PAijpam.eu ON FUZZY INVENTORY MODEL WITH ALLOWABLE SHORTAGE

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1 International Journal of Pure and Applied Mathematics Volume 99 No. 205, 5-7 ISSN: (printed version; ISSN: (on-line version url: doi: PAijpam.eu ON FUZZY INVENORY MODEL WIH ALLOWABLE SHORAGE D. Stephen Dinagar, J. Rajesh Kannan 2 PG. and Research Department of Mathematics.B.M.L College Porayar, INDIA 2 Department of Basic Engineering S.S.P. College Puthur, INDIA Abstract: In this paper, the fuzzy inventory model with allowable shortage has been considered in a fuzzy environment using the new hexagonal fuzzy numbers. Our goal is to determine the fuzzy optimal total cost and fuzzy optimal order quantity for the proposed inventory model. Holding cost, Ordering cost, Shortage cost and Demand are taken as in terms of Hexagonal fuzzy numbers. New arithmetic operations are defined and applied in sensitivity analysis. A relevant numerical example is also included, to justify the proposed notion. AMS Subject Classification: 03E72, 90B05 Key Words: hexagonal fuzzy numbers, fuzzy inventory model, fuzzy optimal total cost, fuzzy optimal order quantity, allowable shortage. Introduction Inventory model was developed by Harris [3], Wilson [9] aroused interest in the EOQ model in academics and industries. In 970, Zadeh et al. [] proposed some strategies for decision making in fuzzy environment. Jain [4] worked on decision making in the presence of fuzzy decision making models. hus, EOQ Received: July 3, 204 c 205 Academic Publications, Ltd. url:

2 D.S. Dinagar, J.R. Kannan model serves a useful approximation to many real life problems. Urgeletti [8] treated EOQ model in fuzzy sense, and used triangular fuzzy number. Chen and Wang [] used trapezoidal fuzzy number to fuzzify the order cost, inventory cost, and back order cost in the total cost of inventory model without backorder. Further, in a series of papers, Yao et al. [0], considered the fuzzified problems for the inventory with or without backorder models. Kacpryzk et al. [5] discussed some long-term inventory policy making through the fuzzydecision making models. Later, Hadley et al [2] analyzed many inventory systems. Parvathi and Gajalakshmi [] investigate a deterministic inventory model with allowable shortage. In this paper, a new fuzzy number called Hexagonal fuzzy numbers is utilized in developing the notion of Inventory model. hough Rajarajesvari et.al[7] proposed the above said fuzzy number without any restrictions of parameter, We have modified the definition of the number by including conditions for the convexity of the number and few more results are also included in the work. he main aim of the authors is to estimate the fuzzy optimal order quantity and fuzzy optimal total cost of an inventory system under study due to irregularities or physical properties of the parameter variables. For this situation we apply fuzzy concepts, shortage is allowed and it is completely backlogged. An algorithm is developed to find the fuzzy total cost. Sensitivity analysis is carried out through the numerical examples. In this article, in Section 2, some basic definitions and arithmetic operations on hexagonal fuzzy numbers are presented. In Section 3, we describe in brief the notions and assumptions used in the developed Fuzzy inventory model Formulation and Analysis of the inventory model in fuzzy sense and algorithm are presented. In Section 4, a numerical example is given to illustrate the model and sensitivity analysis has been made for different changes in the parameter values. In Section 5, the concluding remarks are also given. 2. Definitions and Preliminaries Definition 2. (Fuzzy Set. A Fuzzy set à in a universe of discourse x is defined as the following set of pairs à = {(x,µ à (x : x X}. Here µã : X [0,] is a mapping called the membership value of x X in a fuzzy set Ã. Definition 2.2 (Convex Fuzzy Set. A fuzzy set à = {(x,µ à (x} X is called convex fuzzy set if all à x are convex sets i.e. for every element x A α and x 2 A α for every α [0,]λx +( λx 2 A α λ [0,]. Otherwise the

3 ON FUZZY INVENORY MODEL... 7 fuzzy set is called non convex fuzzy set. Definition 2.3 (Hexagonal Fuzzy Number. A fuzzy number on Ãh is a Hexagonal fuzzy number denoted by Ãh = (a,a 2,a 3,a 4,a 5,a where (a a 2 a 3 a 4 a 5 a are real numbers Satisfying a 2 a a 3 a 2 and a 5 a 4 a a 5 and its membership function µãh (x is given as; Remark , x < a ( x a 2 a 2 a, a x a 2 ( 2 + x a2 2 a 3 a 2, a 2 x a 3 µãh (x =, a 3 x a 4 ( x a4 2, a 4 x a 5 ( a x 2 a a 5, a 5 x a 0, x > a. a 5 a 4 ( he Hexagonal fuzzy numbers Ãh becomes trapezoidal fuzzy numbers if a 2 a = a 3 a 2 and a 5 a 4 = a a 5 (2 he Hexagonal fuzzy numbers Ãh becomes non-convex fuzzy numbers if a 2 a > a 3 a 2 and a 5 a 4 < a a 5 Definition 2.5 (Equality of wo Hexagonal Fuzzy Numbers. wo Hexagonal fuzzy numbers à = (a,a 2,a 3,a 4,a 5,a and B = (b,b 2,b 3,b 4,b 5,b are said to be equal i.e. à = B if and only if a = b,a 2 = b 2,a 3 = b 3,a 4 = b 4,a 5 = b 5,a = b. Definition 2. (Symmetric Hexagonal fuzzy number. A fuzzy number à = (a,a 2,a 3,a 4,a 5,a is said to be a symmetric Hexagonal fuzzy number, if a 3 a = a a 4. Otherwise the fuzzy number is called Non symmetric fuzzy number. Definition 2.7. A Hexagonal fuzzy number à = (a,a 2,a 3,a 4,a 5,a is said to be none negative (non positive i.e. à 0 (à 0 if and only if a 0 (a 0. Definition 2.8 (New Arithmetic Operations. he new arithmetic operations between hexagonal fuzzy numbers proposed are given below. Let us consider à = (a,a 2,a 3,a 4,a 5,a and Ã2 = (b,b 2,b 3,b 4,b 5,b be two hexagonal fuzzy numbers. hen,

4 8 D.S. Dinagar, J.R. Kannan able : Different types of hexagonal fuzzy numbers. ypes of Hexagonal Pictorial fuzzy number Conditions representations à = (a,a 2,a 3,a 4,a 5,a Hexagonal fuzzy numbers à h becomes trapezoidal fuzzy numbers a 2 a = a 3 a 2 a 5 a 4 = a a 5 0 a a 2 a 3 a 4 a 5 a A Convex Hexagonal fuzzy numbers Ãh a 2 a a 3 a 2 a 5 a 4 a a 5 0 a a 2 a 3 a 4 a 5 a A Non convex Hexagonal fuzzy numbers Ãh a 2 a > a 3 a 2 a 5 a 4 < a a 5 0 a a 2 a 3 a 4 a 5 a A Symmetric Hexagonal fuzzy numbers Ãh a 3 a = a a 4 0 a a 2 a 3 a 4 a 5 a A (i he addition of à and Ã2 is à (+Ã2 = (a +b,a 2 +b 2,a 3 +b 3,a 4 +b 4,a 5 +b 5,a +b. (ii he subtraction of à and Ã2 is à ( Ã2 = (a b,a 2 b 5,a 3 b 4,a 4 b 3,a 5 b 2,a b. (iii he multiplication of à and Ã2 is

5 ON FUZZY INVENORY MODEL... 9 able : Continuation: Different types of hexagonal fuzzy numbers. Non symmetric Hexagonal fuzzy numbers type- (l s > r s a 3 a > a a 4 0 a a 2 a 3 a 4 a 5 a A Non symmetric Hexagonal fuzzy numbers type-2 (l s < r s a 3 a < a a 4 0 a a 2 a 3 a 4 a 5 a A Perfect symmetric Hexagonal fuzzy numbers a 2 a = a 3 a 2 a 3 a 2 = a 4 a 3 a 4 a 3 = a 5 a 4 a 5 a 4 = a a 5 0 a a 2 a 3 a 4 a 5 a A à ( Ã2 = ( a σ b, a 2 σ b, a 3 σ b, a 4 σ b, a 5 σ b, a σ b, where σ b = (b +b 2 +b 3 +b 4 +b 5 +b. (iv he division( of à and Ã2 is à ( Ã2 = a σ b, a 2 σ b, a 3 σ b, a 4 σ b, a 5 σ b, a σ b if σ b 0 where σ b = (b +b 2 +b 3 +b 4 +b 5 +b. (v If k { 0 is a scalar kã is defined as kã = (ka,ka 2,ka 3,ka 4,ka 5,ka, if k > 0 (ka,ka 5,ka 4,ka 3,ka 2,ka, if k < 0 (vi à = a,a 2,a 3,a 4,a 5,a = ( a, a 2, a 3, a 4, a 5, a. where a,a 2,a 3,a 4,a 5,a are non zero positive real numbers. Definition 2.9. We define a ranking function R : f(r R which maps each fuzzy numbers to the real line: f(r represents the set of all hexagonal

6 70 D.S. Dinagar, J.R. Kannan fuzzy numbers. ( If R be any linear ranking function, then R(à = a +a 2 +a 3 +a 4 +a 5 +a. Definition 2.0 (Equivalent Hexagonal Fuzzy Numbers. A fuzzy number à is said to be equivalent to a fuzzy number B if its value of the Ranking function are the same. i.e. à B if R(à = R( B. 3. Fuzzy Inventory Model 3.. Notations We define the following symbols: C h : Fuzzy holding cost per unit quantity per unit time C a : Fuzzy Setup cost (or ordering cost per order : Length of the plan C s : Fuzzy Shortage cost per unit quantity D t : Fuzzy otal demand over the planning time period [0,] Q: Fuzzy Order quantity per cycle c : Fuzzy total cost for the period [0,] F(Q : Minimum Fuzzy total cost for [0,] Q d : Fuzzy Optimal order quantity 3.2. Assumptions In the present paper, the assumptions considered are as follows: (i Shortage cost is fuzzy nature (ii otal demand is fuzzy nature (iii ime plan is constant (iv Holding cost, Ordering cost are fuzzy in nature

7 ON FUZZY INVENORY MODEL Formulation and Analysis of the Model We consider the model with allowable shortage in fuzzy environment since the fuzzy total demand with time period [0,], fuzzy ordering cost per order fuzzy carrying cost (or holding cost per unit quantity per unit time and fuzzy shortage cost per unit quantity are in fuzzy nature. Our goal is to determine the fuzzy optimal total cost and fuzzy optimal order quantity for the proposed fuzzy inventory model. Now we fuzzifying total cost is given by c = H Q S 2 2( H + S 2 + S Q ( 2 S H + S 2 + Ã D Q. Our aim is to obtain fuzzy total cost and the optimal order quantity in terms of hexagonal fuzzy numbers by using simple calculus techniques. If C h, C a, C s, D t denote the hexagonal fuzzy numbers defined by C h = (C h,c h2,c h3,c h4,c h5,c h D t = (D t,d t2,d t3,d t4,d t5,d t C a = (C a,c a2,c a3,c a4,c a5,c a C s = (C s,c s2,c s3,c s4,c s5,c s ( C h c = Q( C C s Q s 2 2 ( Ch + C 2 + s C s C h + C s C a Dt Q. By using new arithmetic operations and simplifying we get

8 72 D.S. Dinagar, J.R. Kannan where Q C s a b + Q ( (Ch +C s C s a c 2 d Q C s 2 a b + Q ( (Ch2 +C s2 C s5 a c Q D t, d Q D t 2, Q C s 3 a b + Q ( (Ch3 +C s3 C s4 a + c 2 d Q D t 3, C = Q C s 4 a b + Q ( (Ch4 +C s4 C s3 a + c 2 d Q D t 4, Q C s 5 a b + Q ( (Ch5 +C s5 C s2 a + c 2 d Q D t 5, Q C s a b + Q ( (Ch +C s C s a + c 2 d Q D t c = (a,a 2,a 3,a 4,a 5,a = F( Q ( a = (C s +C s +C s +C s +C s +C s, b = (C h +C h2 +C h3 +C h4 +C h5 +C h [ Ch +C s c = (C h +C s + +C h +C s C ] h +C s (C h +C s + +C h +C s 3 d = (C h +C s + +C h +C s, ( (C h +C s C s = (C h +C s + +C h +C s (C h +C s C s + + (C h +C s + +C h +C s = (C a +C a2 +C a3 +C a4 +C a5 +C a, g = (Q d +Q d2 +Q d3 +Q d4 +Q d5 +Q d. he fuzzy optimal order quantity Q d which is minimize the total inventory cost c = F( Q is obtained as the solution of the first order fuzzy differential

9 ON FUZZY INVENORY MODEL equation d d Q ( c = 0 and it is found as 2D t Q d = a 2D t4 a + Dt b, 2Dt 2 a + Dt 4 b, 2Dt 5 a + Dt 2 b, 2Dt 3 a + Dt 5 b, 2Dt a + Dt 3 b, + Dt b. Q d = (Q d,q d2,q d3,q d4,q d5,q d by using Definition 2.8(vi. Also Q = Q d we have d2 F( Q d Q > 0. 2 his shows that F( Q is minimum at Q = Q d and from (. F(Q = 3 C s a b g + ( (Ch +C s C s a g c 72 d 3 C s 2 a b g + ( (Ch2 +C s2 C s5 a g c 72 3 C s 3 a b g c C s 4 a b g c C s 5 a b g c C s a b g c + 72 d ( (Ch3 +C s3 C s4 a g d ( (Ch4 +C s4 C s3 a g d ( (Ch5 +C s5 C s2 a g d ( (Ch +C s C s a g d + D t g, + D t 2 g, + D t 3 g, + D t 4 g, + D t 5 g, + D t g Algorithm for Finding Fuzzy Optimal otal Cost and Fuzzy Optimal Order Quantity Step : Calculate the model fuzzy total cost for the fuzzy values of C h, C a, C s, Dt. Step 2: Now determine fuzzy total cost using new arithmetic operations fuzzy holding cost, fuzzy ordering cost, fuzzy shortage cost and fuzzy demand taken in terms of hexagonal fuzzy numbers Step 3: Find the fuzzy optimal order quantity which can be obtain by putting the first derivative of F( Q equal to zero and second derivate is positive at Q = Q d.

10 74 D.S. Dinagar, J.R. Kannan able 2: Sensitivity analysis. No Demand Ca = (4,5,9,2,25,2 Ca = (4,5,8,22,25,2 ( D t Q d F( Q Q d F( Q (200, 300, 450, 500, 00, 50 2 (225, 325, 475, 525, 25, 75 3 (250, 350, 500, 550, 50, (275, 375, 525, 575, 75, (300, 400, 550, 00, 700, 750 (38.00, 4.54, 57.00, 0.09, 5.82, 8.5 (40.3, 48.44, 58.57,.57, 7.8, 9.82 (42.49, 50.27, 0.09, 3.02, 8.53, 7.0 (44.5, 52.04,.57, 4.44, 9.82, 72.3 (4.54, 53.74, 3.02, 55.82, 7.0, (25.8, 78.93, , , 454.7, (33.39, 8., 3, 3.3, 43.79, (4.25, 94.0, 34.58, 39.2, 473.2, (48.82, 20.29, , 377.9, , (5., , , 38.0, 49., (38.00, 4.54, 57.00, 0.09, 5.82, 8.5 (40.3, 48.44, 58.57,.57, 7.8, 9.82 (42.49, 50.27, 0.09, 3.02, 8.53, 7.0 (44.5, 52.04,.57, 4.44, 9.82, 72.3 (4.54, 53.74, 3.02, 55.82, 7.0, (25.8, 78.93, , , 454.7, (33.39, 8., 3, 3.3, 43.79, (4.25, 94.0, 34.58, 39.2, 473.2, (48.82, 20.29, , 377.9, , (5., , , 38.0, 49., Let Ch = (,7,,3,7,8, C a = (4,5,9,2,25,2, C s = (,2,5,7,0,, 4. Numerical Example 4.. Fuzzy Model D t = (250,350,500,550,50,700, = days then Q d = (42.49,50.27,0.09,3.02,8.53,7.0 F( Q = (4.25,94.0,34.58,39.2,473.2, From able 2 we observed that (i he fuzzy economic order quantity obtained is very closer to crisp economic order quantity. (ii he fuzzy total cost is very closer to crisp total cost. (iii For different values of fuzzy ordering quantity by changing middle two

11 ON FUZZY INVENORY MODEL spreads, the fuzzy economic order quantity remains fixed. he same is true for fuzzy total cost. 5. Conclusion In this paper, we have studied fuzzy optimal order quantity and fuzzy optimal total cost with aid of hexagonal fuzzy number. o estimate various fuzzy optimal quantities, the demand, holding cost, ordering cost and shortage cost using hexagonal fuzzy numbers have been utilized. A new arithmetic operations of hexagonal fuzzy numbers are proposed to get the expected result. Also it is observed that the fuzzy estimates all closer to the crisp estimates of the real systems. References [] Wang Chan, Backorder fuzzy inventory model under function principle, Information Science, 95 (99, 2, [2] G. Hadley and.m. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 93. [3] F. Harris, Operations and Cost, AW Shaw Co. Chicago, (95. [4] R. Jain, Decision making in the presence of fuzzy variables, IIIE ransactions on Systems, Man and Cybernetics, 7 (97, [5] J. Kacpryzk and P. Staniewski, Long-term inventory policy-making through fuzzy-decision making models, Fuzzy Sets and Systems, 8 (982, [] P. Parvathi and S. Gajalakshmi, An Inventory Model with Allowable Shortage Using rapezoidal Fuzzy Numbers, International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-203, ISSN [7] P. Rajarajesvari and A. SahayaSudha, A New operation on hexagonal fuzzy number, Fuzzy Logic Systems, 3 ( [8] G. Urgeletti inarelli, Inventory control models and problems, European Journal of Operational Research, 4 (983, 2.

12 7 D.S. Dinagar, J.R. Kannan [9] R. Wilson, A scientific routine for stock control, Harvard Business Review, 3 (934, 28. [0] J.S. Yao and J. Chiang, Inventory without back order with fuzzy total cost and fuzzy storing cost defuzzified by centroid and signed distance, European Journal of Operational Research, 48 (2003, [] L.A. Zadeh and R.E. Bellman, Decision Making in a Fuzzy Environment, Management Science, 7 (970, 40.