Fuzzy multi-item economic production quantity model under space constraint: A geometric programming...

Applied Mathematics and Computation 184 (2007) 326–335

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Fuzzy multi-item economic production quantity modelunder space constraint: A geometric programming approach

Sahidul Islam, Tapan Kumar Roy *

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711103, West Bengal, India

Abstract

In this study we have formulated an economic production quantity (EPQ) model with flexibility and reliability consid-eration of production process and demand dependent unit production cost with fuzzy parameters. The model is restrictedwith available limited storage space constraint. The inventory related costs, storage spaces and others parameters are takenhere as triangular fuzzy numbers. The problem is solved by modified geometric programming method. Finally, the model isillustrated by numerical example.� 2006 Elsevier Inc. All rights reserved.

Keywords: Inventory; Storage space; Triangular fuzzy number; Geometric programming

1. Introduction

A basic assumption in the classical EPQ model is that the production set-up cost is fixed [23]. In additionthe model also implicitly assumes that items produced are of perfect quality [10,13]. While the set-up time,hence set-up cost, will be fixed in short term, it will tend to decrease in the long term because of the possibilityof investment in new machineries that are highly flexible, e.g. flexible manufacturing systems [16]. In a paper,Van Beek and Van Puttin [24] addressed extensively the issue of flexibility improvement production and inven-tory management under various scenarios, while the issues of process reliability, quality improvement and set-up time reduction discussed by Porteus [18,19], Rosenblatt and Lee [20] and Zangwill [27]. Cheng [5] proposeda general equation to model the relationship between production set-up cost and process reliability and flex-ibility. Cheng [6] also introduced the demand dependent unit production cost. The economic production quan-tity model for items with imperfect quality discussed by Ben-Daya [3], Goyal et al. [8] and Salameh and Jaber[22]. Cao et al. [4] describe an examination of inventory and production cost in a revised EMQ/JIt production-run model.

It is often difficult to precise the actual costs and others parameters of the inventory problem. They fluc-tuate depending upon different aspects. So the inventory cost parameters such as holding cost, set-up cost,

0096-3003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.05.198

* Corresponding author.E-mail addresses: [email protected] (S. Islam), [email protected] (T.K. Roy).

mailto:[email protected]

mailto:[email protected]

S. Islam, T.K. Roy / Applied Mathematics and Computation 184 (2007) 326–335 327

shortage cost are assumed to be flexible (i.e. fuzzy in nature). In 1965 Zadeh [26] first gave the concept of fuzzyset theory. This theory now has made an entry into the inventory control systems. Park [17] examined theEOQ formula in the fuzzy set theoretic perspective associating the fuzziness with the cost data. Roy and Maiti[21] solved a single objective fuzzy EOQ model using GP technique.

GP method is an effective method to solve a typical non-linear programming problem. It has certain advan-tages over the other optimization methods. Duffin et al. [7] discussed the basic theories on GP with engineeringapplication in their books. Another famous book on GP and its application appeared in 1976 [2]. Worral andHall [25] analysed the inventory models with some constraints and solved by GP technique. Hariri and Abou-el-ata [9] and Abou-el-ata and Kotb [1] used GP to solve multi-item inventory problems. Jung and Klain [14]developed single item inventory problems and solved by GP method.

In this paper we propose an economic production quantity model with flexibility and reliability consider-ation of production process and demand dependent unit production cost with fuzzy parameters. The model isassociated with available limited storage space constraint. The inventory related costs, storage spaces andother parameters are taken here as triangular fuzzy numbers. The problem is solved by modified geometricprogramming method. Finally, the model is illustrated by numerical.

2. Mathematical model

2.1. Notations

TC(Di,Si,qi, ri) total average cost of production and inventory carrying cost per unit timen number of items

Parameters for the ith (i = 1,2, . . . ,n) item are

Si set-up cost per batch (a decision variable)Di demand rate (a decision variable)qi production quantity per batch (a decision variable)ri production process reliability (a decision variable)Hi inventory carrying cost per item per unit timefi(Si, ri) total cost of interest and depreciation for a production process per production cyclepi unit demand dependent production cost

2.2. Assumptions

(i) The demand rate Di is uniform over time.(ii) Shortages are not allowed.

(iii) The time horizon is infinite.(iv) Total cost of interest and depreciation per production cycle is inversely related to a set-up cost and

directly related to production process reliability according to the following equation fiðSi; riÞ ¼aiS�bii rci

i ; where ai; bi; ci ðP 0Þ are shape parameters.(v) The unit production cost is a continuous function of demand D and takes the following form pi ¼ aiD

�bii ,

where bi (>1) and ai (>0) are shape parameters.

2.3. Model formulation

The situation of this inventory model is illustrated in Fig. 1. If qi(t) is the inventory level of the ith at time t

over the time period (0,Ti), then

dqiðtÞdt¼ �Di for 0 6 t 6 T i ð2:3:1Þ

with initial and boundary conditions qi(0) = riqi, qi(Ti) = 0.

q i(t)riqi Di

Ti t

Fig. 1. The situation of the economic production quantity model.

328 S. Islam, T.K. Roy / Applied Mathematics and Computation 184 (2007) 326–335

The solution of this differential equation is obtained as

qiðtÞ ¼ riqi � Dit for 0 6 t 6 T i ð2:3:2Þand T i ¼ ðriqiÞ=Di: ð2:3:3Þ

Now inventory carrying cost ¼ H i

Z T i

0

qiðtÞdt ¼ H ir2i q2

i

2Di: ð2:3:4Þ

Total inventory related cost per production cycle of the ith item = set-up cost + production cost + inven-tory carrying cost + interest as well as depreciation cost

¼ Si þ piqi þ H iq2i r2

i =2Di þ f ðSi; riÞ: ð2:3:5Þ
The total average cost of the inventory system consists of the set-up, production, inventory carrying and
interest and depreciation cost for the n items is given by

TCðDi; Si; qi; riÞ ¼Xn

i¼1

DiSiq�1i r�1

i þ aiD1�bii r�1

i þH iqiri

2þ aiDiS

�bii q�1

i rðci�1Þi

� �
and required storage area ¼
Pni¼1wiriqi.

Hence the inventory model for n-items is as follows:

min TCðDi; Si; qi; riÞ ¼Xn

i¼1

DiSiq�1i r�1


i þHiqiri

2þ aiDiS

�bii q�1

i rðci�1Þi

� �s:t:

Xn

i¼1

wiriqi 6 w; ð2:3:6Þ

Di; Si; qi; ri > 0 ði ¼ 1; 2; . . . ; nÞ:

3. Prerequisite mathematics

Fuzzy sets first introduced by Zadeh [26] in 1965 as a mathematical way of representing impreciseness orvagueness in everyday life.

Fuzzy set: A fuzzy set eA in a universe of discourse X is defined as the following set of pairs eA ¼fðx; leAðxÞÞ : x 2 Xg. Here leA : X ! ½0; 1� is a mapping called the membership function of the fuzzy set eAand leA (x) is called the membership value or degree of membership of x 2 X in the fuzzy set eA. The largerleA (x) is the stronger the grade of membership form in eA.

Fuzzy number: A fuzzy number is a fuzzy set in the universe of discourse X. It is both convex and normal.Following Fig. 2 shows a fuzzy number eA.

a-cut of a fuzzy number: The a-cut of a fuzzy number eA is defined as a crisp set Aa ¼ fx : leAðxÞP a; x 2 Xgwhere a 2 [0, 1].

Aa is a non-empty bounded closed interval contained in X and it can be denoted by Aa = [AL(a),AR(a)].AL(a) and AR(a) are the lower and upper bounds of the closed interval, respectively.

Fig. 2 shows a fuzzy number eA with a-cuts Aa1¼ ½ALða1Þ;ARða1Þ�, Aa2

¼ ½ALða2Þ;ARða2Þ�. It is seen that ifa2 P a1 then AL(a2) P AL(a1) and AR(a1) P AR(a2).

Triangular fuzzy number (TFN): Let F ðRÞ be a set of all triangular fuzzy numbers in real line R. Atriangular fuzzy number eA 2 F ðRÞ is a fuzzy number with the membership function leA : R! ½0; 1�

μÃ (x)

1

α2

α1

AL (α 1) AL (α 2) AR (α 2) AR (α 1) x

Fig. 2. Fuzzy number eA with a-cuts.


parameterized by a triplet (a1,a2,a3)TFN, where a1 and a3 denote the lower and upper limits of support of afuzzy eA (Fig. 3):

leAðxÞ ¼lLeAðxÞ ¼ x�a1

a2�a1; for a1 6 x 6 a2

lReAðxÞ ¼ a3�xa3�a2

; for a2 6 x 6 a3

0; otherwise:

8>><>>:
3.1. Approximated value of TFN
3.1.1. Removal area (RA) method

According to Kaufman and Gupta [15], ‘let us consider an ordinary number k 2 R, and a fuzzy number eA(as illustrated in Fig. 4). The left side removal of eA with respect to k;RlðeA; kÞ (Fig. 4a), is defined as the areabounded by x = k and the left side of the fuzzy number eA. Similarly, the right side removal, RrðeA; kÞ (Fig. 4b),is defined. The removal of the fuzzy number eA with respect to x = k is defined as the mean of RlðeA; kÞ andRrðeA; kÞ. Thus RðeA; kÞ ¼ 1

2ðRlðeA; kÞ þ RrðeA; kÞÞ.

Ã (x)1

O a1 a2 a3 x

μ

Fig. 3. TFN.

Fig. 4a. Left removal area RlðeA; kÞ of eA.

Fig. 4b. Right removal area RrðeA; kÞ of eA.


If eA � ða1; a2; a3Þ is the TFN then the removal number of eA with respect to origin is defined as the mean oftwo areas, RlðeA; 0Þ ¼ a1þa2

2and RrðeA; 0Þ ¼ a2þa3

2, where RlðeA; 0Þ (Fig. 5a) denotes the area bounded by left side ofeA and x = 0 and RrðeA; 0Þ (Fig. 5b) denotes the area bounded by the right side of eA and x = 0.

Let bARA be the removal area of eA w.r.t. x = 0.So, bARA ¼ 1

2ðRlðeA; 0Þ þ RrðeA; 0ÞÞ ¼ a1þ2a2þa3

4.

According to Kaufmann and Gupta [15], this removal area bA is defined as an approximated value ofeA � ða1; a2; a3Þ.

3.1.2. Mean of expected interval (MEI) method

The a level set of eA is defined as

Aa ¼ ½ALðaÞ;ARðaÞ� ¼ ½a1 þ aða2 � a1Þ; a3 � aða3 � a2Þ�:
According to Heilpern [11], the expected interval of fuzzy number eA, denoted as EIðeAÞ is EIðeAÞ ¼R 1
0ALðaÞda;

R 1

0ARðaÞda

h i.

The approximated value of eA is given by bAMEI ¼ 12

R 1

0ALðaÞdaþ

R 1

0ARðaÞda

h i¼ a1þ2a2þa3

4:

3.1.3. Weighted mean (WM) method

For any TFN eA ¼ ða1; a2; a3Þ, weighted mean of a1, a2, a3 with respective weights k, 2k, k (for any k 2 R+),denoted by bAWM and is given by bAWM ¼ a1þ2a2þa3

4.

Fig. 5a. Left removal area RlðeA; 0Þ of TFN eA ¼ ða1; a2; a3Þ.

Fig. 5b. Right removal area RrðeA; 0Þ of TFN eA ¼ ða1; a2; a3Þ.

1

a0-λ a0 a0+λ

Fig. 6. Symmetric triangular fuzzy number.


3.1.4. Nearest symmetric triangular defuzzification (NSTD) method

Let eA be a fuzzy number with a-cut is [AL(a),AR(a)] (Fig. 6).A triangular symmetric triangular fuzzy number B[a0,k] centered at a0 with basis 2k

BðxÞ ¼

x�a0þkk ; a0 � k 6 x 6 a0

a0þk�xk ; a0 6 x 6 a0 þ k

0; otherwise:

8><>:
Its a-cuts are
BL½a0; k�ðaÞ ¼ a0 � kþ ka; BR½a0; k�ðaÞ ¼ a0 þ k� ka:

To obtain a symmetric triangular fuzzy number which is the nearest to eA, we minimize

D eA;B½a0; k��

¼Z 1

0

ðALðaÞ � BL½a0; k�ðaÞÞ2 daþZ 1

0

ðARðaÞ � BR½a0; k�ðaÞÞ2 da

with respect to a0, k. If B[a0,k] minimizes D eA;B½a0; k��

, B[a0,k] provides a defuzzification of eA with a defuzz-ifier a0 and fuzziness k.

In order to minimize D eA;B½a0; k��

, we have

a0 ¼1

2

Z 1

0

ðALðaÞ þ ARðaÞÞda and k ¼ 3

2

Z 1

0

ðARðaÞ � ALðaÞÞð1� aÞda:

If a triangular fuzzy number eA ¼ ða1; a2; a3Þ, the nearest symmetric triangular defuzzification of eA is givenby the defuzzifier is denoted by eANSTD. i.e. eANSTD ¼ a1þ2a2þa3

4.

4. Fuzzy model

In general the shape parameters for cost of interest and depreciation and unit production cost, cost param-eter for the objective function and storage spaces are not precisely known. So, here we have assumed that allthe parameters (ai,bi,ci,ai,bi,Hi) and storage spaces (wi,w) are fuzzy in nature. Hence, these are expressed astriangular fuzzy number. Then the above crisp inventory model (2.3.6) reduces to


i¼1

DiSiq�1i r�1

i þ ~aiD1� ~bii r�1

i þfH iqiri

2þ ~aiDiS

�~bii q�1

i rð~ci�1Þi

!

s:t:Xn

i¼1

~wiriqi 6 ~w; ð4:0:1Þ

Di; Si; qi; ri > 0 ði ¼ 1; 2; . . . ; nÞ:

Using approximated value of TFN, ai; bi; ci; ai; bi; bH i; wi and w are the approximated values of the TFNparameters ~ai ¼ ðai1; ai2; ai3Þ, ~bi ¼ ðbi1; bi2; bi3Þ, ~ci ¼ ðci1; ci2; ci3Þ, ~ai ¼ ðai1; ai2; ai3Þ; ~bi ¼ ðbi1; bi2; bi3Þ, eH i ¼ðHi1;Hi2;H i3Þ, ~wi ¼ ðwi1;wi2;wi3Þ and ~w ¼ ðw1;w2;w3Þ. The above fuzzy model (4.0.1) reduces to



i¼1

DiSiq�1i r�1


i þbH iqiri

2þ aiDiS

�bii q�1

i rðci�1Þi

!

s:t:Xn

i¼1


Di; Si; qi; ri > 0 ði ¼ 1; 2; . . . ; nÞ:

5. Mathematical analysis

5.1. Geometric programming (GP) technique

The method of geometric programming for solving the problem is described in the following form.

Primal program:


i¼1

DiSiq�1i r�1


i þbH iqiri

2þ aiDiS

�bii q�1

i rðci�1Þi

!

s:t:Xn

i¼1


Di; Si; qi; ri > 0 ði ¼ 1; 2; . . . ; nÞ:

The coefficient of each term of the objective function is positive. Hence this is the posynomial primal geometricprogramming problem.

Dual program:Applying modified geometric programming (GP) technique according to Hariri and Abou-El-Ata [9], the

dual problem of (5.1.1) is

max dðaÞ ¼Yn

i¼1

1

a1i

� �a1i ai

a2i

� �a2i bH i

2a3i

!a3iai

a4i

� �a4i wi

wa5i

� �a5i Xn

j¼1

a5j

!a5i

ð5:1:2Þ

s:t: a1i þ a2i þ a3i þ a4i ¼ 1;

a1i þ ð1� biÞa2i þ a4i ¼ 0;

a1i � bia4i ¼ 0;

� a1i þ a3i � a4i þ a5i ¼ 0; ð5:1:3Þ� a1i � a2i þ a3i þ ðci � 1Þa4i þ a5i ¼ 0:

0 6 aji 6 1 ðj ¼ 1; 2; 3; 4; 5; i ¼ 1; 2; . . . ; nÞ:

Hence the solutions becomes

a�1i ¼ biðbi � 1Þ=ðci þ 2bibi þ 2bi � cibiÞ;a�2i ¼ ðbi � 1Þ=ðci þ 2bibi þ 2bi � cibiÞ;a�3i ¼ ðbibi þ bi þ ci � cibiÞ=ðci þ 2bibi þ 2bi � cibiÞ;a�4i ¼ ðbi � 1Þ=ðci þ 2bibi þ 2bi � cibiÞ;a�5i ¼ �ðbi þ ci � cibi þ 1Þ=ðci þ 2bibi þ 2bi � cibiÞ ði ¼ 1; 2; . . . ; nÞ:

ð5:1:4Þ

So the dual objective value is given by

dða�Þ ¼Yn

i¼1

1

a�1i

� �a�1i ai

a�2i

� �a�2i bH i

2a�3i

!a�3i ai

a�4i

� �a�4i wi

wa�5i

� �a�5i Xn

j¼1

a�5j

!a�5i

:


Thus we have the minimum value of the total average cost is

TC�ðD�i ; S�i ; q�i ; r�i Þ ¼ nffiffiffiffiffiffiffiffiffiffiffidða�Þn

p:

(see [12]).To find the optimal values of D�i ; s

�i ; q

�i ; r�i we have

D�1S�1q��11 r��1

1

a�11

¼ a1D�1�b11 r��1

1

a�21

¼bH 1q�1r�12a�31

¼ a1D�1S��bi1 q��1

1 r�ðc1�1Þ1

a�41

¼ D�2S�2q��12 r��1

2

a�12

¼ a2D�1�b22 r��1

2

a�22

¼bH 2q�2r�22a�32

¼ a2D�2S��b22 q��1

2 r�ðc2�1Þ2

a�42

¼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

¼ D�nS�nq��1n r��1

n

a�1n

¼ anD�1�bnn r��1

n

a�2n

¼bH nq�nr�n2a�3n

¼ anD�nS��bnn q��1

n r�ðcn�1Þn

a�4n

¼Pn

i¼1ðD�i S�i q��1i r��1

i þ aiD�1�bii r��1

i þ bH iq�i r�i2þ aiD�i S��bi

i q��1i r�ci�1Þ

i ÞPni¼1ða�1i þ a�2i þ a�3i þ a�4iÞ

¼ TCðD�i ; S�i ; q�i ; r�i Þn

¼ nffiffiffiffiffiffiffiffiffiffiffidða�Þn

pn

¼ffiffiffiffiffiffiffiffiffiffiffidða�Þn

p:

So,

D�i S�i q��1i r��1

i ¼ a�1i

ffiffiffiffiffiffiffiffiffiffiffidða�Þn

p; aiD

�ð1�biÞi r��1

i ¼ a�2i


p;bH iq�i r�i ¼ 2a�3i


p; aiD�i S��bi

i q��1i r�ðci�1Þ

i ¼ a�4i


pðfor i ¼ 1; 2; . . . ; nÞ: ð5:1:5Þ

Solving the above four non-linear equation (5.1.5), we get optimal values of D�i ; S�i ; q

�i ; r�i ði ¼ 1; 2; . . . ; nÞ as

follows:

D�i ¼ðbibi � biÞbiðbi þ 1Þciðbibi þ bi þ ci � cibiÞðbiþ1Þðbi � 1Þ

aiaciibH ðbiþ1Þ

i

nffiffiffiffiffiffiffiffiffiffiffidða�Þn

pci þ 2bibi þ 2bi � cibi

!ð2biþciþ2Þ24 35

1ð1þbiþci�ci biÞ

;

S�i ¼ðbibi � biÞðbibi þ bi þ ci � cibiÞ

D�i bH i



!2

;

q�i ¼ðbi þ 1Þðbibi þ bi þ ci � cibiÞbH iaiD�i ð1� biÞ



!2

;

r�i ¼aiD�i ð1� biÞðbi þ 1Þ

ci þ 2bi þ 2bibi � cibi


p !ðfor i ¼ 1; 2; . . . ; nÞ:

6. Numerical example

A manufacturing company produces two types of machines. The machines are produced in lots. Thedemand rate of each machine is uniform over time and can be assumed to be deterministic. The pertinent datafor the machines is given in Table 1.

With imprecise shape parameters ð~ai; ~bi; ~ai; ~bi and ~ciÞ are given in Table 2.Determine the demand rates (D1,D2), set-up cost (S1,S2), production quantity (q1,q2) and production pro-

cess reliability (r1, r2) of each machines and also find optimal total average cost (TC) of the production system.

Table 1Input data

Types ofmachines (i)

Productioncost

Interest anddepreciation cost

Carrying cost,Hi ($)

Storage space areaper unit item (wi sq. ft.)

Total storagespace area (w sq. ft.)

1 D�b1

1 a1S�b1

1 rc1

1 (8.5,10,12) (8,10,12) (1500,2000,3500)2 D�b2

2 a2S�b2

2 rc2

2 (11,12.9,13) (9,11,13)

Table 2Input imprecise data for shape parameters

Types of machines (i) ~ai~bi ~ai

~bi ~ci

1 (10,000,12,000,140,000) (3.4,3.6,3.8) (1200,1500,1800) (1.3,1.6,1.9) (0.7,1,1.3)2 (9500,12,500,15,500) (3.5,3.8,4.1) (1150,1250,1350) (1.4,1.8,2.2) (0.8,1.1,1.4)

Table 3Optimal solutions for the problem

Methods D�1 S�1 ($) q�1 r�1 D�2 S�2 ($) q�2 r�2 TC* ($)

GP(modified) 2664.3 4.97 144 0.73 1557.8 4.98 88.5 0.83 665.9


The optimal solutions of the above problem are given in Table 3. From Table 3, it is observed that to obtainoptimal solutions of the model by geometric programming (GP) technique.

7. Conclusion

We have solved fuzzy multi-item EPQ model with fuzzy parameters by modified GP techniques. This tech-nique can be applied to solve the different decision making problems in inventory control and other engineer-ing and management sciences.

Acknowledgement

This research was supported by CSIR junior research fellowship in the Department of Mathematics, BengalEngineering and Science University, Shibpur. This support is greatfully acknowledged.

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Fuzzy multi-item economic production quantity model under space constraint: A geometric programming...

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