Post design modeling for cellular manufacturing system with cost uncertainty

*Corresponding author. Tel.: (011) 6861733, 6861977; fax: 9111 6862037; e-mail: [email protected].

Int. J. Production Economics 55 (1998) 97—109

Post design modeling for cellular manufacturing systemwith cost uncertainty

Ravi Shanker, Prem Vrat*

Mechanical Engineering Department, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India

Received 9 June 1997; accepted 19 February 1998

Abstract

Post-design decisions in a cellular manufacturing system are modeled in this paper as an interval programming modelin which the coefficients of the objective function are expressed in range rather than a point estimate. Uncertainties inestimations for costs related to exceptional elements and bottleneck machines have been modeled. The interval objectivefunction involves order relations, which represents the decision makers’ preference between interval costs. The objectivefunction is minimized by converting it into a multi-objective problem using order relations. A quantitative estimate of‘risk by gain ratio’ is proposed to understand the model behavior and to facilitate the selection of appropriatestrategies. ( 1998 Elsevier Science B.V. All rights reserved.

Keywords: Cellular manufacturing system; Interval analysis; Multiple objective programming

1. Introduction

Cellular manufacturing system (CMS) is a manu-facturing system in which parts are grouped intopart families (PF) and machines are grouped intomachine cells (MC). Similarity in part design ormachining operation is used to facilitate the clus-tering. For each part family, one dedicated machinecell is conceived so that all the processing require-ments of the part family are completed in the corre-sponding machine cell. In practice, however, someparts are allowed to be processed in two or morecells. Similarly, some machines are required by two

or more part families. These parts and machines arereferred as exceptional elements (EE) and bottle-neck machines (BM), respectively. To generate theeffective grouping of parts and machines into PFand MC, all possible EE or BM may be temporar-ily removed at the clustering stage. In the post-design modeling of CMS, we evolve the strategiesto deal with them. When an EE is allowed to haveintercell transfer, there would be no need to duplic-ate the BM responsible for this intercell transfer,provided machine capacity is adequate. Otherwise,either the EE has to be subcontracted or corre-sponding BM has to be duplicated [1]. There arethree more options for eliminating the EE: (i) re-route the EE, (ii) re-design the EE and (iii) forma remainder cell having an independent cluster ofEE [2]. A survey of CMS industries in US suggests

0925-5273/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reservedPII S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 0 4 3 - 7

Table 1Reasons for considering cost uncertainty in post design CMS modeling

SN Cost element Reasons for considering uncertainty in cost estimate

Internal factor External factor

1 Acquisition cost ofbottleneck machine

1. Time gap between design, implementationand operation

2. Change in management perception towardmake/capacity of the machine

3. In-house R&D breakthroughs4. Impact of cost reduction due to better

scheduling after machine duplication5. Insufficient market survey6. Possibilities of getting discount, if duplicate

machines are ordered7. Installation cost not known, as layout is

not finalised

1. Change in government policies2. Unwillingness of suppliers to quote prices

without orders in near future3. Inflation4. Change in import restrictions5. Change in interest rate for capital acquired

to purchase machines6. High cost in acquiring data with precision7. Arrival of new technology8. Obsolescence of software and hardware by

the time design is implemented9. Global exchange of technology

10. Possibility of joint ventures and transferof technology from partner firms

2 Intercell transfercost

1. Material handling equipments—yet undecided2. Likely changes in plant layout (traveling

distance between each cells — yet notfinalised)

3. Number of part type traveling between cells— yet not finalised

4. Production volume and number of timeeach part travels — yet not finalised

5. Location and size of stores — yet notfinalised

6. Undecided process plan and sequence ofoperations

1. Inflation2. Lack of statistical observations

3 Subcontracting costof exceptionalelements

1. Time gap between design, implementationand operation

2. Issues related to transfer of technology tothe subcontractor — yet not finalised

1. Market uncertainty due to competition2. Unwillingness of subcontractor to quote

prices without possibility of orders innear future

3. Inflation4. Subcontractor finalised at this stage fails

to cope up with the requirement

that 20% of companies with manned cells and 14%with unmanned cells had machines shared by twoor more cells. The survey also concludes that it israre to find completely independent cells [3]. Inanother survey Pullen [4] reported that only 36%cells were without an EE. Therefore, the strategyselection problem in CMS is an important practicalissue with many interrelated factors involving EEand BM. The crucial aspect however, is their costestimation, in terms of the acquisition cost (AC) of

BM, intercell transfer cost (IC) of EE and subcon-tracting cost (SC). It is observed that in real-lifesituations, none of these costs can be estimatedaccurately when the cells are designed and strat-egies for dealing EE and BM are decided [5]. Thisis mainly because of four reasons: (i) long-time gapsbetween design, implementation and operations, (ii)high cost in acquiring these figures with precision,(iii) lack of statistical observations and (iv) uncer-tainty in external factors (Table 1). Generally,

98 R. Shanker, P. Vrat/Int. J. Production Economics 55 (1998) 97—109

decision makers tend to give the cost figuresin a range estimate due to the dynamics of marketforces involving competitors, subcontractors,collaborators, financial institutions, governmentagencies, etc. Some typical examples of range esti-mates are: ‘unit subcontracting cost of part 3 wouldbe between $7.5 and $2.5’, ‘acquisition cost of ma-chine 2 would be in the range of $60 000—70 000’,‘cost of intercell transfer for part 8 would be $2.6per unit with $50% variation’, etc. Exactcost estimates are more difficult particularly whenthe economy is volatile or if there are politicaluncertainties. The point estimate, normally asso-ciated with the deterministic models is thus a speci-fic value in the set of feasible values for thecorresponding range estimate. The chance thata particular value in the corresponding range setwould be valid over the life cycle of a manufactur-ing system is quite remote. It is therefore importantto capture the cost uncertainty and incorporatethese while modeling the problem to decide aboutEE and BM.

This paper deals with the non-deterministic in-formation of range type by constructing an intervalobjective function for cost minimization. This takescare of uncertainty in the cost coefficients of themodel for strategy selection in CMS. In the contextof CMS, this is perhaps the first attempt to capturethis type of uncertainty in a mathematical pro-gramming framework.

2. Literature review

Cell formation is an area of extensive researchduring recent years. A large variety of approacheshave been adopted for this purpose. Productionflow analysis [6] has been used to form machinecells by using routing or operation sequence in-formation. Many clustering algorithms have alsobeen adopted for formatting the cell structure. Ar-ray-based clustering techniques such as rank-orderclustering [7], direct clustering analysis [2], bondenergy analysis [8], etc. use procedures to rear-range rows and columns of part-machine incidencematrix. These hierarchical clustering approachescompute similarity/dissimilarity between eachpair of parts or machines. On the other hand,

non-hierarchical clustering techniques, whichare iterative methods to form cells, have alsobeen used [9]. Mathematical programming ap-proaches such as linear programming [10], integerprogramming [11], goal programming [12], etc.have been adopted for this problem. Comprehens-ive reviews of such works exist in literature[13—15].

Most of these cell formation approaches eithercompletely ignore EE/BM from consideration, ortemporarily remove them from the problem andreinstate them on the conclusion of the process [1].This might be done to keep the problem size withincomputational capacity of software/hardware used.Some approaches to CMS design treat EE/BMseparately, which we call in this paper as the post-design strategy selection problem. Shafer et al. [16]have proposed a mathematical programmingmodel for such a strategy selection problem.Amirahmadi and Choobineh [17] have evolveda two-stage procedure for economically identifyingthe cell composition. In the first stage an ortho-gonal set of EE/BM is identified along with the cellformation while in the second stage strategy to dealwith EE/BM is evolved. An advantage of the mod-els used in these two papers is their small problemsize, as the computational effort in integer pro-grams used in these models depends on thecardinality of exceptional set(s). Since these car-dinalities are generally small, commercial optimiza-tion packages are quite adequate for many real-lifeproblems [17]. Strategy to deal with bottleneckmachine in the presence of budgetary limitation hasbeen considered by Logendran [18] througha two-phase methodology. Seifoddini [19] com-pared cost of BM duplication with cost of materialhandling created by intercellular transfer of the EE,but ignored subcontracting option for the EE. Kernand Wei [1], however, considered subcontractingoption with machine duplication for eliminatingthe EE. For eliminating EE, cost for each avoidedintercellular transfer is calculated and this informa-tion is used in the process of strategy selection.King and Nakornchi [7], McAuley [20] andSeifoddini and Wolfe [21] used BM duplication asa strategy to eliminate intercellular transfer of anEE. However, they ignored the subcontracting op-tion to achieve the same goal. Kumar and Vannelli

R. Shanker, P. Vrat/Int. J. Production Economics 55 (1998) 97—109 99

[22] and Kusiak and Chow [23] compared sub-contracting cost with cost required for duplicationof BM for evaluating EE that remains in the solu-tion of the cell formation problem.

Very little work has been reported to deal withthe uncertainty in situational parameters of CMS.Seifoddini [24] proposed a probabilistic model forselecting the cell configuration which uses differentproduct mixes with associated probabilities to cal-culate the expected inter-cellular material handlingcost. Some simulation-based works also exist inliterature [25,26]. Shanker and Vrat [27,5] haveused chance constrained and fuzzy programmingapproaches to deal with uncertainty.

3. Interval objective function

The objective function of a mathematical pro-gramming problem employed in this paper cap-tures uncertainty by considering its coefficients inrange interval. The form of such an objective func-tion is

min: Z(x)"(A1x1#A

2x2#2#A

nxn:

x3SLRn), (1)

where ‘S’ is the feasible region of x and Aiis the

interval representing the uncertainty in the costinformation for x

i. Thus, the problem becomes that

of the inexact programming type [28]. Lowercaseand uppercase letters are used in this section torepresent the real-number and closed-interval, re-spectively. As proposed by Falk [29], the conserva-tive strategy to get optimal solution against theworst case scenario would be a mini—max case orthe right limit of Z(x) as below:

min(maxai|Ai

z(x)"(a1x1#a

2x2#2#a

nxn) :

ai3A

i& x3SLRn). (2)

We have adopted the modified model ofIshibuchi and Tanaka [30], where in addition tothe right limit of interval, we consider the left limit,i.e. minimum of the range, center and width. Forclarity, before reformulating objective equation (1)as a multi objective model, we would review the

interval arithmetic nomenclature, order relationsand their application to the minimization problemin the next three subsections.

3.1. Interval nomenclature

The closed interval as denoted by ‘R’ representsthe set of all real numbers in the range. The leftlimit, right limit, center and width of the intervalA are denoted as a

L, a

R, a

Cand a

W, respectively. The

ordered pair aLand a

Rfor an interval are defined as

A"[aL, a

R]"Ma : a

L)a)a

R: a3RN. (3)

Also,

A"SaC, a

WT

"Ma : (aC!a

W))a)(a

C#a

W) : a3RN. (4)

The center and width of the interval are cal-culated as follows:

aC"0.5(a

R#a

L), (5)

aW"0.5(a

R!a

L). (6)

Alefeld and Herzberger [31] and Ishibuchi andTanaka [30] have discussed interval arithmetic ingreater detail.

3.2. Order relation for minimization problem

The decision makers’ preferences between inter-val costs are represented by order relations(symbolically as )*

LR, )*

CWor )*

RC) between un-

certain costs from two alternatives represented bysets of intervals A and B. Ishibuchi and Tanaka[30] defined the following order relations:

A)*LR

B if and only if aL)b

Land a

R)b

R, (7)

A(*LR

B if and only if A)*LR

B and AOB, (8)

A)*CW

B if and only if aC)b

Cand a

W)b

W,

(9)

A(*CW

B if and only if A)*CW

B and AOB,

(10)


A)*RC

B if and only if aR)b

Rand a

C)b

C,

(11)

A(*RC

B if and only if A)*RC

B and AOB,

(12)

Order relation )*LR

represents the decisionmakers’ preference for alternatives with lower min-imum cost and maximum cost. Order relation)*

CWrepresents the decision makers’ preference for

the alternatives with lower expected cost and lessuncertainty. Order relation )*

RCrepresents the de-

cision makers’ preference for the alternatives withlower maximum cost and lower expected cost. Incase that either ‘A)*

LRB’ or ‘A)*

CWB’ holds good,

A is preferred to B. Following properties of thisorder relation is relevant to the minimization prob-lem:

If A)*CW

B; then aR)b

R. (13)

If aW"b

W

"0; then

)*CW

reduces to an inequality relation

)on a set of real numbers. (14)

3.3. Minimization of the interval objective function

The interval objective function can be refor-mulated into a multi-objective problem with solu-tion set defined by the preference relations betweenintervals. Using order relations defined earlier,which represent decision makers’ preference be-tween interval costs, solution set of (1) is defined asnon-dominated solutions. The following proposi-tions have been proved by Ishibuchi and Tanaka[30]:

A)*RC


B or A)*CW

B,

(15)

A(*RC


B or A(*CW

B.

(16)

The solution set of minimization problem (1)may be defined by x3S if and only if there is no

x@3S, which satisfies Z(x@)(*LR

Z(x) orZ(x@)(*

CWZ(x). Using Eqs. (15) and (16), this solu-

tion set may also be defined by x3S if and only ifthere is no x@3S, which satisfies Z(x@)(*

RCZ(x).

Thus, the solution set of (1) can be obtained as thePareto optimal solution which gives non-inferior,non-dominated solution of the following multi-ob-jective problem:

Min((zR(x), z

C(x): x3SLRn). (17)

For xi*0, the right limit z

R(x) of the Z(x) may be

calculated as

zR(x)"(a

C1x1#2#a

Cnxn)

#(aW1

x1#2#a

Wnxn), (18)

where, aCi

is the center and aWi

is the width ofcoefficient A

iof interval objective function Z(x).

Similarly, the center zC(x) of Z(x) is

zC(x)"(a

C1x1#2#a

Cnxn). (19)

Therefore, with interval coefficients Aiin (1), be-

ing denoted by SaCi, a

WiT, the right limit and center

of Z(x) may be calculated from Eqs. (18) and (19).

4. Risk by gain ratio (a) in the analysis ofuncertainty in the range estimate

To study the impact of uncertainty associatedwith the range estimate on decision variables,a concept of risk-by-gain ratio (a) is defined asfollows:

a"zW

(x)%MzM(x)!z

C(x)N, (20)

where

zW(x)"0.5Mz

R(x)!z

L(x)N. (21)

zM(x) is the maximum of all z

R(x) values or budget

for total cost in the objective function.Numerator in a reflects the degree of spread or

uncertainty in the range estimate. In the limitingcase, when spread of an interval is zero, i.e. we havea point estimate, a would be zero. It is clear fromEq. (21) that it increases with the spread of interval.The denominator in a provides a measure of the


expected savings from a budget provision. We havenotionally taken the maximum of all z

R(x) values as

the budget provision for a cost minimization prob-lem, because the purpose of proposing this factor islimited only to compare various alternatives.A lower value of a indicates a low risk per unit gain.Hence, from a solution set, only that decision ispreferred, which has the lowest value of a.

5. Modeling CMS with the interval objectivefunction

A mathematical programming model is for-mulated for the optimum selection of exceptionalparts to be subcontracted and bottleneck machinesto be duplicated in the CMS environment. Theobjective function minimizes the total cost asso-ciated with subcontracting EE, the intercellulartransfer cost and the acquisition cost of BM. All thecost coefficients in the objective function are ex-pressed as a range of upper and lower limit of theexpected cost. The model presented here is an ex-tension of the deterministic model of Shafer et al.[16], incorporating the range objective function tocapture uncertainty of cost parameters.

5.1. Model formulation

Three strategies to deal with EE and BM areconsidered in this model:

1. if an EE is allowed to have intercell transfer, thenthe BM which is responsible for this is not du-plicated in the cell containing this EE;

2. if an EE is not allowed to have intercell transfer,then the BM which is responsible for this isduplicated in the cell containing this EE, and

3. if neither intercell transfer of EE nor duplicationof BM is possible then the EE is subcontracted.

It is assumed that the BM has adequate capacityand adequate budget is available to implementthese strategies. Re-design and re-routing optionsfor EE are not considered in this model. Indexingsets, variables and parameters used in the modelare as follows:

Indexing sets:

f, i, j index for machine cells, parts and ma-chines, respectively

¸, R, C, ¼ index for lower, upper, center and widthof range estimate

Decision variables:

Xi

units of part i to be subcontracted½

jfnumber of machines of type j to bepurchased for cell f

Zij

number of intercellular transfers re-quired by part i as a result of machinetype j not being available within part’smachine cell

Mij

number of machines of type j dedicatedto production of part i

Parameters:

[ajL

, ajR

] lower and upper range of annual cost(a

j) for acquiring a machine of type j

[siL, s

iR] lower and upper range of incremental

cost (si) for subcontracting unit part i

[tiL, t

iR] lower and upper range of incremental

cost (ti) for moving part i outside a cell

as opposed to moving it within the cellD

iannual forecasted demand for part i

Cj

annual capacity of machine type jPij

processing time to produce part i ona machine type j

Gf

set of exceptional parts of cell fH

fset of bottleneck machines required byparts of cell f

5.2. The model

A mathematical programming model for optimalselection of strategies in CMS with interval costestimate is formulated as follows:

Objective function:

Min: z(x)"+fC +i|Gf

[siL, s

iR]X

i# +

j|Hf

[ajL

, ajR

]½jf

# +i|Gf

[tiL, t

iR]Z

ijD, (22)


Table 2Incidence matrix and data set of the illustrated problem

s.t.:

Zij#X

i#(C

jM

ij/P

ij)"D

ifor all EE, (23)

+i|Gf

Mij)½

jffor all j, f, (24)

Xi"0 for all ‘i’ that should not be subcontracted,

(25)

Xi, ½

jf, Z

ijare all integers. (26)

Objective function (22) minimizes the costs asso-ciated with EE and BM, such as: subcontractingcost of EE, intercellular transfer cost of EE anddiscounted acquisition cost of BM. It is assumedhere that only these cost informations are vagueand available in the form of interval estimates.

Constraint set (23) puts restriction on each EE sothat their production should conform to the corre-

sponding part demand. This is achieved by satisfy-ing demand of an EE by intercell part processing,subcontracting and intracell part processing. Con-sequently, it evaluates the number of intercellulartransfers between machine type j and part i thatremain in the solution. Constraint set (24) ensuresthat number of machines of type j purchased for cellf are the same as the number assigned to variousEEs found in constraints (23). Constraint set (25)puts restrictions on parts not to be subcontractedgoing into solution. Constraint set (26) takes care ofinteger restrictions.

5.3. Conversion of interval objective function intomulti-objective model

Employing concepts discussed in Section 3, theoriginal range objective function (22) is converted


Tab

le3

Par

eto

optim

alre

sults

obta

ined

by

the

mode

l

Ran

geofes

tim

ate

(in%

of$

cent

erofin

terv

al)

wPar

tssu

b-co

ntr

acte

dX

iO0

Mac

hin

edu

plic

ated

½jf"

1

Inte

r-ce

lltr

ansfer

ZijO

0

Obje

ctiv

efu

nct

ion

[zL(x

),z R(x

)]Sz

C(x),

z 8(x

)T

a"z 8

(x)%

MzM(x

)!z C(x

)N

S ia j

t i(’0

0000

)(’0

0000

)

50%

2%10

%0—

0.07

0½

13,½

22,

Z28"

2759

8[4

.472

,4.8

84]

S4.6

8,0.

21T

0.08

8½

41,½

43,

Z57"

1870

7½

61Z

62"

2276

Z84"

940

0.08

0½

13,½

22,

Z28"

5050

[4.5

27,4

.837

]S4

.68,

0.15

T0.

066

½41

,½

43,

Z57"

1870

7½

61,½

81Z

62"

2276

Z84"

940

0.09

—10

½13

,½

22,

Z28"

2759

8[4

.572

,4.7

99]

S4.6

8,0.

12T

0.04

8½

41,½

43,

Z57"

1870

7½

61,½

72Z

62"

2276

½81

Z84"

940

50%

10%

2%0—

10

½13

,½

22,

Z28"

2759

8[4

.322

,5.0

34]

S4.6

8,0.

35T

0.15

2½

41,½

43,

Z57"

1870

7½

61,½

72Z

62"

1704

0½

81Z

42"

1134

0Z

84"

940

10%

50%

2%0—

0.3

0½

13,½

22,

Z28"

2759

8[3

.01,

6.34

]S4

.68,

1.67

T0.

727

½41

,½

43,

Z57"

1870

7½

61,½

72Z

62"

1704

0½

81Z

42"

1134

0Z

84"

940

0.4—

0.6

X1"

3212

8½

13,½

22,

Z28"

2759

8[3

.744

,5.9

7]S4

.86,

1.1T

0.51

5½

43,

Z57"

1870

7Z

62"

2276

Z84"

940

0.7—

0.8

X1"

3212

8½

13,½

43,

Z28"

2759

8[4

.28,

5.87

]S5

.08,

0.79

T0.

411

Z42"

1134

0Z

57"

1870

7Z

62"

1704

0Z

84"

940

0.9—

1X

1"32

128

½43

Z28"

2759

8[4

.718

,5.8

33]

S5.2

7,0.

56T

0.31

9Z

42"

1134

0Z

57"

1870

7Z

62"

1704

0Z

84"

940


10%

2%50

%0

0½

13,½

22,

Z28"

2759

8[3

.912

,5.4

4]S4

.68,

0.76

T0.

325

½41

,½

43,

Z57"

1870

7½

61Z

62"

2276

Z84"

940

0.1—

0.2

0½

13,½

22,

Z28"

5050

[4.4

7,4.

897]

S4.6

8,0.

21T

0.09

½41

,½

43,

Z62"

2270

½61

,½

72,

Z84"

940

½81

0.3—

1X

6"22

76½

13,½

22,

Z28"

5050

[4.5

1,4.

87]

S4.6

9,0.

18T

0.07

7½

41,½

43,

Z84"

940

½61

,½

72,

½81

2%50

%10

%0—

0.2

0½

13,½

22,

Z28"

2759

8[2

.900

,6.4

57]

S4.6

8,1.

78T

0.76

1½

41,½

43,

Z57"

1870

7½

61Z

62"

2276

Z84"

940

0.3—

0.8

X1"

3212

8½

13,½

22,

Z28"

2759

8[3

.76,

5.96

]S4

.86,

1.09

T0.

509

½43

Z57"

1870

7Z

62"

2276

Z84"

940

0.9—

1.0

X1"

3212

8½

13,½

43Z

28"

2759

8[4

.22,

5.93

]S5

.08,

0.85

T0.

438

Z42"

1134

0Z

57"

1870

7Z

62"

1704

0Z

84"

940

2%10

%50

%0

0½

13,½

22,

Z28"

2759

8[3

.65,

5.70

]S4

.68,

1.03

T0.

44½

41,½

43,

Z57"

1870

7½

61Z

62"

2276

Z84"

940

0.1—

0.2

0½

13,½

22,

Z28"

5050

[4.1

19,5

.25]

S4.6

8,0.

57T

0.24

3½

41,½

43,

Z62"

2276

½61

,½

72,

Z84"

940

½81

0.3—

1X

6"22

76½

13,½

22,

Z28"

5050

[4.1

67,5

.22]

S4.6

9,0.

53T

0.22

7½

41,½

43,

Z84"

940

½61

,½

72,

½81


into a multi-objective mixed integer programming[MOMIP1] problem, as follows:

[MOMIP1]

Min(zR(x), z

C(x)), s.t. (23)—(26). (27)

Using the weighing method, we can obtain thePareto-optimal solution of [MOMIP1] from thefollowing mixed integer linear programming prob-lem [MILP1];

[MILP1]

Min[wzR(x)#(1!w)z

C(x)] s.t. (23)—(26). (28)

The Pareto-optimal solution of [MOMIP1] isobtained from [MILP1] by gradually increasing‘w’ in (28) from 0 to 1 and observing changes indecision variables and objective function at everystage.

6. Illustrative example

To illustrate the model proposed here, anexample with marginal modification in data set isadopted from literature [16]. All costs are ex-pressed as upper and lower value of interval.

For comparing results with the deterministicmodel of Shafer et al. [16], the average of upper andlower limit of each cost interval is maintained ata value which is the same as its deterministiccounterpart. Table 2 gives the part-machine inci-dence matrix along with the central value for vari-ous cost intervals. The entries at machine-partintersection in the incidence matrix represent theprocessing time in minutes. For example, part4 needs 3.89 min on machine 2. Last two columnsare machine acquisition cost and machine capacityin hour per year, respectively. Last three rows inthis table contain the unit subcontracting cost, an-nual demand of parts and unit intercellular transfercost, respectively. The left and right values of thesecost intervals are varied upto $50% around thecentral value. For example, subcontracting cost ofpart 1, s

1is expressed as [7.5, 2.5] or S5, 2.5T for

a $50% variation around the central value 5.

7. Results and discussions

[MILP1] is employed to the problem defined inthe model (22)—(26). For a range of weight factor ‘w’,number of Pareto optimal solutions are obtainedusing LINGO package on a personal computer.These are given in Table 3.

Here, a $2% variation around the central valueis treated as a nearly deterministic cost estimatewhile $50% variation is taken as quite uncertaincost information. Thus, a sequence of (2, 2, 2) wouldrepresent a variation of $2% in all the three costestimates and is nearly deterministic situation,whereas (50, 50, 50) would represent nearly uncer-tain system parameters. In between, various combi-nations of these are considered and results areshown in Table 3. Table 4 shows the results forvarious situations involving same degree of uncer-tainty in all the three parameters.

It can be seen from Table 3 that for a combina-tion of (50, 2, 10), the expected cost minimizationcriterion would prefer solution 1, while a worst-case minimization criterion would prefer solution 3.A risk-by-gain ratio minimization criterion will alsoprefer solution 3.

In case of multiple solutions, the decision makermay go for a solution with lowest risk-by-gain ratio.For example, in Table 3, in case of range (50, 2, 10),out of the three alternatives, the solution at weight1.0 provides the same set of strategies to deal withEE and BM for lowest risk-by-gain ratio, lowestzR(x) and highest z

C(x). Neither of the three solu-

tions in this set satisfies both objectives of worstcase (i.e. z

R(x)) minimization and expected cost (i.e.

zC(x)) minimization. Ishibuchi and Tanaka [31] do

not provide any way out to resolve the decisionmakers’ dilemma at this stage. However, lowestrisk-by-gain ratio has the potential to capture thecombined effect of both objectives. Therefore, un-der this set of cost uncertainty, the prudent strat-egies would be the solution set with minimum a.For this, no part is subcontracted (as all X

iare

zero). Machines 4, 6 and 8 are duplicated in cell 1;machine 2 is duplicated in cell 2 and machines1 and 4 are duplicated in cell 3, respectively, as Y

41,

Y61

, Y81

, Y22

, Y13

and Y43

are equal to 1. Othersets of results in Table 3 may be similarly inter-preted.


Table 4‘Risk-by-gain ratio’, (a) for various scenarios of informational uncertainty

SN Range of estimate(in % of$the centerof interval)

Nature ofinformationaluncertainty in the model

Objective function (’00000)

[zL(x), z

R(x)] Sz

C(x), z

8(x)T

a"z8(x)%

MzM(x)!z

C(x)N

Si

aj

ti

1 00% 00% 00% Deterministic [4.678, 4.678] S4.678, 0T 02 2% 2% 2% Nearly deterministic [4.58, 4.77] S4.678, 0.0935T 0.043 10% 10% 10% Fairly deterministic [4.21, 5.14] S4.678, 0.4678T 0.24 20% 20% 20% Moderately deterministic [3.742, 5.61] S4.678, 0.935T 0.3995 30% 30% 30% Moderately uncertain [3.27, 6.08] S4.678, 1.40T 0.5986 40% 40% 40% Fairly uncertain [2.8, 6.55] S4.678, 1.87T 0.7997 50% 50% 50% Nearly uncertain [2.339, 7.017] S4.678, 2.339T 1.0

Note: For all the calculations of a in Tables 3 and 4, we have taken zM(x) as z

R(x) in SN 7 of Table 4, i.e. 7.017. It is the maximum among

all options.

Fig. 1. Effect of uncertainty in machine acquisition cost due tothe range estimate.

The model is tested for validation and the resultsare shown in Table 4. For the present data set,when all estimates have zero range, (0, 0, 0), i.e.a deterministic case, the results are the same asthose obtained from the deterministic model ofShafer et al. [16]. It is observed that as the totalsystem uncertainty increases, the risk-by-gain ratioalso increases, which is quite logical even intuitively.

The system behavior is also observed with re-spect to the level of informational uncertainty inmachine acquisition cost (AC) (Fig. 1), intercelltransfer cost (IC) (Fig. 2) and subcontracting cost ofEE (SC) (Fig. 3). It can be seen from Figs. 1—3 thatat a given level of cost uncertainty of one kind, thesituation aggravates (resulting in increased value ofrisk-by-gain ratio) when the other two informa-tional uncertainties increase. This effect is evenmore demonstrative at a higher level of cost uncer-tainty. It can be seen from Fig. 1 that the impact ofuncertainty of machine acquisition cost is the mostsignificant. This is because of its major contributionto the objective function. The impact of informa-tional uncertainty in subcontracting cost is relative-ly insignificant, particularly when the other twouncertainties are in the narrower range (Fig. 3).

The interval objective function models need to becompared with its deterministic counterpart. It isclear that when there is informational vagueness inthe design parameters of the objective function, themodel proposed in this paper is very insightful ascompared to the deterministic model. Alternative

solutions may be analyzed and risk associated withany set of decisions may be assessed. Managementmay look into various options which they may liketo adopt. Some of these options are: (i) solution setwith low expected cost but high risk (for risk


Fig. 2. Effect of uncertainty in intercell transfer cost due to therange estimate.

Fig. 3. Effect of uncertainty in subcontracting cost due to therange estimate.

takers), (ii) solution set with low risk but highexpected cost (for risk avoiders) and (iii) solution setwith low risk per unit gain (for risk neutrals). Suchflexibility of alternative scenarios may not be avail-able in conventional LP formulations. Moreover,deterministic model provides only one of the solu-tions generated by the model discussed here. Im-pact of some information, for example, acquisitioncost in our model, is more significant as comparedto the rest. Therefore, decision maker should focuson getting more reliable data for these parametersthrough intense market research and sophisticatedforecasting techniques. For the remaining informa-tional inputs no major effort except, expert opinionabout the interval estimate may be required. Thiscan be exploited to prioritize the time, effort andcost required in generating a reliable data base.

Computational effort for each run is nearly thesame in deterministic and the proposed model, be-cause the number of variables and problem size isnearly the same. However, more runs are requiredin interval model due to large enumeration in-

volved in Pareto optimal solutions with two objec-tive functions.

8. Conclusions

This paper has attempted to model a more realis-tic situation for the post design strategy selectionproblem in the CMS environment. Informationaluncertainty associated with cost estimates pertain-ing to EE and BM has been considered. In thedynamic business situation, exact estimates of costsare unlikely to be available at the stage of suchdecision making. Hence, the range objective func-tion proposed here is a very useful way to capturethese uncertainties. Concept of proposed risk-by-gain ratio is insightful to the decision makers inweighing the risk and gain associated with a feas-ible set of solutions. Eventually, the model is solvedby the conventional LP package and hence nospecial purpose software is needed to operational-ize the proposed model.


Acknowledgements

The authors gratefully acknowledge the twoanonymous referees of this paper for their valuablesuggestions for the improvement of this paper.

References

[1] G.M. Kern, J.C. Wei, The cost of eliminating excep-tional elements in group technology cell formation, Inter-national Journal of Production Research 29 (8) (1991)1535—1548.

[2] H.M. Chan, D.A. Milner, Direct clustering algorithm forgroup formation in cellular manufacture, Journal ofManufacturing System 1 (1) (1982) 56—75.

[3] U. Wemmerlov, N.L. Hyer, Cellular manufacturing in USindustries — a survey of users, International Journal ofProduction Research 27 (1989) 1511—1530.

[4] R.D. Pullen, A survey of cellular manufacturing cells, TheProduction Engineer 56 (1976) 451—454.

[5] R. Shanker, P. Vrat, Some design issues in cellular manu-facturing using fuzzy programming approach, Interna-tional Journal of Production Research, (1998), in-process.

[6] J.L. Burbidge, Production flow analysis, The ProductionEngineer 42 (12) (1963) 742—752.

[7] J.R. King, V. Nakornchi, Machine-component groupformation in group technology: review and extension, In-ternational Journal of Production Research 25 (1982)1715—1728.

[8] W.T. McCormick, P.J. Schweitzer, T.W. White, Problemdecomposition and data recognition by a cluster tech-nique, Operation Research 20 (5) (1972) 993—1009.

[9] M.P. Chandrasekharan, R. Rajagopalan, An ideal seednon-hierarchical clustering algorithm for cellular manu-facturing, International Journal of Production Research24 (2) (1986) 451—464.

[10] G.F.K. Purcheck, A linear programming method for thecombinatorial grouping of an incomplete power set, Jour-nal of Cybernetics 4 (3) (1975) 51—76.

[11] K. Raja Gunasingh, R.S. Lashkari, Machine-groupingproblem in cellular manufacturing systems — an integerprogramming approach, International Journal of Produc-tion Research 27 (9) (1989) 1465—1473.

[12] S.M. Shafer, D.F. Roger, A goal programming approach tothe cell formation problem, Journal of Operations Man-agement 10 (1) (1991) 28—43.

[13] C.J. Chu, Cluster analysis in manufacturing cell formation,Omega 17 (3) (1989) 289—295.

[14] S.S. Heragu, Group technology and cellular manufactur-ing, IEEE transactions on Systems, Man and Cybernetics24 (2) (1994) 203—215.

[15] N. Singh, Design of cellular manufacturing system: aninvited review, European Journal of Operations Research69 (1993) 284—291.

[16] S.M. Shafer, G.M. Kern, J.C. Wei, A mathematical pro-gramming approach with exceptional elements in cellularmanufacturing, International Journal of Production Re-search 30 (5) (1992) 1029—1036.

[17] F. Amirahmadi, F. Choobineh, Identifying the composi-tion of cellular manufacturing system, International Jour-nal of Production Research 34 (9) (1996) 241—248.

[18] R. Logendran, A model for duplicating bottleneck machin-es in the presence of budgetary limitations in cellularmanufacturing, International Journal of Production Re-search 30 (3) (1992) 683—694.

[19] H. Seifoddini, Duplication process in machine cells forma-tion in group technology, IIE Transactions 21 (4) (1989)382—388.

[20] J. McAuley, Machine grouping for efficient production,The Production Engineer 51 (1972) 53—57.

[21] H. Seifoddini, P.M. Wolfe, Application of similarity coef-ficient method in group technology, IIE Transactions 18(1986) 271—277.

[22] K.R. Kumar, A. Vannelli, Strategic subcontracting forefficient disaggregated manufacturing, International Jour-nal of Production Research 25 (12) (1987) 1715—1728.

[23] A. Kusiak, W.S. Chow, Efficient solving of the grouptechnology problem, Journal of Manufacturing Systems6 (2) (1987) 117—224.

[24] H. Seifoddini, A probabilistic model for machine cellformation, Journal of Manufacturing System 9 (1) (1990)69—75.

[25] J.S. Morris, R.J. Tersine, A simulation analysis offactors influencing the attractiveness of group technologycellular layout, Management Science 36 (12) (1990)1567—1578.

[26] N.C. Suresh, Partitioning work centers for group techno-logy: analytical extension and shop-level simulation in-vestigation, Decision Sciences 23 (1992) 267—289.

[27] R. Shanker, P. Vrat, Design of cellular manufacturingsystem: a chance constrained programming approach, in:A. Mubeen, S.A.H. Rizvi (Eds.), CAD, CAM, Automation,Robotics and Factories of Future, Narosa PublishingHouse, New Delhi, India, 1996, pp. 110—116.

[28] A.L. Soyster, Inexact linear programming with generalizedresource sets, European Journal of Operations Research3 (1979) 316—321.

[29] J.E. Falk, Exact solutions of inexact linear programs, Op-erations Research 24 (1976) 783—787.

[30] H. Ishibuchi, H. Tanaka, Multi objective Programming inoptimization of interval objective function, EuropeanJournal of Operations Research 48 (1990) 219—225.

[31] G. Alefeld, J. Herzberger, Introduction to Interval Compu-tations, Academic Press, New York, 1983.


Post design modeling for cellular manufacturing system with cost uncertainty

Documents

Transcript of Post design modeling for cellular manufacturing system with cost uncertainty