Chapter 3 A Fuzzy Inventory Model Without Shortages Using Triangular Fuzzy Number 3.1 Introduction In this chapter, an inventory model without shortages has been considered in a fuzzy environment. Triangular fuzzy numbers have been used to consider the ordering and holding costs. For defuzzification, signed-distance method has been used to, compute the optimum order quantity. In Syed et al[95] EOQ model has been prepared without shortage cost by using triangular fuzzy number, then the total cost has been computed by using signed-distance method. But unfortunately in this paper, the optimal 66
order quantity in fuzzy sense gives the wrong result in calculation, i.e., the formula for q d is not correct. The result in that paper is also not correct. This chapter is an attempt to produce a formula of the optimal order quantity in fuzzy environment, i.e., q d * without considering the shortage cost. Accordingly the associated result in [95] has been changed and the error has been analyzed in the sensitivity analysis part. Also, a new example has been introduced for numerical and graphical presentation. 3.2 Formulation of Inventory Model without Shortages We consider an inventory model, without shortages, both in crisp and fuzzy environment. First, we make a model of inventory problem without shortages in crisp environment and thereafter we consider the same problem in fuzzy environment using fuzzy parameters. Particularly we have paid our attention to ordering and holding costs as uncertain quantities. We have represented them by triangular fuzzy numbers and the total cost is obtained by applying signed distance method for defuzzification. Later on, the result has been compared with the crisp obtained values and the sensitivity analysis has been done on EOQ as well as on total cost. 67
3.2.1 Notations Used The following symbols are used in the inventory models in connection with the model in this chapter: c - carrying or holding cost per unit quantity per unit time; o - ordering or set-up cost per order; T - length of the plan; q - order quantity per cycle; t q length of a cycle; D total demand over the planning time period [0, T]; c fuzzy carrying cost; o fuzzy ordering cost. 3.2.2 Mathematical Model for Inventory in Crisp Environment In this model, the economic lot size is obtained by the following model equation q = 2oD ct, where q t q = D T. (3.1) The total cost for the period [0, T] is given by The crisp optimal solution is: TC = F(q) = carrying cost + ordering cost F q (c, o) = F (q) = ct q 2 + od q. (3.2) 68
Figure 3.1: Variation in quantity q w.r.t time optimal order quantity q = 2oD ct, (3.3) minimum total cost F (q ) = 2ocDT. (3.4) Figure 3.1 is the diagrammatic representation of the above inventory model. Example 3.2.1. Let, o = 20; c = 12; D = 500; T = 6. Then, q* = 16.67, TC = 1200. 3.2.3 Mathematical Model for Inventory in Fuzzy Environment Within this model, we consider the carrying and ordering costs as imprecise quantities and they are represented as triangular fuzzy numbers. The total demand and time of plan are taken as constants. Therefore, the fuzzy total cost becomes F T C = c T q 2 + o D q. (3.5) 69
Suppose c and o denote the fuzzy carrying cost and fuzzy ordering cost which are characterized by the triangular fuzzy numbers (c, c 1, c 2 ) and (o, o 1, o 2 ) respectively in L-R form. F T C = F q ( c, o) = [ c T q 2 ] [ o D q ] = [(c c 1, c, c + c 2 ) ( T q 2 )] [(o o 1, o, o + o 2 ) ( D q )] [ ( c1 T q = F q (c, o) + o ) ( 1D c2 T q, F q (c, o), F q (c, o) + 2 q 2 + o )] 2D, q Applying defuzzification d(f q ( c, o), 0) = F q (c, o) + 1 [( c2 T q + o ) ( 2D c1 T q + o )] 1D 4 2 q 2 q = F q (c, o) + 1 ( T q 4 2 (c 2 c 1 ) + D ) q (o 2 o 1 ) = F d (q). Computation of q d at which F d(q) is minimum F d (q) is minimum at F d (q) =0, where F d (q) is positive F d (q) = d dq F d(q) = ct 2 od q 2 + 1 4 ( ) T (c 2 2 c 1 ) D (o q2 2 o 1 ) = 0. Thus, Also, q d = 2D T ( 4o + o2 o 1 4c + c 2 c 1 ). (3.6) d 2 dq F d(qd) = 2o ( D + 1 ) 2 q 3 4 (c 2 c 1 ) > 0. (3.7) 70
This shows that F d (q) is minimum at q d. Algorithm for finding fuzzy total cost and fuzzy optimal order quantity Step 1: First calculate total cost TC of the crisp model, for given crisp values of c, q, T & D. TC =F q (c, o) = F(q) = ct q 2 + od q. Step 2: Next calculate fuzzy total cost FTC by applying fuzzy arithmetic operations on the fuzzy carrying cost c and fuzzy ordering cost o, considered as triangular fuzzy numbers. F T C = F q ( c, o) = [ ( c1 T q F q (c, o) + o ) ( 1D c2 T q, F q (c, o), F q (c, o) + + o )] 2D 2 q 2 q Step 3: Now to defuzzify F q, ( c, o) apply ranking of fuzzy numbers then find fuzzy optimal order quantity q d (eqn. 3.6), which can be obtained by putting the first derivative of F d (q) equal to zero and at this value of q d, the second derivative of F d (q) is positive. 3.2.4 Numerical illustration Example 3.2.2. In crisp environment Let o = Rs 20 per unit, c = Rs 12 per unit, D = 500 units, T = 6 days. Then, the economic order quantity q* = 16.67 units and total cost TC = Rs 1200 In fuzzy environment 71
Table 3.1: Sensitivity analysis of example 3.2.2 S. No. Demand qd TC 1 450 16.07 1138.6 2 475 16.51 1169.8 3 500 16.95 1200.2 4 525 17.36 1229.8 5 550 17.78 1258.8 Table 3.2: Error analysis of example 3.2.2 S. No. Demand qd TC Old New %age error Old New %age error 1 450 15.56 16.07.0317 1214.16 1138.6.0663 2 475 15.99 16.51.0315 1247.43 1169.8.0664 3 500 16.73 16.95.0130 1279.00 1200.2.0663 4 525 16.81 17.36.0316 1311.41 1229.8.0664 5 550 17.20 17.78.0326 1342.31 1258.8.0663 Let o = (20, 4, 5), c = (12, 2, 1), D = 500 days, T = 6 days. Then, qd = 16.95, TC = 1200.2 The sensitivity analysis is done in tables 3.1 and error analysis with respect to [95] in table 3.2. Example 3.2.3. A company uses 24000 units of raw material which costs 1.25 rupees per unit. Placing each order costs rupees 2.5 and carrying cost 5.4% per year of the inventory. Find EOQ and the total inventory in both crisp and fuzzy environments. Solution: 72
Table 3.3: Sensitivity analysis of example 3.2.3 S. No. Demand qd TC 1 23250 897.1157 902.2051 2 23500 901.9260 907.0427 3 23750 906.7108 911.8547 4 24000 911.5160 916.6410 5 24250 916.2054 921.4030 6 24500 920.9160 925.8011 7 24750 925.6489 930.8384 8 25000 930.2656 935.5432 In crisp environment: o = 22.5, c = 1.25 5.4% = 0.675, D = 24000, T = 1. We obtain, q = 4000 units & TC = 853.82 rupees. In fuzzy environment: o = 22.5, c = 0.675, D = 24000, T = 1, o 1 = 4.5, o 2 = 6, c 1 =.008, c 2 =.009. By the signed distance method we obtain, q d * = 911.5160, TC = 916.6410. The graphical representation for above set of values is done in figure 3.2 and sensitivity analysis is done in table 3.3. 73
3.3 Observations Figure 3.2: Variation in Demand and EOQ - The economic order quantity obtained by signed-distance method is closer to crisp economic order quantity. - Total cost obtained by signed distance method is more than crisp cost. - Economic order quantity is more sensitive to demand. - Total cost increases as demand increases. 3.4 Conclusion In this chapter, the EOQ i.e., q* has been calculated in both crisp and fuzzy environment. First in crisp sense, the EOQ has been calculated from inventory model and later on the same has been calculated in fuzzy sense. By using triangular fuzzy number, defuzzification has been done and the 74
relative changes have been computed. It is seen that q* increases/decreases with small amount which is under consideration and is within our level of expectation. 75