A New Method for Solving Deterministic Multi-Item Fuzzy Inventory Model with Three Constraints

IJSRD - International Journal for Scientific Research & Development| Vol. 3, Issue 09, 2015 | ISSN (online): 2321-0613

A New Method for Solving Deterministic Multi-Item Fuzzy Inventory Model with Three Constraints A.Faritha Asma1 E.C.Henry Amirtharaj2 1 Assistant Professor 2Associate Professor 1,2 Department of Mathematics 1 Government Arts college, Trichy-22.Tamilnadu, India 2Bishop Heber college, Trichy-17.Tamilnadu, India Abstract— The aim of this paper is to present a method in which fuzzy multi item inventory model together with the constraints is reduced to crisp inventory using ranking function and then the crisp multi item inventory problem (MIIP) with constraints is solved using NLPP technique.

Key words: Multi–Item Inventory Problem, Fuzzy Ware House Capacity, Fuzzy Maximum Investment, Fuzzy Maximum Average Number of Units, Fuzzy Multi-Item Inventory Problem, Trapezoidal Fuzzy Numbers, Fuzzy Ranking I. INTRODUCTION Most of the real world problems is inherently characterized by multiple, conflicting and incommensurate aspects of evaluation. These axes of evaluation are generally operationalized by objective functions to be optimized in framework of multiple objective linear programming models. Furthermore, when addressing real world problems, frequently the parameters are imprecise numerical quantities. In order to deal with such uncertain situations fuzzy model is used [2] in such cases, fuzzy set theory, introduced by Zadeh is acceptable. There are several studies on fuzzy EOQ model. Lin et al. [9] have developed a fuzzy model for production inventory problem. Katagiri and Ishii[7] have proposed an inventory problem with shortage cost as fuzzy quantity. This paper discusses a fuzzy multiitem inventory model subject to ware house capacity constraint, an investment constraint, maximum average number of units in all the items constraint. We introduced a method in which a fuzzy multi item inventory problem is reduced to crisp MIIP using ranking function GMIR [11] and the resulting one is solved by crisp NLPP technique using LINGO software. The costs namely holding, setup of all items as well as the constraints is fuzzy in nature. A numerical example is given to illustrate the procedure.

II. REVIEW OF LITERATURE In recent years inventory problems in fuzzy environment have received much attention Lin et al. [9] have developed a fuzzy model for production inventory. Katagiri and Ishii [7] have proposed an inventory problem with shortage cost as fuzzy quantity. Roy and Maiti (1995) considered a fuzzy inventory model with constraint. Srinivasan and Dhanam (2006) have considered cost analysis on a deterministic single item fuzzy inventory model with shortage. Also they have considered a multi-item EOQ inventory model with three constraints in a fuzzy environment. The model is solved by fuzzy non-linear programming method using Lagrange multiplies. Different kinds of inventory problems are solved in the papers[4] zimmermann[16] has introduced fuzzy programming approach to solve crisp multi objective

linear programming problem. Recently H.M.Nehi et al.[10] used ranking function suggested to solve fuzzy MOLPP. III. PRELIMINARIES A. Definition 1: The characteristic function µA of a crisp set A ⊆ X assigns a value either 0 or 1 to each member in X. this function can be generalized to a function  A such that the value assigned to the element of the universal set X fall within a specified range i.e.  A : X→[0,1]. The assigned value indicates the membership grade of the element in the set A. The function  is called the membership function A

and the set

A=

{(x,

 A (x)); x ϵ X} defined by  A (x) for

each x ϵ X is called a fuzzy set.

B. Definition 2: A fuzzy set A , defined on the universal set of real numbers R, is said to be a fuzzy number if its membership function has the following characteristics: 

A



A

 

A

:R→[0,1] is continuous. x  (, a] [d , ) (x) =0 for all .

(x) Strictly increasing on [a,b] and strictly decreasing on [c,d]. A (x) =1 for all x  [b, c] , where a < b < c < d.

C. Definition 3: [4] A fuzzy number A =(a1,a2,a3) is said to be a triangular fuzzy number if its membership function is given by  x  a1   a  a ; a1  x  a2  1  2   a3  x   A ( x)   ; a2  x  a3  a  a 2  3  0; otherwise     

IV. MULTI-ITEM MULTI OBJECTIVE INVENTORY PROBLEM A. Assumptions: A multi-item, multi-objective inventory model is developed under the following notations and assumptions.  The inventory system pertains to multi- items.  Demand rate is deterministic.  The inventory is replenished in single delivery for each order.  Replenishment is instantaneous.  There is no lead time.

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A New Method for Solving Deterministic Multi-Item Fuzzy Inventory Model with Three Constraints (IJSRD/Vol. 3/Issue 09/2015/132)



1

Shortage is not allowed.

=  h  (a  (b  a)h)  (1   )(c  (c  b)h) dh

B. Notations: n - total number of items being controlled simultaneously.

Qi - number of units ordered per order for i th item(a decision variable) Di - Annual demand for ith item si - setup cost per order for ith item. hi - holding cost per unit quantity per unit time for ith item.

B - maximum available storage space to store all items.

F - maximum investment N - maximum average number of units for all items Ai - Storage space required per unit of item i (i=1,2,3,…..n) ci - Price per unit of item i (i=1,2,3……..n) si - Fuzzy setup cost per order for ith item.

0 1

 hdh 0

=  a h2   (b  a)h3  (1   )ch2  (1   )(c  b)h3 1  2 

3

3

Now applying the ranking function(GMIR) to (1)  n D  Q   R  m in T  Qi    min T *  Qi   R   i si  i hi   2   i 1  Qi  

= = =



n

S.to:

D Q  min T  Qi     i si  i hi  2  i 1  Qi n  Ai Qi  B  S.to:  (1) i 1  n  Ci Qi  F   i 1   1 n Qi  N   2 i 1  n

as hi  (h1i , h2i , h3i ) ; si  (s1i , s2i , s3i ) ; B  ( B1 , B2 , B3 ) ; F  ( F1 , F2 , F3 ) ; N  ( N1 , N2 , N3 ) V. RANKING OF TRIANGULAR FUZZY NUMBERS A. The graded mean Integration Representation (GMIR) A triangular fuzzy number A   a, b, c  defined by its

ba

R( x) 

cx ;b  x  c c b

The inverse functions L1 and R1 can be analytically expressed as given below: L1 (h)  a  (b  a)h ; R1 (h)  c  (c  b)h Then the graded mean integration representation (GMIR) of membership function of A is 1

R( A) 

 h  L

1

(h)  (1   ) R 1 (h)  dh

0 1

 hdh



i



i

i

i

i

 n  R   Ai Qi  B    i 1   n  R   Ci Qi  F    i 1  1 n  R   Qi  N    2 i 1  

 (2) (i.e)  A Q  R  B    n

i 1

i

i

 

 C Q  R  F  n

i

i

i 1   1 n Qi  R N   2 i 1 

 

 ( s1i  s3i )  s3i  2s2i 3

 (h1i  h3i )  h3i  2h2i

R( B) 

3  ( B1  B3 )  B3  2 B2

R( F ) 

 ( F1  F3 )  F3  2 F2

R( N ) 

are triangular fuzzy numbers represented

membership function as follows: xa And L( x )  ;a  x  b

n

R(hi ) 

D. Multi- item Fuzzy EOQ model with three constraints

hi , si , B, F and N

  Di   Qi    R  si   R  hi     2  i 1   Qi 

R( si ) 

n

Here



i

Qi  hi  2 

n

Where

  Ci Qi  F   i 1   1 n Q  N  i  2 i 1 i 1



si 

D Q   Q R  s   2 R  h  i 1



S.to:  AiQi  B 

 Di

n

R Q i 1

C. Multi-item Crisp eoq Model with three Constraints n Minimize D Q  T  Qi     i si  i hi  2

 0

=  (a  c)  c  2b

item.

B - Fuzzy maximum average number of units for all items F - Fuzzy maximum investment N - Fuzzy maximum average number of units for all items

 Qi

3

1

h  2  0

hi - Fuzzy holding cost per unit quantity per unit time for ith

i 1

2

2

3

3

 ( N1  N3 )  N3  2 N 2 3

B. Numerical Example: Let h1= Rs.2100; h2=1600; D1=7; D2=17; S1=2100; S2=1300; A1=25 sq.units; A2=25 sq.units; C1=4500; C2=2000; B=250; F=17,000; N=5.

Suppose these are considered as triangular fuzzy numbers,

h1 =(1950,2075,2150) h2 =(1400,1550,1700) s1 =(1800,2100,2175) s2 =(1125,1200,1300) B =(175,225,300) F =(15,000, 18,000, 20,000) N =(3,5,8) Then using GMIR to these fuzzy numbers, R(h1 ) 

 (200)  6300 3

 (300)  4800 R(h )  2

3

 (375)  6375 R( s )  1

3

 (175)  3700 R( s )  2

3

0

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A New Method for Solving Deterministic Multi-Item Fuzzy Inventory Model with Three Constraints (IJSRD/Vol. 3/Issue 09/2015/132)

 (125)  750

R( B) 

S.to: 25Q1  25Q2  229.17

3

4500Q1  2000Q2  17833.33

3

Q1 Q2  5 2 2

 (5000)  56000 R( F )  R( N ) 

 (5)  18

So, R  C* (Q)   min c(Q) = = D1 Q1

min C (Q) 

2



i

i 1



i



D Q   Q R  s   2 R  h 

R  s1  

i

i

i

3

 

 

Q1 D Q R h1  2 R  s2   2 R h2 2 Q2 2

Q1   (200)  6300  17   (175)  3700  Q2   (300)  4800    2    Q    2   3 3 3 2

S.to:

25Q1  25Q2 

4500Q1  2000Q2  17000 Q1 Q2  4 2 2

(3)

 (125)  750

VI. RESULTS AND DISCUSSION

3

The solutions obtained from (1),(2),(3) are given in table 1, 2, 3 respectively.

4500Q1  2000Q2

 (5000)  56000 

Q1 Q2  (5)  18   2 2 3

3

Q1* 2

Q* 4

Min C(Q) 17188.04

Table 1: Results

1) Case (i): when α = 0

14875 Q1 20966.66 Q2 min C (Q)   (2100)   (1600) Q1 2 Q2 2 4500Q1  2000Q2  18666.67 Q1 Q2  6 2 2

Q1* 2

Q* 4

Min C(Q) 17022.94

Table 2: Results Q1* 2

S.to: 25Q1  25Q2  250

Q* 4

Min C(Q) 16889.56

Table 3: Results (1)

2) Case (ii): when α = 0.5 min C (Q) 

14000 Q1 19975 Q2  (2033.33)   (1500) Q1 2 Q2 2

S.to: 25Q1  25Q2  208.33



= 7  (375)  6375  Q1 

(2)

3) Case (iii): when α =1

3

VII. COMPARISON TABLE

14437.5 Q1 20470.89 Q2  (2066.67)   (1550) Q1 2 Q2 2

Model Crisp Crisp Crisp Crisp Crisp Fuzzy Case-1 Fuzzy Case-2 Fuzzy Case-3

h1 2100 2000 1900 2100 2100 2100 2066.67 2033.33

h2 1600 1500 1500 1500 1600 1600 1550 1500

s1 s2 B F N Q1* Q2* Min C(Q) 2100 1300 250 17000 5 2 4 18055.55 2000 1200 260 18000 6 2 4 16585.83 1900 1100 250 17500 7 2 4 15879.50 2000 1200 260 17000 5 2 4 17081.75 1800 1000 250 17500 6 2 4 15506.86 2125 1233.33 250 18666.67 6 2 4 17188.04 2062.5 1204.17 229.17 17833.33 5 2 4 17022.94 2000 1175 208.33 17000 4 2 4 16889.56 Table 4: Comparison Table [2] R.E.Bellman,L.A.Zadeh, “Decision making in fuzzy VIII. CONCLUSION environment” management science, Vol 17(1970) pp B 141-B164.

In this paper, the costs and the constraints are considered as [3] L.Campos and A.Munoz,A subjective approach for imprecise numbers described by fuzzy numbers which are ranking fuzzy sets and system 29(1989) 145-153 more realistic and general in nature. Moreover, the multi [4] Dhanam, K. and Srinivasan, A., (2007). Cost analysis item fuzzy inventory model has been transformed in to crisp on a deterministic multi item EOQ inventory model multi item inventory problem using GMIR ranking method with three constraints in a fuzzy environment , [10].Numerical example shows that by this method we can communicated to journal of combinatorics, information have the optimal total cost. By using GMIR [10] ranking and system sciences. method we have shown that the total cost obtained is [5] P.Fortemps and F.Roubens, ranking and defuzzyfication optimal. Moreover, one can conclude that the solution of methods based on area compensation, Fuzzy sets and fuzzy problems can be obtained by GMIR ranking method Systems 82(1996) 319-330. effectively.

[6] G.Hadley, T.M.Whitin, Analysis of inventory systems, Prentice-Hall, Englewood Cliffs, NJ, 1958.

REFERENCE [7] Katagiri,H.,Ishii,H.; Some inventory problems with [1] S.Abbasbandy and T.Hajjari, A new approach for fuzzy shortage cost. Fuzzy Sets and Systems; ranking of trapezoidal fuzzy numbers, computers and 2000;111(1),87-97 mathematics with applications, vol. 57. 2009, pp. 413[8] H.R.Maleki, ranking functions and their applications to 419. fuzzy linear programming, Far EastJ. Math. sci. (fjms) 43(2002) 183-301.

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A New Method for Solving Deterministic Multi-Item Fuzzy Inventory Model with Three Constraints (IJSRD/Vol. 3/Issue 09/2015/132)

[9] Lin, D.C., Yao, J.S.; Fuzzy economic production for production inventory. Fuzzy Sets and Systems; 2000;111, 465-495 [10] H.Mishmast Nehi and Hamid Hajmohamadi A ranking function method solving fuzzy multi objective linear programming , Annals of fuzzy Mathematics and Informatics Vol x, No x ppl - xx ISSN-2093-9310(5 Jun 2011) www.afmi.or.kr.

[11] A. Nagoor Gani, and S.Abbas, A method for solving intuitionistic fuzzy transportation problem, Applied Mathematical Sciences, Vol 7, 2013, no.28, 1357-1365.

[12] K.S.Park, fuzzy set theoretic interpretation of economic order quantity, IEEE Trans. Systems Man Cybernet.

17(6)(1987) 1082-1084.

[13] E.A.Silver, R.Peterson, Decision systems for Inventory Management and Production Planning, John Wiley, New York, 1985.

[14] G.Sommer, Fuzzy inventory scheduling, in: G.Lasker (Ed.),Applied systems and Cybernetics, Vol.VI, Academic Press. New York,1981.

[15] H.A.Taha, Operations Research an introduction, Prentice-Hall of India (2005) 725-728.

[16] H.J.Zimmermann, “Description and optimization of fuzzy systems,” International Journal of general systems, Vol 214(1976), p.p209-215.

[17] H.J.Zimmermann, “Fuzzy programming and linear programming with several objective functions”, fuzzy sets and systems, Vol (1978) p.p 45-55.

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