fuzzy programming for optimal product mix decisions based on expanded abc approach..rtf

7/27/2019 Fuzzy programming for optimal product mix decisions based on expanded ABC approach..rtf

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International Journal of Production Research

Vol. 48, No. 3, 1 February 2010, 729744

RESEARCH ARTICLE

Fuzzy programming for optimal product mix decisions basedon expanded ABC approach

E. Karakas*, M. Koyuncu, R. Erol and A. Kokangul

Faculty of Engineering and Architecture, Department of Industrial Engineering, CukurovaUniversity, 01330, Adana, Turkey

(Received 21 January 2008; final version received 29 August 2008)

In this paper, we deal with the problem of determining the optimal product mix andproduction quantities based on expanded ABC (activity-based costing) approach inthe presence of obscure estimation of parameters for the capacities of theactivities and the demands of each product. First, we give the mixed zero-oneprogramming model such that the profit is maximised subject to capacities ofactivities and demand of products. Second, to handle vagueness of capacity anddemand in real-life production systems in fuzzy environments, fuzzy programmingis presented. We employ linear membership functions for the capacity of eachactivity and demand of each product. Using the data in the study of Kee (1995),we show the usefulness of the fuzzy model.

Keywords: fuzzy methods; expanded ABC; mathematical programming;product-mix

1. Introduction

The product mix problem is one of the most well-known applications of linearprogramming. The problem includes determining both the quantity and theidentification of each product to produce. The main structure of the problem is tomaximise profit from the mix of manufactured products subject to constraints on theavailable capacity of resources (Malik and Sullivian 1995).

Activity-based costing (ABC) and the theory of constraints (TOC) represent alternative

paradigms for evaluating the economic consequences of production-related decisions.Both paradigms are designed to overcome limitations of traditional cost-based systemsand, thereby, provide more relevant information for evaluating the economic consequencesof resource-allocation decisions. While their objectives are similar, the means used toachieve these objectives differ significantly (Kee and Schmidt 1998).

ABC models the causal relationship between products and the resources used in theirproduction. This enables ABC to provide more accurate product-cost information forevaluating the profitability of the firms product lines and customer base (Cooper et al.1992). ABC differs from traditional cost systems in two important respects. First it tracesindirect cost to cost objects such as products and customers on the basis of factor (costdrivers) that cause or correlate highly with indirect cost. Second, ABC traces indirect costs


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DOI: 10.1080/00207540802471249

http://www.informaworld.com


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730 E. Karakas et al.

on the basis of the structural or hierarchical level at which costs are incurred in theproduction process. For example many indirect costs are incurred at the batch,product, and facility levels (Cooper 1990). The use of multiple cost drivers and tracingcost at the hierarchical level enables ABC to more accurately model the relationshipbetween the resources used in production and the products. Therefore, ABC providesa better estimate of product cost (Kee 2005). As a result, ABC enables the manager topredict the economic consequences of production related decisions.

Conversely, the TOC represents an application of general systems theory for optimising

production. It uses the most constrained of the firms activities to guide production and

process improvement decisions (Kee and Schmidth 1998). In the early 1990s, Goldratt

(1993) demonstrates the concept of TOC. Later, it is shown that the product-mix decision

problem under TOC can be mathematically tackled as a linear programming (LP) model.

Many researches have discussed the product mix problem and its solution through TOC

(e.g., Luebbe and Finch 1992, Lee and Plenert 1993, Plenert 1993, Frendall and Lea 1997,

Balakrishan and Cheng 2000, Coman and Ronen 2000). However, it is revealed that the

algorithm is inefficient in handling two types of problems. The first type includes problemsassociated with adding new product alternatives to an existing production line. The second

type includes problems concerning more than one bottleneck in which the algorithm could

not reach the feasible optimum solution (Lee and Plenert 1993). Fredendall and Lea (1997)

revise the TOC product-mix heuristic to identify the optimal product-mix under conditions

where the original TOC heuristic failed. Instead of TOC heuristic, a tabu search-based

algorithm is reported by Onwubolu (2001). Onwubolu and Mutingi (2001) present a genetic

algorithm-based TOC procedure for solving combinatorial problems encountered in

practice which cannot be solved using linear integer programming or similar techniques.

In all of the studies mentioned above, it is assumed that there is not any

uncertainty in TOC product mix decision. Conversely, there are some studies in theliterature that consist of a certain degree of fuzziness in TOC product mix decision(Vasant et al., 2005, Bhattacharya et al. 2006, Bhattacharya and Vasant 2007).

In addition to these studies, there are many works in the literature about deciding which

paradigm to select for production-related decisions (Bakke and Hellberg 1991, MacArthur1993, Spoede et al. 1994, Holmen 1995, and Lea and Frendall 2002). The complementary

nature of the TOC and ABC has also been examined (Kee 1995, Kee and Schmidt 1998,Kee 2003, Kee 2004). Kee (1995) discusses how the principles of ABC and TOC may be

used in conjunction with one another. His paper demonstrates that ABC and TOC reflectdifferent aspects of the production process and that concepts of both models may be

integrated to provide deeper insights into the firms underlying production process. Hedevelops a mixed integer programming (MIP) model to integrate ABC with physical usage

and capacity of production activities, which is called Expanded ABC. In this approach, thecost and physical usage of resources by production is modelled at unit level activities,

batch and product level activities.

Having carried out an examination of a vast range of literature on product mixdecision and two alternative paradigms (ABC and TOC) for this decision, we havereached the following conclusions:

(1) The strengths of ABC and TOC are complementary in nature. The strengths ofeach model overcome a major limitation of the other. To form a larger model

that links the costs and physical attributes of production structure, Kee (1995)International Journal of Production Research 731


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developed a MIP that expands the framework of ABC to incorporate theresource usage of products and the capacities of the processes used inproduction (expanded ABC).

(2)Although, there are some studies in the literature that consist of a certain degree offuzziness in TOC product mix decision, none of the studies based on ExpandedABC takes into account possible fuzziness in some information about productionsuch as demand and production capacity. However, most of the real-life problemsand models contain linguistic variables and constraints or imprecise data.

The main purpose of this paper is to develop a mathematical modelling approachbased on Expanded ABC to define optimal product mix under the conditions wherefuzziness in demand of the products and capacity of the activities exist.

2. Product mix determination based on expanded ABC approach

Spoede et al. (1994), Kee (1995) and Kee and Schmidt (1997) examined how theprinciples of ABC and TOC may be used in conjunction with one another. It isdemonstrated that ABC models the economic aspects of how resources at the unit, batch

and product-level activities are transformed into the firms products. ABC represents a longterm perspective of how costs vary with production (Kee 1995). On the other hand, ABC

has generally been criticised for its failure to incorporate the physical usage of resourcesby production activities and the capacity constraints of the activities (Spoede et al. 1994).

Conversely, the principles of the TOC reflect how the physical resources consumedby production activities and their production capacity play a critical role in theproduction process. This means that the TOC is based on managing productionconstraints. However, one of the limitations of the TOC concerns using throughputmaximisation as decision criteria. This may lead to sub optimal decision in some

circumstances (Kee 1995).Because of these limitations, the studies of Spoede et al. (1994), Kee (1995) and

Kee and Schmidt (1997) are based on integrating ABC and TOC to form a largermodel that simultaneously links the cost and physical constraints of a firmsproduction. Kee (1995) expands the ABC model, which is called Expanded ABCexplicitly to recognise the physical usage of resources and the capacity of theproduction activities. Unlike the study of Spoede et al. (1994), in the study of Kee(1995) product mix decision is made within the framework of the ABC model.

As noted earlier in our study, the proposed mathematical model is based on theMIP model developed by Kee (1995). The main reason for taking this model as a base

in this study is its capability of capturing the integration between the costs, physicalresources, and the capacity of production activities. The model enables an optimalproduction mix to be determined from simultaneous evaluation of ABC data andphysical attributes of the production process (Kee 1995).

However, a product mix based on the solution of the model developed by Kee(1995) results in success only when each activity capacity is stable and the demand ofeach product is precise. As known, most of the real-life problems and models containlinguistic variables and constraints or imprecise data. Therefore, we present a fuzzyprogramming for product mix selection in the light of obscure estimation of parametersfor the capacities of the activities and the demands of each product.


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In Section 2.1, we give the formulation of the MIP model based on Expanded ABCapproach. Problem formulation incorporating fuzzy constraints is then explained inSection 3.

2.1 Problem formulation

In this approach, three activity levels; the unit-level, the batch-level, and the product-level, are included in the product-mix decision model by using three different kinds ofdecision variables described below:

Xi: the number of units of product i produced in a given time period, i

1, 2, 3 . . . N.

Yi: the number of batches of product i produced in a given time period, i

1, 2, 3, . . . N

1, if product i is produced in a given time period, i 1, 2, 3, . . . , N:

Zi 0,if product i is not produced in a giventime period, i 1, 2, 3, . . . , N:

In this model, Xi and Yi

are integer variables, Zi isa binary variable.

The costs of unit-level activities

are assigned in proportion to

the number of productsproduced. Therefore, as the

volume of production

increases, the costs of this kind

of activities increase. In batch

related activities, resourceconsumption is proportional to

the number of batches

processed. A batch driver

assigns the cost of an activity

to a batch. Product-sustaining

activities are performed inorder to continue to produce

and sell individual products.

The costs of these activities

can be traced to each product

but should not be allocated

based on the number of units

or batches produced, because

they are not affected by the

level of production volume. The

only way to eliminate the costof product sustaining activities

i t di ti th d t

(

K

e

e

1

99

5

,

G

u

r

s

es

1

9

9

9

)

.

Thepara

meter

of the problem are:

ai

u:p

erunitusage

ofactivityu

by

product

iai

b:per

batchusageof

act


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ivityb

byproducti

ai

p:perpro

duct

usage ofactivity pby producti crof unit levelactivity ufor eachproduct i

crbatchactivity bfor eachproduct i perbatchcrproduct-sustainingactivity pfor eachproduct iper batchCof unit-levelactivity u

Cof batch-level activitybCof product-level activitypperformedbssize ofproduct isiproduct i

dmmaterialcost ofproduct idl

laborcos

tofproducti

Di:demandof

produc

tiNU: the totalnumber of unitlevel activities,

NB:thetotalnumberofunitlevelandbatch

levelactivities,NP:thetotalnumber

ofactivities,


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International Journal of Production Research 733

The problem, then, can be written in MIP problem (Model 1) form as follows:Model 1

NNU N NB N NP N

Maxs dm

d

l X a crXi

a crY i a crZi

i i i iiu u

NU1

ib b ip p

i 1 u 1 i 1

bi 1 p NB 1 i 1

X XX X X X X

1

Subject to

N

Xi1

aiuX

iC

u u 1, . . . NU

2

N

Xi1

aibY

iC

b b NU 1, . . . NB

3

N

Xi1

aipZ

iC

p p NB 1, . . . NP

4

bsYXi

1, . . .N 5

i i i

Xi MZi i 1, . . . N

6

Xi

Dii 1, . . . N

7

All variables are greater than or equal to zero.

Xi and Yi are integer variables; Zi is a binary variable.M is a very big number.

Equation (1) or objective function reflects the goal of maximising the profit.Constraints (2) to (4) reflect the capacity-related constraints for unit level, batch leveland product level activities, respectively. Constraint (5) must be included in the model

for each product manufactured in the system to define the batch sizes. This constraintguarantees that whenever one unit of a product i is manufactured, the relevant batch-level activity costs are incurred in the objective function Constraint (6) ensures that


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whenever product i is produced, the relevant product-sustaining activity costs areincurred in the objective function.

3. Problem formulation incorporating fuzzy constraints

As noted earlier, a product mix model in Section 2.1 can give an optimal solution when

each activity capacity is stable and the demand of each product is precise. As an optimalsolution for product mix problem is determined according to the capacities of each activity

and the number of forecasted demand, the optimal solution is not appropriate to implementwhen it is difficult to forecast or to estimate the parameters precisely. Also, in real

production problems, it is supposed that parameters involved in an objective function


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and constraints, such as demands or production capacities, are forecasted or estimated byexperts judgements and therefore such values are not always precise (Skawa et al. 2001).

In this section, for cases where demand and capacity parameters cannot beforecasted precisely, we incorporate fuzzy constraints for formulating the product mixdetermination problem based on ABC.

For production capacity constraints and demand constraints of the fuzzy model

N

Xi1

a Xi5Cu u

1, . . . NU 8

iu

N

Xi1

a Y

i

5C

bb

1, . . . NB 9

ib

N

Xi1

a Z

i

5

Cp p 1, . . . NP

10

ip

Xi5Di i 1, . . . N

11

We employ linear membership function as depicted in Figure 1 and Figure 2 forcapacity and demand constraints respectively.

Suppose that there exists a fluctuation 100(1 vi)% in maximum annual demand of each

product, and there is a possibility that an actual capacity of each unit level activity is 100(1

tu)% of the estimated capacity, an actual capacity of each batch level activity is

Cu (Mu)

1

0 (1tu)Cu C M


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u

u

Figure 1. A membership function of fuzzy constraints for the unit level activity capacities.

Di(xi)

1

(1ni)Di

Xi

Di

Figure 2. A membership function of fuzzy constraints for the demand of each product.International Journal of Production Research 735

100(1 b)% of the estimated capacity, and an actual capacity of each product level

activity is 100(1 p)% of the estimated capacity.Then, the membership function of the fuzzy capacity constraints for unit level

activities is defined as

8

1,C

u

M

u

CuM

u > ,

> t C

>

:where

XN

Mu

a

i

u

X

i

i

1

1


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13


12/26

tu: tolerancelimit for eachof the unitlevel activity

capacityUsing the samemembership function,fuzzy constraints for thebatch level and product

level activities are as follows,respectively

C

W

b

where

b: tolerance

limit for each

of the batch

level activitycapacity


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81, Kp 1 pCp

>

C

p

K

p

C Kp>

1 p Cp Kp Cp16

>

Cp

>

>

>

>

where:

N

Kp Xi1a

ipZ

i 17

p: tolerance the product level activity capacity


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The membership function of the fuzzy demand constraints is defined as;

8 1, Xi 1 iDi

D

X

i >

DiXi , 1 i

D

i

Xi

Di

18

i >

D

>

i < i

Xi4Di> 0,

>

>

:

i: tolerance limit for demand constraints of each product.

A linear programming problem with fuzzy constraints in this form can be solved by amulti-objective optimisation problem (Rommelfanger 1996). Therefore, the membershipfunction of objective function (z(x)) must be described. Then, the membership functions

z(x) are described as (Zimmermann

1978)

z x 8 0, z z 19>z z = z z , z 5z 5z

>

1,

>

:

While z is computed based on the lowest value of demand for each product and thelowest capacity for each activity, z is computed based on the highest value of them.

Consequently, a linear programming problem with fuzzy constraints can be solved by aclassical linear programming problem using the method as follows (Zimmermann 1978)

max 20as:t zx 20b

ix i 1 . . . m1 20c

x 2 XU

and 2 0,

1&: 20d

4. A numerical example

To illustrate the integration of fuzzy demand and capacity with expanded ABC basedproduct mix determination problem, consider the example provided by Kee (1995).

The example is presented in Table 1 through 3. XYC Inc. is a medium-sized firm withtwo production departments, assembly and finishing, and three support departments,

t h i d i i Di t t i l d l b t d di tl t


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the individual products. Unit level overhead is estimated using direct labour cost as thecost driver. Set-up and purchasing costs are incurred at the batch level, engineering isincurred at the product-sustaining level (Kee 1995).

To make the proposed model consistent with the data by Kee (1995), some slightmodifications in the model are conducted in this section. First of all, as seen fromTable 2, there exist two batch-level activities whose batch sizes are different from eachother. So, the model must include two different decision variables to reflect these

activities instead of the general decision variables Yi.

STi: the number of set-ups of product i produced in a given time periodPRi: the number of batch of product i processed by purchasing

department in a given time period.International Journal of Production Research 737

Table 1. Unit level activity cost-capacity and operating structure.

Product

Product1

Product2

Product3

Product4

Capacity

Assembly labour hours 1 1 2 5200,000

Finishing labour hours 0.50 0.50 2 4180,000

Direct material cost $4 $7 $15 $28

Direct labour cost $12 $12 $32 $72

Overhead cost $36 $36 $96 $216

Price $70 $80 $223 $516

Maximum annual expecteddemand 100,000 100,000 30,000 20,000

Table 2. Batch -level activities.

Product

Set-up departmentX1 X2

X3 X4

Batch size (units) 1000 100050

0 200Hours/batch 2 2 4 5

Cost per set-up hour

$400

Expected capacity(hours)

500

Purchasing department

Batch size (units) 4000 4000

10

00 500Orders/batch 5 8 12 15Cost per purchase order $


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200

Expected Capacity(orders)

800

Table 3. Product-level activities.

Product

Engineering DepartmentX1 X2 X3 X4

Drawings/product100 100 300

500

Expected capacity 1000Cost per engineering Drawing $1000

Second, in this example, unit level cost is estimated using direct labour cost as thecost driver in accordance with the study of Kee (1995). So the objective function of theModel 1, which is denoted by Equation (1), is revised as seen in Equation (21).

Ovr: Taken as a percentage of direct labour cost (in this example, 300% ofdirect labour cost was accepted as unit level cost driver, so ovr 3for this numerical example.)


So, the proposed fuzzy model can be rewritten as follows (Model 2)

4 4 4 4

MaxX

si dmi dliXiX

ovrdli XiX

ai3cr3STiX

ai4cr4PRi

XN

ai5cr5Zi

i1i1

i1 i1 i1

Subject to21

N

Xi

1

a Xi5Cu i

1 . . .4; u

1,2

22

iu


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N

a STi5C3 i

1 . .. 4

23

Xi1

i3

N

a

PR

i

5C

4 i

1 . .. 4

24

Xi1

i4

N

X

i1

aZi5C5 i

1 . . .4

25

i5

bsY X

i

i

1 . .. 4

26

i

i

Xi MZii 1 . . . 4 27

X

i 5Dii 1 . . . 4

28

Using the approach denoted by Equations (20a) to (20d), we transformed model 2 asfollows

Max ;

Subject to

4 4 4X

i1 si dmi dli ovrdliXiX

i1a

i3cr

3ST

iX

i1 ai4cr4PRi 29

4

X

i1 ai5cr5Zi z z z 30

4X

i1 aiuXi Cu tuCu u 1, 2 31

Cu MutuCu Cu Mu u 1, 2 32International Journal of Production Research 739

N

Mu Xi1a

iX

i

u 1,2

33


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4

Xi1 ai3STi C3 3C3 b 3

34

C3 W33C3 C3 W3 b 3

3

54

W3

X

i1a

i3ST

i b 336

4

Xi1 ai4PRi C4 4C4 b 4

37

C4 W44C4 C4 W4 b 438

4

W4

X

i1a

i4

PRi b 4

39

4

Xi1 ai5Zi C5 5C5 p 5

40

C5 K55C5 C5 K5 p 541

4

K5

X

i1a

i5Z

i p 5

42

X D i

D

1 . . .4 43

i

Di XiiDi Di Xi i 1 . . . 444

bsY X

i

i

1 . . . 4 45

i i

Xi MZii 1 . . .

446

Equations (32) to (33), Equations (35) to (36), Equations (38) to (39), Equations (41) to

(42) and Equation (44) enable us to calculate the value of the membership function offuzzy unit-batch-product level capacities and membership function of fuzzy demand ofeach product respectively The transformed capacity constraints and demand


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constraint are given in Equations (31), (34), (37), (40) and (43).

Suppose that there exists a fluctuation 20% in demand at each product and there is apossibility that actual capacities of unit level, batch level and product level activities are


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80%, %90 and %90 of estimated capacity respectively. Then, z and z are computed as3,732,000 and 4,620,000, respectively.

The proposed model is solved using CPLEX 300 solver in MPL Package Program. The

solution of mixed integer programming model (Kee 1995) and fuzzy model are given in

Table 4 and Table 5. In mixed integer programming model, the optimal production strategy

consists of producing 30,000 units of product 1, 100,000 units of product 2, 30,000 units of

product 3. The product mix is computed by ranking each product in terms of its profitability

and producing the products with the highest profitability at the limit of the capacities. The

proposed fuzzy model uses the same ranking method with ABC cost data, however it offers

a product mix strategy avoiding output at the limit of activities capacity and producing more

quantity of products than %80 of the forecasted maximum demand. As seen in Table 4,

fuzzy application contains 35,668 units of product 1, 89,163 units of product 2, and 26,748

units of product 3. As production of product 2 and 3 decreases in

Table 4. Solution of the models.

Production quantityof each product Outputs of fuzzy model Outputs of MIP*

X1 35,668 30,000X2 89,163 100,000X3 26,748 30,000X4 0 0,000

The number of batch for each activityST1 36 30ST2 90 100ST3 54 60PR1 9 8PR2 23 25PR3 27 30Z (Profit) 4,260,000 4,227,809

*Kee (1995)

Table 5. The values of the membership function in fuzzyapplication.

Membership functionCapacityconstraints

value of thecapacities

Assemblydepartment 0.541Finishingdepartment 1Set-up department 0.64Purchasingdepartment 1Engineering department 1Production demand constraints

Product 1 1Product 2 0.541Product 3 0 542


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Product 4 1Generalsatisfaction 0.542


fuzzy application in order not to make products at the limit of the capacity, the profit ofthe fuzzy application is 4,227,809 while the profit of MIP is $ 4,260,000.

Because membership function of each fuzzy constraint for the capacity of eachactivity is identified such that a degree of satisfaction becomes 0 if the output run up toa limit of the capacity and it becomes 1 if the output is smaller or equal to %80 of thecapacity, as seen in Table 5, generally each output is %80 of the activity capacityexcept assembly activity and set-up activity. Similarly, the membership function ofeach of the fuzzy constraints for demand of the products is identified such that adegree of satisfaction becomes 0 if the production runs up to forecasted demand andit is 1 if the production is smaller or equal to %80 of the forecasted demand, as seen inTable 5, a degree of satisfaction is equal to 1 for product 1 and 4 and it is equal to0.541 and 0.542 for product 2 and product 3 respectively.

Table 6 shows the slack capacity of non-constrained activities according to the solutionof fuzzy model and MIP model. The slack variables for demand constraints in Table 6indicate the units of products that the firm will be unable to supply. These values are70,000 units of demand for product 1 and 20,000 units of demand for product 4 in MIP and53,445 units for product 1 and 17,832 for product 4 in fuzzy application. In fuzzy

application, the slack values for demand constraints are lower because fuzzy model offersa product mix strategy taking %80 of the forecasted maximum demand as actual demand.

As indicated, assembly and finishing will have 10,000 and 55,000 excess labour hours

of capacity in MIP model, respectively. However, in the fuzzy model finishing will have

44,583 excess labour hours and assembly will have no excess labour hours. Also, in the

MIP model solution, purchase and engineering will have excess capacity of 200 purchasingorders and 500 engineering drawings. However, excess capacities of each of these

activities will be 203 purchasing orders and 445 engineering drawings, respectively in fuzzy

application. As a result, while set-up department is a constrained activity in MIP model

solution, both assembly and set-up are constrained activities in fuzzy application.

As we know, constrained or bottleneck activities are critical for maximising production and

profitability and minimising excess resources of the other activities. Improving a constrained

activity relieves a bottleneck, thereby increasing throughput. (Kee 1995). According to fuzzy

model solution there are two constrained activities (assembly and set-up activities) as different

from the study of Kee (1995). The impact of expanding the capacity of set-up can be evaluated

by increasing the amount of set-up time in the fuzzy model while assembly capacity is keptconstant. In the same way, the impact of expanding the capacity of assembly can be evaluated

by increasing the amount of assembly time while

Table 6. The values of the slack variables.

Slack variables Fuzzy application MIP application**

Excess capacity in the assemblydepartment 10,000Excess capacity in finishing department 44,583 55,000Excess purchase order capacity 203 200Excess capacity in engineering 445 500Excess demand for product 1 53 445 70 000


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Excess demand for product 4 17,832 20,000

**Kee(1995)742

E. Karakas etal.

49,00,00

0

48,00,000

47,00,00

0

p

r

o

f

i

t

)

46,00,00

0

45,00,00

0

Z

(

$

44,00,00

0

43,00,00

0

42,00,00

0

41,00,00

0

450 500 550 600

6

5

0 700 750 800

Set-up capacity (hours)

Figure 3. The impact of increasing set-up capacity on the profit.

Z

43,20,000

43,10,000

43,00,000

42,90,000

42,80,000

42,70,000

42,60,000

42,50,000

42,40,000

42,30,000


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42,20,000 1,95,0

00

2,00,000 2,05,000 2,10,000 2,15,000

2,20,000 2,25,000 2,30,000

Assembly

capacity

(hours)

Figure 4. The impact ofincreasing assembly capacity

on the profit.

set-up capacity is keptconstant. Figure 3 andFigure 4 show the impact ofexpanding the resources inset-up and assembly,relatively.

As seen in Figure 3,increasing set-up capacityhas an important impact onthe profit. However, whenset-up time has beenincreased up to 700, any

addit

ionalincreasi

ng in this activity will notprovide moreimprovement in profit. Atthis point, assembly willhave a key role to expand

production and profitfurther, additionalresources would need tobe added to the assemblydepartments.

As seen in Figure 4,expanding the resourcesin the assemblydepartment will increaseprofitability. However,

when compared to theimpact of expanding set-up activity on profit withthe impact of expanding

assembly capacity, it can beunderstood that increasing thecapacity of set-up activity canprovide more improvement.As seen in Figures 3 and 4,

the maximum value for profitwill be $ 4,853,000 if set-upcapacity is increased while itwill be $ 4,310,800 if assembly departmentcapacity is increased. Themaximum value for profit willbe obtained at 22,000 hoursassembly capacity.

As a result, the fuzzy model

for product mix determinationbased on the expanded ABCmodel can work out an optimalproduct mix plan when demandcannot be estimated


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precisely and the limits of capacities are not stable. Also, using the sensitivity analysis,the bottleneck constraint can be identified and therefore management can begincontinuous improvement process under these circumstances.

5. Conclusion

In this paper, we have considered product mix determination problem based on theexpanded ABC approach proposed by Kee (1995). We have presented a fuzzyprogramming for product mix selection in the light of obscure estimation of parameters forthe capacities of the activities and the demands of each product. The proposed model,thereby, can handle impreciseness in some parameters involved in an objective functionand constraints in real-life production problems, such as demands or production capacities.

The differences between the solution of the MIP (based on precise informationabout the capacity and demand) and the solution of the fuzzy model indicate that theuncertainties should be taken into account under the circumstance where impreciseinformation about demand and capacities of activities exist. Otherwise a firmsmanager can give a wrong production related decision and can be faced with theunaffordable cost of this decision.

In addition to these benefits, the solution of the fuzzy model also identifies non-constraint activities with their excess resources and constrained activities. Therefore,the model may be used to identify the constraint that limits production beforeproduction has begun and then constraint identification provides a starting point formanaging bottleneck activity.

References

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