 American Journal of Oper ations Research, 2011, 1, 155-159 doi:10.4236/ajor.2011.13017 Published Online September 2011 (http://www.SciRP.org/journal/ajor) Copyright © 2011 SciRes. AJOR Multi-Item Fuzzy Inventory Model Involving Three Constraints: A Karush-Kuhn-Tucker Conditions Approach R. Kasthuri, P. Vasanthi, S. Ranganayaki*, C. V. Seshaiah Department of Mathematics, Sri Ramakrishna Engineering College, Coimbatore, India E-mail: {kasthuripremkumar, *rakhul11107}@yahoo.com, {vasdev1066, cvseshaiah}@gmail.com Received July 7, 2011; revised July 22, 2011; accepted August 22, 2011 Abstract In this paper, a multi-item inventory model with storage space, number of orders and production cost as con-straints are developed in both crisp and fuzzy environment. In most of the real world situations the cost pa-rameters, the objective functions and constraints of the decision makers are imprecise in nature. This model is solved with shortages and the unit cost dependent demand is assumed. Hence the cost parameters are im-posed here in fuzzy environment. This model has been solved by Kuhn-Tucker conditions method. The re-sults for the model without shortages are obtained as a particular case. The model is illustrated with numeri-cal example. Keywords: Multi-Item Inventory Model, Membership Function, Karush-Kuhn-Tucker Condition 1. Introduction The literal meaning of inventory is the stock of goods for future use (production/sales). The control of inventories of physical goods is a problem common to all enterprises in any sector of an economy. The basic objective of in-ventory control is to reduce investment in inventories and ensuring that production process does not suffer at the same time. In general the classical inventory problems are de-signed by considering that the demand rate of an item is constant and deterministic and that the unit price of an item is considered to be constant and independent in na-ture. But in practical situation, unit price and demand rate of an item may be related to each other. When the demand of an item is high, an item is produced in large numbers and fixed costs of production are spread over a large number of items. Hence the unit cost of the item decreases. .i.e., the unit price of an item inversely relates to the demand of that item. So demand rate of an item may be considered as a decision variable. Zadeh  first gave the concept of fuzzy set theory. Later on, Bellman and Zadeh  used the fuzzy set the-ory to the decision-making problem. Zimmerman  gave the concept to solve multi objective linear pro-gramming problem. Fuzzy set theory has made an entry into the inventory control systems. Sommer  applied the fuzzy concept to an inventory and production sched-uling problem. Park  examined the EOQ formula in the fuzzy set theoretic perspective associating the fuzzi-ness with the cost data. Hence we may impose ware-house space, cost parameters, number of orders, produc-tion cost etc, in fuzzy environment. The Kuhn-Tucker conditions  are necessary condi-tions for identifying stationary points of a non linear constrained problem subject to inequality constraints. The development of this method is based on the La-grangean method. These conditions are also sufficient if the objective function and the solution space satisfy the conditions in the following Table 1. The conditions for establishing the sufficiency of the Kuhn-Tucker conditions  are summarized in the fol-lowing Table 2. Kuhn-Tucker conditions, also known as Karush-Kuhn- Tucker (KKT) conditions was first developed by W. Ka-rush in 1939 as part of his M.S. thesis at the University of Chicago. The same conditions were developed inde-pendently in 1951 by W. Kuhn and A. Tucker. In this paper, a multi-item, multi-objective inventory problem with shortages along with three constraints such as limited storage space, number of orders and produc-tion cost has been formulated. The unit cost is considered here in fuzzy environment. The problem has been solved by KKT conditions method. This model is illustrated by numerical example [8-11]. 156 R. KASTHURI ET AL. Table 1. The objective function and the solution space. Required conditions Sense of optimization Objective function Solution space Maximization Concave Convex Set Minimization Convex Convex Set Table 2. The sufficiency of the Kuhn-Tucker conditions. Problem Kuhn-Tucker conditions 1) Max z = f(X) subject to hi(X) ≤ 0 X ≥ 0, 1, 2,,im  100,0,1,2,,0,1,2,,miiijjiiiifX hXxxhXhX imim 2) Min z = f(X) subject to hi(X) ≥ 0 X ≥ 0, 1, 2,,im  100,0,1,2,,0,1,2,,miiijjiiiifX hXxxhXhX imim 2. Assumptions and Notations A multi-item, multi-objective inventory model is devel-oped under the following notations and assumptions. 2.1. Notations n = number of items t = number of orders W = Floor (or) shelf-space available B = Total investment cost for replenishment For ith item: (i = 1, 2, , n) Di = Di (pi) demand rate [function of unit cost price] Qi = lot size (decision variable) Mi = Shortage level (decision variable) Si = Set-up cost per cycle Hi = Inventory holding cost per unit item mi = Shortage cost per unit item pi = price per unit item (decision variable) wi = storage space per item TC(p, Q, M) = expected annual total cost 2.2. Assumptions 1) replenishment is instantaneous 2) lead time is zero 3) demand is related to the unit price as iiiiiiADApip where Ai (>0) and βi (0 < βi < 1) are constants and real numbers selected to provide the best fit of the estimated price function. Ai > 0 is an obvious condition since both Di and pi must be non-negative. 3. Formulation of Inventory Model with Shortages Let the amount of stock for the ith item (i = 1, 2, , n) be Ri at time t = 0. In the interval (0, Ti(= t1i + t2i)), the inventory level gradually decreases to meet demands. By this process the inventory level reaches zero level at time t1i and then shortages are allowed to occur in the interval (t1i, Ti). The cycle then repeats itself (Figure 1). The differential equation for the instantaneous inven-tory qi(t) at time t in (0, Ti) is given by 11for 0foriiiidq tDtdtDttTiit (1) with the initial conditions qi(0) = Ri(= Qi – Mi), qi(Ti) = –Mi, qi(t1i) = 0. For each period a fixed amount of shortage is allowed and there is a penalty cost mi per items of unsatisfied demand per unit time. From (1) 111for 0.foriii iii iqtRDtt tDtttt T i So Dit1i = Ri, Mi = Dit2i, Qi = DiTi Holding cost = 120d2itiiiiiiHQ MiHqt tTQ Shortage cost = 12d2iiTiiiiitmMmqtt TQi Production cost = piQi 22The total costProduction costSet up costHolding costShortage cost22ii iiii iiiiiiQM mMpQ SHTTQQ  The total average cost of the ith item is  22,, 22iiiiii iii iiiiiHQ MSD mMTCpQMp DQQ iQ 122,,22iiiiiii iiiiiii iiiiiASTCpQMA ppQHQ MmMQQ (2) for 1,2, 3,,in. Copyright © 2011 SciRes. AJOR R. KASTHURI ET AL. Copyright © 2011 SciRes. AJOR 157 Figure 1. Inventory level of the ith item. There are some restrictions on available resources in inventory problems that cannot be ignored to derive the optimal total cost. 1) There is a limitation on the available warehouse floor space where the items are to be stored i.e. 1niiiwQ W; 2) Investment amount on total production cost cannot be infinite, it may have an upper limit on the maximum investment. i.e. ; 1niiipQB3) An upper limit on the number of orders that can be made in a time cycle on the system (i.e.) 1niiiDtQ. The problem is to find price per unit item, the lot size, the shortage amount so as to minimize the total average cost function (2) subject to the total space and total pro-duction cost restrictions. It may be written as Min TCi(pi, Qi, Mi) for all 1,2,3,,in. Subject to the inequality constraints 1niiiwQW 1niiipQB 1niiiDtQ 4. Fuzzy Inventory Model with Shortages When sip are fuzzy decision variables, the above crisp model under fuzzy environment reduces to 1122,, 22iiniiii iiiii iiiiiASMinTCpQMA ppQHQ MmM tQQ subject to the constraints 1niiiwQ W 1niiipQB 111iniiiipAQt [Here cap ‘~’ denotes the fuzzification of the parame-ters.] The above fuzzy non-linear programming can be solved using Kuhn-Tucker conditions. 4.1. Membership Function The membership function for the fuzzy variable pi is de- fined as follows 1,,0,iiiiiiiiLLipLLLiLpLUpXLpULpUiiLU Here ULi and LLi are upper limit and lower limit of pi respectively. 158 R. KASTHURI ET AL. 4.2. Fuzzy Inventory Model without Shortages When sip are fuzzy decision variables, the above crisp model without shortages under fuzzy environment re-duces to 11Min ,2iiniii iii iiiASHTCpQA ppQQ subject to the constraints 1niiiwQW 1niiipQB 11iniiiipAQ t 5. Numerical Example To solve the above non-linear programming using Kuhn- Tucker conditions, the following values are assumed. n = 1, t = 3 A1 = 100, S1 = \$100, H1 = \$1, w1 = 2 sq. ft, W = 150 sq. ft, B = \$1200, m1 = \$1 and \$10 ≤ p1 ≤ \$20 By the method of Kuhn-Tucker conditions, consider the four cases 1) λ1 = 0, λ2 = 0 2) λ1 ≠ 0, λ2 = 0 3) λ1 = 0, λ2 ≠ 0 4) λ1 ≠ 0, λ2 ≠ 0 Here Kuhn-Tucker conditions are used as trial and er-ror method by taking different values for β1 until an op-timum result is obtained. Optimal solutions for the fuzzy model with shortages β1 p1 µP1 value Q1 D1 M1 Expected Total cost0.88 10.179 0.982 72.04612.978 36.023168.1330.89 12.406 0.759 65.22010.85 32.609164.4840.90 15.405 0.4595 58.4288.533 29.214160.6640.91 19.561 0.0439 51.6936.681 25.847156.533 From the above table it follows that 10.179 has the maximum membership value 0.982. Hence the required optimum solution is p1 = 10.179, Q1 = 72.046, M1 = 36.023 Minimum expected Total cost = \$168.133. Optimal solutions for the fuzzy model without short-ages β1 p1 µP1 value Q1 D1 Expected Total cost0.89 10.788 0.921 75.000 12.042 183.4600.85 11.178 0.882 50.693 12.850 194.3270.92 16.00 0.400 75.000 07.802 172.733 From the above table it follows that 10.788 has the maximum membership value 0.921. Hence the required optimum solution is p1 = 10.788, Q1 = 75 Minimum expected Total cost = \$183.460. 6. Conclusions In this paper we have proposed a concept of the optimal solution of the inventory problem with fuzzy cost price per unit item. Fuzzy set theoretic approach of solving an inventory control problem is realistic as there is nothing like fully rigid in the world. By solving the above fuzzy inventory model using Kuhn-Tucker condition method we have the values of imprecise variable for decision making. The above discussed model can be developed with many limitations, such as their inventory level, Warehouse space and budget limitations, etc. 7. References  L. A. 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