt

ð~HitÞxit þX

i

Xt

ð~CitÞxit

x 2 UðxÞð1Þ

Xi

Pi

Xt2Sj

xit � Bj þ oj�1 � oj 8 j ð2Þ

Xi

Pi

Xt

xit � TB ð3Þ

Pt

xit ¼ 1 8i

xit 2 ½0;1� \ I 8i; tð4Þ

oj � 0 8j ð5Þ

The auxiliary model after converting fuzzy costs coefficientsinto equivalent crisp possibilistic mean values using Carlsson andFukker (2001) is given below.

Z1 ¼ minimizeX

i

Xt

ðHoit þ 4 � Hm

it þ HpitÞ � xit=6

þX

i

Xt

ðCoit þ 4 � Cm

it þ CpitÞ � xit=6

x 2 UðxÞ

ð6Þ

The output of this auxiliary model provides ordering scheduleof items by taking into consideration the uncertainties prevalentduring planning scenario and imposed budget constraints. Theobjective function gives the value of penalty incurred due toimpreciseness in requirement and lead times of items and limitedcash availability in the stages.

5. Procurement scheduling in a shipbuilding project

5.1. Optimization model

The model shown in Section 4 is applied for procurementscheduling of items of a shipbuilding project ‘‘Anchor HandlingTug Supply Vessel’’. The data for the project was collected by visit-ing a leading shipyard. We conducted interviews of project manag-ers, materials managers and referred relevant documents liketechnical specifications of shipbuilding contract and purchase or-ders of items to collect data. The project comprised of 85 activities;interlinked through complex precedence rules. It had requirementof 341 items, with variety of items ranging from simple standard-ized items like gaskets and fasteners obtained from local suppliersto highly engineered customized items like engines and genera-tors, procured from different suppliers across the world. Giventhese interdependencies, crisp accurate data of activities’ durationsand lead times of items during planning are not available. How-ever, managers have past experience using which they can indicateoptimistic, most likely and pessimistic values of the required datawhich can be represented as L–R TFN.

Appendix A shows the data set of items with their total price,shortage cost coefficient, holding cost coefficient, fuzzy require-ment and lead time. Fuzzy order time ~Oi is obtained by fuzzy sub-traction of requirement and lead times. For long projects, orderplacement starts before the project commences therefore; we havenegative time units and it continues till the last stage. In this pro-ject, we have 41 time units (1 time unit = 15 days) for order place-ment ranging from �10 to 30 and four stages: Stage 1 from t = �10to 5, Stage 2 from t = 6 to 15, Stage 3 from t = 16 to 21 and Stage 4


from t = 22 to 30. Where the lowest time unit �10 is equal to theminimum of optimistic value of order time of items and highest30 is equal to maximum of pessimistic value. Thus, the numberof binary decision variables i.e. whether order of item i is placedon time unit t is equal to 341 * 41 = 13981.

According to shipbuilding contract terms, cash inflows takeplace at the key events like at starting of project, keel laying, at firstblock erection, at launching and at delivery. This defines the num-ber of stages, their time periods and budget allocation. Therefore,the model has 4 budget parameters B1, B2, B3, B4 and 3 over cashavailability variables o1, o2, o3. Table 1, shows the budget allocationof the 4 stages.

The procurement scheduling model for the shipbuilding projectis as follows:

Z1 ¼ MinimizeX341

i¼1

X30

t¼�10

ðHoit þ 4 � Hm

it þ HpitÞ � xit=6

þX341

i¼1

X30

t¼�10

ðCoit þ 4 � Cm

it þ CpitÞ � xit=6

UðxÞ :

X341

i¼1

Pi

X5

t¼�10

xit � 1:924� o1

X341

i¼1

Pi

X15

t¼6

xit � 0:612þ o1 � o2

X341

i¼1

Pi

X21

t¼16

xit � 1:95þ o2 � o3

X341

i¼1

Pi

X30

t¼22

xit � 2:672þ o3

X341

i¼1

Pi

X30

t¼�10

xit � 7:158

X30

t¼�10

xit ¼ 1 8i ¼ 1 to 341

xit 2 ½0;1� \ I 8i ¼ 1 to 341; t ¼ �10 to 30



Fig. 4.2. Sensitivity analyses of costs with degree of uncertainty.

Fig. 4.1. Variation of degree of uncertainty of data set.

Fig. 5. Sensitivity analyses of stage budget re



o1; o2; o3 � 0

The optimal objective function value Z1 = 15.066 milion USD givesthe sum of holding and shortage costs incurred due to inherentuncertainty and imposed budget constraints. Appendix A showsthe output order time Oi of items obtained from the results of abovemodel.

As a next step, we remove the budget availability constraints ofall the stages by declaring all Bj’s as variables. Thus, the model isfree to optimally determine the budget to be assigned to eachstage. Being free from budget constraints, the model now mini-mizes the total inventory holding and shortage costs by orderingeach item on the time unit corresponding to its total cost minimaas shown in Fig. 3.1 Following Table 2 shows the optimal budgetrequirement for each stage:

The objective function value Z2 = 9.24 million USD gives themagnitude of the penalty incurred solely due to uncertainty ininformation during planning. It can be seen that it is 9.24 ⁄ 100/15.066 = 61% of the total cost of budget constraint scenario solvedabove. Thus, penalty incurred due to uncertainty forms a signifi-cant portion of total cost and managers should take steps to reduceit by exercising greater control on activity durations during projectexecution and frequent follow ups with suppliers to reduce vari-ability in lead times of items. This finding also has significantimplications for how a project’s financing needs are determined.The current practice of allocating budgets at each stage withoutconsidering the uncertainties significantly increases costs. Our re-sults suggest that companies manufacturing complex productsshould determine the budget allocation in each satge by consider-ing the uncertainties in requirement and lead times of items duringplanning.

quirements with degree of uncertainty.



Fig. 7.1. Sensitivity analyses of total cost with stage budget constraints.


5.2. Sensitivity analyses

To generate further insights, two types of sensitivity analyseswere performed: first was to observe the effect of variation of de-gree of uncertainty of the data set on total cost and on stage budgetrequirement variables and second was to observe the effect ofchanges in allocated stage budget parameters on total cost. Forthe first type of sensitivity analysis, total z1 = 0 to 60 independentinstances of the problem were solved. We start with the first prob-lem instance z1 = 0 having zero degree of uncertainty i.e. fuzzy or-der optimistic and pessimistic values are equal to most likely value(Om). And then the degree of uncertainty of data set was increasedin each subsequent instance of problem. This was achieved byshifting optimistic values towards left by an offset equal to0.25 * Om and pessimistic values towards right by 0.0045 * Om.Therefore, for each item i in each instance of problem z1 we haveOo

i ðz1Þ ¼ Omi � 0:025 � z1 � Om

i and Opi ðz1Þ ¼ Om

i þ 0:0045 � z1 � Omi

(Refer Fig. 4.1). The magnitude of left and right hand offsets andnumber of instances to be performed were decided such that thefuzzy order time period range covered by any item in last instancefalls within the range of �10 to 30. Fig. 4.2 shows the results,objective function value of total cost increased from zero to highvalues with increase in degree of uncertainty of the data set. Itclearly depicts the severity of uncertainty in inventory costs, evenwithout any budget constraints.

Uncertainty affects the budget requirement variables of stagesas well. Fig. 5 shows the fluctuations with degree of uncertainty.Stage 1 budget requirement increased from instances z1 = 21 to31. This increase is because the number of items, having their fuzzyorder time period ranges lying partially or fully inside Stage 1,

Fig. 6.2. Stage budgets with degree of uncertainty.

Fig. 7.2. Holding cost with stage budget constraints.

Fig. 6.1. Number of items with degree of uncertainty.


increases from instances z1 = 21 to 31 as shown in Fig. 6.1. Lefthand offset in each instance pulls the fuzzy triangle towards lowervalues inside Stage 1. However, after instance no. z1 = 31 it be-comes stable as there are no more items left with their most likelyorder values low enough to get inside Stage 1 even till the last in-stance with maximum degree of uncertainty. Stage 2 budget fluc-tuations are again the reflection of number of items in Stage 2which rise till instance no. z1 = 16. However, due to subsequent lefthand offsets, the items whose fuzzy order time period range wasup to Stage 2, now begin to reach Stage 1 and thus, number of

Fig. 7.3. Shortage cost with stage budget constraints.




items with optimal order date in Stage 2 decreases, causing de-crease in budget requirement.

Stage 3 budget and corresponding number of items also showsimilar pattern. Stage 4 budget requirement pattern is mostly stablebecause most of the shifting of items between instances occursacross in Stages 1, 2 and 3. The optimal ordering scheme in each in-stance hardly delays any item to an extent that its optimal orderingpoint shifts to Stage 4. Only the items having their fuzzy order time

Fig. 8. Summary of model formula


period range entirely inside Stage 4 account for its budget require-ment. From above observations, it can be concluded that uncertaintysignificantly affects budget allocation decisions and makes procure-ment planning more challenging (see Fig. 5 and Fig. 6.2).

Second set of sensitivity analyses were performed on the samedata set of the shipbuilding project, as shown in Appendix A. Westarted with initial values of stage budget parameters equal to thatcalculated by unconstraint model i.e. equal to the values shown in

tion and sensitivity analyses.




Table 2. We divide the sensitivity analyses into four sets each com-prising of 150 independent instances of problem to be solved. Thestage budget parameters were changed across instances in a sys-tematic way within each set as shown below.

Set 1: Initial stages budgets decreased and that of later stagesincreased. Stage 1, 2 budgets decrease and Stage 3, 4 budgetsincrease in each instance for z2 = 0 to 150.

Stage 1: B1 = 1.426–0.0006 ⁄ z2

Stage 2: B2 = 3.686–0.0004 ⁄ z2

Stage 3: B3 = 2.036–0.0006 ⁄ z2

Stage 4: B4 = 0.010 + 0.0004 ⁄ z2

Set 2: Periphery stages budgets increased and that of middlestages decreased. Stage1, 4 budgets increase and Stage 2, 3decrease in each instance for z2 = 0 to 150.

Stage 1: B1 = 1.426 + 0.0004 ⁄ z2

Stage 2: B2 = 3.686–0.0004 ⁄ z2

Stage 3: B3 = 2.036–0.0006 ⁄ z2

Stage 4: B4 = 0.010 + 0.0006 ⁄ z2

Set 3: Initial stages budgets increased and that of later stagesdecreased. Stage 1, 2 budgets increase and Stage 3 budget decreasein each instance for z2 = 0 to 150.

Stage 1: B1 = 1.4258531 + 0.0002 ⁄ z2

Stage 2: B2 = 3.685984786 + 0.0004 ⁄ z2

Stage 3: B3 = 2.035389392–0.0006 ⁄ z2

Stage 4: B4 = 0.00952864135

Set 4: Stage1 budget increase and Stage 2 decrease in eachinstance for z2 = 0 to 150.

Stage 1: B1 = 1.426 + 0.0004 ⁄ z2

Stage 2: B2 = 3.686–0.0004 ⁄ z2

Stage 3: B3 = 2.036Stage 4: B4 = 0.010

Figs. 7.1–7.3 show the plot of total, holding and shortage costsin 150 instances of Sets 1, 2, 3, 4. Sets 1 and 4 show spike in totalcost while Set 2 shows incremental increase and Set 3 is constant.In Sets 1, 2 and 4, Stage 2 budget is decreased in each instance,which increases the shortage cost as shown in Fig. 7.3 and the ef-fect is more severe if it is combined with decrease in Stage 1 bud-get as well (as shown by the plot of Set 1). The spike in total cost inSets 1 & 2 is because the costliest item ‘‘Main Engine’’ with highestshortage cost was shifted from Stage 1 to 2 leading to higher short-age cost. If we closely analyze Figs. 5.2 and 6.2 it is evident thatStage 2 budget is most sensitive to uncertainty and its magnitudeis always much higher than other stage budgets. All these observa-tions reveal that Stage 2 is the most important stage of the projectand during budget allocation decisions, reduction in budget ofStage 2 should be avoided. If managers have to make adjustmentsin stage budget allocations the best pattern to be followed is givenby Set 3 where the total cost remains constant, equal to the mini-mum value. If this cannot be achieved, changes similar to Set 2 canbe accepted. However, if managers are forced to decrease budget ininitial stages, the acceptable range as shown in sensitivity analysisin Set 1 is from point 1 to 31. Thus, Stage 1 budget can be variedbetween 1.24 and 1.426 million USD and Stage 2 budget can varybetween 3.562 and 3.686 million USD.

Fig. 8 shown below summarizes the model formulation andsensitivity analyses.

Caron and Marchet (1998) and Yeo and Ning (2006) have broad-ened the concept of the insertion of appropriate buffers between


construction and procurement activities based on prevalent uncer-tainties. However, they have addressed the procurement schedul-ing problem at an aggregate planning level and are more suitedfor broad level planning. They did not drill down to the executionlevel to finally decide the ordering dates for every item based on itsindividual characteristics. Our work overcomes this gap. The aboveproposed model obtains an optimal order date for every individualitem, based on the uncertainty in its requirement date and leadtime and relevant budget constraints. It also performs two typesof quantitative sensitivity analyses: first to understand the effectof variation of degree of uncertainty on total cost and on budgetrequirement of stages and the second to study the effect of changesin allocated stage budget parameters on total cost. These sensitiv-ity analyses can help practicing managers to identify the mostsensitive stage of the project and determine the ranges in whichstage-wise budgets can be varied.

6. Conclusion and limitations

Material procurement scheduling is a vital aspect of projectmanagement, particularly when uncertainty is involved duringplanning period. In this work, we capture uncertain/ impreciseinformation available with practicing managers about activitydurations and lead times of items using fuzzy numbers. Due tothese uncertainties the inventory holding and shortage costs corre-sponding to a particular order time unit cannot be crisply definedand thus, they are also computed as fuzzy numbers. The procure-ment model developed minimizes the sum of fuzzy inventoryholding and shortage costs and provides optimal order time unitsof items under the uncertainties and budget constraints prevalentduring the planning stage.

Due to the uncertainties involved and the associated complexi-ties, material procurement in manufacturing of complex productslike ships are not aligned with the PWBS based scheduling fol-lowed in production. Results of our model show that companieswill also be better off, if they determine the stage-wise budget allo-cations based on the model output considering the associateduncertainties rather than determining the budget upfront. Sensi-tivity analyses also reveal that degree of uncertainty in data setgreatly impacts the total cost and stage budget allocation deci-sions. If the budget requirement at stages cannot be met, the sen-sitivity analyses of our results indicate the ranges by which thebudget of certain stages can be varied. For the shipbuilding pro-ject’s budget allocation decisions, Stage 2 was identified to be themost important and sensitive stage.

Our research has certain limitations. To solve fuzzy optimiza-tion problem the fuzzy cost coefficients are converted into equiva-lent crisp numbers using fuzzy number ranking methods like Yager(1978), Yager (1981), Chang (1981), Goetschel and Voxman (1986),Liou and Wang (1992), Herrera and Verdegay (1995). However,this approach has two shortcomings. First, it converts fuzzy costcoefficients into equivalent crisp numbers at the very first step it-self; which leads to loss of information available in the form of fuz-zy number. Second, the value of equivalent crisp number dependson the method applied. In this study, we have used method givenby Carlsson and Fukker (2001). However, the output of the modelcan vary with the fuzzy ranking method being followed. But theadvantages of this approach are its computational simplicity andease in performing sensitivity analyses. Lai and Hwang (1992),Wang and Fang (2001), have provided multiple approaches whichconserve the fuzzy number information till the last stage of optimi-zation. They retain the membership functions of fuzzy cost coeffi-cients and shift the whole function towards right or left dependingon maximization or minimization problem. But these approachesinvolve multiple crisp optimization problems to be solved before




final solution of the fuzzy problem is obtained. During our compu-tational experience, we found them to be very sensitive to the dataset. These computational complexities hinder rigorous sensitivityanalyses and make these approaches incompatible for usage in a

Appendix A

Note: Due to large size of data set only few items are listed below.

Item Shortagecostcoefficient

Holdingcostcoefficient

Price(MillionUSD)

Fuztim

ci hi Pi Ro

Mid ship Steel1 0.8 0.5 0.015329 12Mid ship bar1 0.7 0.2 0.003716 12Mid ship Angle1 0.6 0.3 0.005249 12Boss aft 0.7 0.7 0.019071 15Aft ship Steel1 0.7 0.5 0.047691 15Aft ship bar1 0.7 0.2 0.011561 15Aft ship Angle1 0.6 0.3 0.016329 15Aft ship Steel6 0.7 0.5 0.020439 20Aft ship bar6 0.5 0.2 0.003716 20Aft ship Angle6 0.5 0.3 0.005249 20Boss fwd 0.7 0.7 0.019071 17Fore ship Steel5 0.8 0.5 0.015329 19Fore ship bar5 0.7 0.2 0.003716 19Fore ship Angle5 0.6 0.3 0.005249 19Fore ship Steel6 0.8 0.5 0.015329 19Mast Steel 0.7 0.5 0.006813 23Mast bar 0.6 0.2 0.001652 23Mast Angle 0.6 0.3 0.002333 23Main Engine 0.6 0.8 1.238738 14Main Generating Set 0.6 0.7 0.187115 15Oil Water Separator 0.4 0.4 0.0119 16FO Pumps 0.3 0.5 0.012681 16Rotary gear pump 0.2 0.5 0.000157 16Fuel Oil Purifier 0.4 0.3 0.01428 16FO Heating system 0.2 0.3 0.00126 16FO Heat exchanger 0.2 0.3 0.0338 16Sea water pumps 0.5 0.5 0.012 16Fresh Water pumps 0.4 0.5 0.008454 16LO Pumps 0.3 0.5 0.010567 16LO Purifiers 0.5 0.3 0.01428 16Lube Oil Heating 0.2 0.3 0.00126 16Air Compressor 0.5 0.6 0.0095 16Air Receiver Tank 0.4 0.6 0.003364 16Compressors 0.3 0.5 0.123814 16Control Gear Box 0.6 0.6 0.004413 21Shaft Alternator 0.7 0.7 0.00545 21Stern tube 0.7 0.7 0.00545 21Shaft 0.6 0.8 0.002425 21Propeller System 0.6 0.8 0.51884 21Steering Gear 0.6 0.8 0.0555 21Flat Type Rudder 0.7 0.7 0.133824 21Dovetail seal 0.5 0.5 7.16E�05 21Tunnel Thruster 0.7 0.8 0.31027 20Deck Machinery 0.6 0.8 0.65 20Cement System 0.6 0.8 0.145 20Hatch Cover 0.4 0.6 0.00198 20Foundation Steel 0.6 0.5 0.002725 15Foundation bars 0.5 0.3 0.000661 15Foundation Angles 0.5 0.2 0.000933 15FD Accomm Panel 0.7 0.7 0.05425 21


real industrial problem. As a future scope of work, different fuzzyranking and fuzzy optimization methods can be used to comparethe outputs and generate further insights regarding applicabilityof the methods for different classes of problems.

Full data can be provided on request.

zy requiremente

Fuzzy leadtime

Fuzzy ordertime

Outputordertime

Rm Rp Lo Lm Lp Oo Om Op Oi

12 12 2 3 10 2 9 10 912 12 1 2 6 6 10 11 1012 12 1 2 6 6 10 11 1015 17 4 6 12 3 9 13 915 17 2 3 10 5 12 15 1215 17 1 2 6 9 13 16 1315 17 1 2 6 9 13 16 1621 26 2 3 10 10 18 24 2221 26 1 2 6 14 19 25 2221 26 1 2 6 14 19 25 2217 20 4 6 12 5 11 16 1120 24 2 3 10 9 17 22 1720 24 1 2 6 13 18 23 2220 24 1 2 6 13 18 23 2220 25 2 3 10 9 17 23 1724 30 2 3 10 13 21 28 2224 30 1 2 6 17 22 29 2224 30 1 2 6 17 22 29 2214 15 10 12 24 �10 2 5 216 17 10 12 24 �9 4 7 417 19 4 6 12 4 11 15 1617 19 4 6 12 4 11 15 2217 19 4 6 12 4 11 15 2217 19 6 10 16 0 7 13 717 19 4 6 12 4 11 15 2217 19 4 6 12 4 11 15 2217 19 4 6 12 4 11 15 1117 19 4 6 12 4 11 15 1617 19 4 6 12 4 11 15 2217 19 6 10 16 0 7 13 717 19 4 6 12 4 11 15 2217 19 4 8 12 4 9 15 917 19 4 8 12 4 9 15 917 19 2 4 6 10 13 17 2222 28 10 12 18 3 10 18 1022 28 10 12 18 3 10 18 1022 28 10 12 18 3 10 18 1022 28 10 12 18 3 10 18 1023 29 10 12 18 3 11 19 1123 29 10 12 18 3 11 19 1123 29 10 12 18 3 11 19 1123 29 6 8 12 9 15 23 1621 27 6 8 12 8 13 21 1621 27 4 6 10 10 15 23 2221 27 4 6 12 8 15 23 2221 27 3 4 8 12 17 24 2216 17 2 3 10 5 13 15 1616 17 1 2 6 9 14 16 1616 17 1 2 6 9 14 16 1622 27 5 6 10 11 16 22 16

(continued on next page)



Appendix A (continued)

Item Shortagecostcoefficient

Holdingcostcoefficient

Price(MillionUSD)

Fuzzy requirementtime

Fuzzy leadtime

Fuzzy ordertime

Outputordertime

ci hi Pi Ro Rm Rp Lo Lm Lp Oo Om Op Oi

FD Furniture 0.5 0.4 0.001094 21 22 27 5 6 10 11 16 22 16FD Windows 0.5 0.2 0.015511 21 22 27 5 6 10 11 16 22 16FD Laundry 0.4 0.4 0.003355 21 22 27 5 6 10 11 16 22 22UFD Accomm Panel 0.6 0.5 0.05425 22 23 29 5 6 10 12 17 24 17Cable trays1 0.6 0.1 0.01737 15 16 17 4 5 8 7 11 13 7Cable hangers1 0.5 0.1 0.020262 15 16 17 4 5 8 7 11 13 7Cable trays2 0.4 0.1 0.01737 18 19 21 4 5 8 10 14 17 16Cable hangers2 0.3 0.1 0.020262 18 19 21 4 5 8 10 14 17 22Electrical Equipment 0.7 0.7 0.12623 22 23 26 2 4 8 14 19 24 22Alarm monitoring 0.6 0.6 0.033 22 23 26 2 4 8 14 19 24 22Navigational equip 0.6 0.6 0.09416 22 23 26 5 6 10 12 17 21 17Navigational lights 0.5 0.2 0.008855 22 23 26 1 2 4 18 21 25 22Navigational displays 0.5 0.2 0.001809 22 23 26 2 4 8 14 19 24 22Communication Sys. 0.7 0.6 0.000997 22 23 26 4 6 10 12 17 22 17DP System 0.8 0.7 0.199 25 26 30 4 6 10 15 20 26 22Penetration sleeves 0.7 0.2 0.002063 17 18 20 4 5 8 9 13 16 13Gland packing 0.6 0.2 0.008973 17 18 20 4 5 8 9 13 16 16FO Pipe 0.6 0.6 0.026392 22 23 26 6 8 12 10 15 20 16FO Pipe fittings 0.4 0.4 0.033833 22 23 26 3 4 6 16 19 23 22Paint for Outer Hull 0.7 0.6 0.031262 23 25 31 2 4 8 15 21 29 22Thinner Outer Hull 0.7 0.6 0.001194 23 25 31 2 4 8 15 21 29 22Anodes 0.6 0.6 0.003058 23 25 31 4 6 10 13 19 27 22Paint Superstructure 0.7 0.6 0.026052 23 24 30 2 4 8 15 20 28 22


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