An inventory model of flexible demand for price, stock and reliability with deterioration under inflation incorporating delay in payment

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1 ISSN (Online) : ISSN (Print) J.Mech.Cont.& Math. Sci., Vol.-13, No.-5, November-December (2018) Pages An inventory model of flexible demand for price, stock and reliability with deterioration under inflation incorporating delay in payment Sudip Adak, G.S. Mahapatra Department of Mathematics, National Institute of Technology Puducherry, Karaikal , India Tel , Fax: g_s_mahapatra@yahoo.com Abstract This paper presents an inventory model for deteriorating items with a constant rate of deterioration and the demand rate is flexible which depends on the price, stock as well as the reliability of the products. This model allowing the shortage under inflation, and delay in payment is also taken into account. We consider situation of the credit period is less than or greater than the cycle time for settling the account. Numerical example is given for different cases and sensitivity analysis is carried out to analyze the effect of the parameters on the optimal solution. Keywords : Deterioration, Reliability, Credit period, Inflation, Delay payment. I. Introduction The study of inventory management is spread over all fields of research of science, engineering, management and commerce. The inventory control till date, has been considering the various topics of real phenomenon such as supply-demand relation, the natural fact of deterioration of items, settlement of account with in or extra time period etc. The traditional inventory model assumes the demand rate to be constant. But in real life, demand rate is not a fixed quantity and should change in dynamical nature. Keeping in view the rapid change of environment and technology, all such parameters are vital for consideration of the study of inventory control. Deterioration of items due to time, storage, transportation etc. is a natural phenomenon, and it is unavoidable. The study of deterioration is very important due to different deterioration rates of many products. Misra [XXIV] developed an economic order quantity (EOQ) model with Weibull deterioration rate incorporating inflationary effects. Elsayed and Teresi [V] presented inventory systems with deteriorating items. Guria et al. [I] studied an inventory policy for an item with inflation induced purchasing price, selling price and demand with immediate part payment. Chung and Ting [XVIII] developed a heuristic EOQ for replenishment for deteriorating items with a linear trend in demand. Benkherouf and Balkhi [II] 127

2 proposed an inventory model for deteriorating items with time-varying demand. Liao et al. [XIII], Chang and Dye [XIV], Chung and Lin [XVII], Manna and Chaudhuri [XXXV], Widyadana and Wee [VII]presented inventory models for deteriorating items. Sicilia et al. [XXII] investigated an inventory model where backordered demand ratio is exponentially decreasing with the waiting time. Liao and Huang [XX] presented an inventory model for deteriorating items with two levels of trade credit taking account of time discounting. Janakiram et al. [VI] developed a comparison of the optimal costs of two canonical inventory systems. Pal et al. [XXVIII] presented EOQ model with Weibull deterioration under inflation and finite horizon. Mahapatra et al. [XII] considered an inventory model for deteriorating items with time and reliability dependent demand and partial backorder. In this paper, we present deterioration on the on-hand inventory per unit time and there is no repair of the deterioration item within the cycle, in classical inventory model, it is generally assumed that the demand rate is independent of factors like stock availability, price of items and reliability of the products etc. However, in practical practice, it is observed that the demand for certain items is heavily influenced by the stock level, price as well as reliability of items. Levin et al. [XXV] proposed that selling of items is proportional to the inventory displayed such that large piles of goods displayed in a supermarket will tempt the customer to buy more. Pal et al. [XXVII] presented a deterministic inventory model for deteriorating items with stock dependent demand rate. Along with this track, several studies where been carried out and some preliminary findings were reported by Chen [XXI], Wee and Law [XV], Jin and Liao [XXXVII], Kim et al. [XXXIII], Roy and Chaudhuri [XXXVIII], Mahapatra et al. [XI], and Pal et al. [XXIX]. Pal et al. [XXX] studied production inventory model for ramp type demand allowing inflation and shortages. Pal et al. [XXXI] presented economic production quantity (EPQ) model for price and stock dependent stochastic demand. Pal and Mahapatra [XXXII] studied production inventory model of stochastic demand. Tripathy et al. [XXIII] studied an EOQ model with process reliability. Shah and Soni [XVI] developed a multi-objective production inventory model with backorder for fuzzy random demand under flexibility and reliability. Mahapatra et al. [IX] presented fuzzy EPQ model under flexibility and reliability consideration. Mahapatra et al. [VIII] introduced production inventory model with fuzzy coefficients using parametric geometric programming approach. Mahapatra et al. [X] studied an EPQ model with imprecise space constraint based on intuitionistic fuzzy optimization technique. In many practical situations, the supplier allows the inventory manager a certain fixed period oftime to settle the accounts. No interest is charged during this period but beyond it, the manager has to pay an interest to the supplier. Goyal [XXXIV] first developed the EOQ model under conditions of permissible delay in payment. Jaggi et al. [IV] discussed the optimal replenishment and credit policy in EOQ model under two-levels of trade credit policy when demand is influenced by credit period. Chung and Wee [III] studied Scheduling and replenishment plan for an integrated deteriorating inventory model with stock dependent selling rate. Datta and Pal [XXXIX] developed a deterministic inventory system for deteriorating items with inventory level-dependent demand rate and shortages. Liao et al. [XIII] developed an inventory model of deteriorating items under inflation and delay in payment. Hou 128

3 [XIX] presentedan inventory model for deteriorating items with stock dependent demand rate and shortage under inflation. Sana and Chaudhuri [XXXVI] proposed various types of demand rates with delay in payment and price discounting. Khanra et al. [XXVI] developed an EOQ model for deteriorating item with time dependent quadratic demand rate under permissible delay in payment. In our real life we have seen that not only the amount of stock but also the price of the items as well as reliability of the products also affect the inventory model. We attempt to develop a deterministic inventory model of deteriorating items with constant rate of deterioration where we consider the demand rate as price, stock and reliability dependent demand under inflation. We also allow the delay in payment as well as shortage. The rest of the paper is organized as follows: Assumptions and notations for the proposed inventory model are presented in section 2, followed by the detail s description of the inventory model. In section 3, the mathematical analysis of the inventory model in two different cases is presented. Section 4 present the optimization of the proposed model for both the cases. To support the inventory model, we present numerical example and sensitivity analysis insection 5 and 6 respectively. Finally concluding remarks are given in section 7. II. Mathematical Formulation of Inventory Model The proposed inventory model is developed using the following assumptions and notations. Notation: ( ( ) ) demand rate where (t) is inventory level at time t, r is the reliability, p is price of the stock, q inflation rate, constant rate of deterioration, the time when the stock level vanishes, replenishment cycle, M credit period settled by the supplier to the retailer, holding cost per unit item, g shortage cost per unit item, f deterioration cost per unit item, C 1 purchase cost per unit item, p selling price per unit item, I e rate of interest earned, I p rate of interest payable or charged due to delay in payment. Assumption: a) The deterioration rate is constant on the on-hand inventory per unit time and there isno repair of the deterioration item within the cycle. b) Demand rate is D I t ; r, p = k p apr b + βi t where k p = γe ( pδ r ) is the price factorwhere a, b, δ > 0are the parameters.β is stock dependent consumption rate parameter 0 β 1 and r is the reliability. 129

4 c) Shortages are allowed and these are fully backlogged. d) Supplier does not charge any interest if retailer pays within the offered credit period. e) Retailer has to pay interest at the rate I p to the supplier if he pays after offer period. The proposed inventory model based on above assumptions is formulated in mathematical form.a typical behavior of the inventory in a cycle is depicted in thefigure 1 as follows: Figure 1: Graphical representation of the proposed inventory system Therefore, the differential equation of the proposed inventory system can be written in the form of mathematical model under consideration based on the above assumptions is as follows: di t + θi t = k p apr b + βi t, 0 t T dt 1 di t = k p apr b, T dt 1 t T where the conditions are I 0 = Q and I = 0. III. Mathematical Analysis of the Proposed Inventory Model The solutions of the differential equations are as follows: k(p)ap r b e I t = k T 1 if 0 t T 1 Where = θ + βk(p) k p apr b T if t T Therefore, the maximum inventory is I 0 = Q k p aprb Q = I 0 = e 1 The maximum shortage I(T) is I T = k p apr b T 130

5 Now the present value of inventory holding cost ( ) is HC = I t e qt k p apr b dt = e t 1 e qt dt 0 0 k p aprb = q( + q) qe + e q + q Now the present value of shortage cost ( ) is T SC = g I t e qt dt = gk p apr b t e qt dt gk p aprb = q 2 e q e q + k T The present value of purchase cost ( ) is PC = c 1 Q + c 1 e qt 0 T Number of deteriorating item ( ) is DI = Q 0 D I t ; r, p dt T k p apr b dt = c 1 Q + c 1 e qt k p apr b T = Q k p apr b + 0 βk p aprb e t 1 dt = k p aprb θ 2 e 1 The deterioration cost ( ) is DC = fk p aprb θ 2 e 1 Ordering cost ( ) is OC = A Now two cases are considered to study the situations i.e., when the credit period is less than or equal to the cycle time and also when the credit period is greater than the cycle time for settling the account. Following two subsections present the details discussion of the situations of the inventory system. III. a. Inventory situation when credit period less than time for stock vanish Here we discuss the inventory situation of the proposed system when credit period less than time for stock vanish (M T₁). Therefore the interest as per the proposed inventory model for the time T₁ should be considered. Interest earned IE 1 due to sale up to 1 is given by IE 1 = c 1 I e 0 t D I t ; r, p dt = c 1 I e t k p apr b βk p aprb + 0 e t 1 dt = c 1I e k p apr b 2 3 2βk p e θ 2 2βk p 131

6 Interest payable (IP 1 ) due to arrival of supplier before the stock ends is as follows T IP 1 = c 1 I p I t dt = c 1 I p I t dt + c 1 I p I t dt M k p apr b T = c 1 I p e t 1 dt + c 1 I p k p apr b T t dt M k p apr b = c 1 I p 2 e M 1 + M c 1I p 2 k p aprb T 2 It is evident that the total cost is the sum of the set-up, production, inventory carrying, interest and depreciation costs, therefore the total cost (TC 1 ) per unit item per unit time TC 1 = 1 T OC + HC + DC + SC + PC + IP 1 IE 1 = 1 T A + c 1k p apr b e 1 + c 1 e qt k p apr b T gk p aprb + q 2 e q e q + k T hk p aprb + q( + q) qe + e q + q + fk p aprb θ 2 e 1 + c 1 I p k p apr b e M 1 + M 2 T 2 c 1I e k p apr b 2 3 2βk p e θ 2 2βk p We have to optimize the total cost (TC 1 ) per unit time for situation of credit period less than time to stock gets empty. III. b. Inventory situation for credit period greater than time for stock vanish The proposed inventory model when credit period greater than time for stock vanish M The having the additional properties with the following form Interest earned (IE 2 ) due to arrival of supplier after the stock ends is IE 2 = c 1 I e 0 M t D I t ; r, p dt + M 0 T D I t ; r, p dt = c 1I e k p apr b 2 3 2βk p e T 2 1 θ + 2 M θ + βk p e 1 Here the retailer has sold Q unit during [0, ] and is paying c 1 Qto the supplier in full at timem so, the retailer does not have to pay any interest so interest charge is zero i.e.,ip 2 = 0. Therefore, the total expected cost (TC 2 ) of the inventory system when the credit period is lessthan the time of stock vanishes is given by TC 2 = 1 T OC + HC + DC + SC + PC + IP 2 IE 2 132

7 = 1 T A + c 1k p apr b e 1 + c 1 e qt k p apr b T gk p aprb + q 2 e q e q + k T k p aprb + q( + q) qe + e q + q + fk p aprb θ 2 e 1 c 1I e k p apr b 2 3 2βk p e T 2 1 θ + 2 M θ + βk p e 1 IV. Optimization of Proposed Inventory Model In management point of view the production/manufacturing cost should be optimum and hence minimizing the total cost per unit time. So, optimizing both the cases of the proposed model by differentiation method is as follows: For Case I: To minimize the total cost per unit time TC 1 from the equation (12) for a givenvalue of when credit period less than the time for stock vanish (M ) is given by dtc 1 = 0 and d 2 TC 1 d dt2 > 0. 1 Now dtc 1 = 0 gives d c 1 e e qt + g q 2 ke qt qe qt 1 + ( + q) e e q + fθ e 1 + c 1I p e M 1 T c 1I e 2 βk p e 1 + θ = 0 The optimum value is at = t 1 provided it satis.es the following condition given below c 1 + fθ e + ge qt 1 + ( + q) e + qe q + c 1 I p e M > c 1 I p + c 1I e βk p e + θ From the equation (15),t 1 can be obtained which satisfy the conditions given in equation (16).The optimum value is attending for = t 1 since d 2 TC 1 d 2 > 0. For Case II: To minimize TC 2 from the equation (14) for a given value of when credit period greater than the time for stock vanish (M ) is given by dtc 2 d = 0 and d 2 TC 2 d 2 >

8 Again here dtc 2 = 0 gives d c 1 e e qt + g q 2 ke qt qe qt 1 + ( + q) e e q + fθ e 1 c 1I e M βk p e + θ = 0 The optimum value of = t 2 provided the following condition holds true c 1 + fθ e + ge qt 1 + ( + q) e + qe q + c 1I e βk p e + θ > c 1 I e M βk p e The minimum value of TC 2 is attend for = t 2 which is obtained from equation (17) and satisfythe condition (18) and d 2 TC 2 d 2 > 0. V. Numerical Example Here we present an example for numerical exposure of the presented inventory model. In a supermarket the demand rate not only depends upon the amount of the stock but also depends upon the reliability as well as the price of the item so that demand rate is D I t ; r, p where γ = 200, δ = 1.4, a = 200, p = 6, r = 0.75, b = 3, β = Let us consider that the item deteriorates at constant rate 0 1 part of the total inventory. Let the shortages cost be $2per unit item and $250 to orderthe total inventory. Let the cost of each item is $3, selling price is $6 to hold the item it requires $0.6 per unit and the reliability of the item is The system considered under inflation rate of 12% and let the retailer earn 15% of interest and pays 20% interest where the total inventory system is considered for a full year. Let the supplier come (I) Monthly, (II) Bimonthly, (III) Quarterly and (IV) Half yearly. Now we have to minimize the total cost per unit item per unit time for the above situations of inventory system. We consider the following information as input parameters for the proposed inventory model, we have p = 6, θ = 0.1, A = 250 per order, = 0.6 per unit, q = 0.12, c 1 = 3 per item, f = 3,g = 2 per item, I e = 0.15 per year, I p = 0.20 per year, T = 1. Using the equation (12), (14), (15), and (17), we get the optimal solution of the proposed inventory system. We present optimal solutions of the inventory model in table 1. Table 1. Optimal expected total cost for different delay in payment Delay of Payment t 1 (year) TC 1 ( t 1 )($) t 2 (year) TC 2 ( t 2 )($) 1 month months months months Table 1 confirms that initially when delay of payment is less i.e., for one month M = t 1, t 2 > M so, the case II contradicts and only case I holds and minimum average cost is attends at TC 1 Again when payment to supplier after two months (i.e., M = ) inventory modelfollows the fact t 1, t 2 > M so here also case II contradicts and only case I hold and minimum average cost TC 1 For third case i.e. payment to the supplier after four 134

9 months (i.e., M = ) inventory model follows the fact t 1, t 2 > M so here also case II contradicts and only case I holds and minimum average cost is TC 1 For last case the payment to the supplier after six months (i.e., M = 0.5) inventory model follows the fact t 1, t 2 < M so both the cases hold and in this case TC 1 TC 2, so the minimum average cost is found in case I. VI. Sensitivity Analysis We discuss the effect of changes of different parameterθ, p, q, c 1, T, r in different levels of percentage. For individual analysis we consider one parameter at a time and the other parameters are unchanged. The sensitivity analysis is done on the basis of delay in payment for two months (M = ). Table 2. Sensitivity analysis of inventory system based on changes of parameters Parameter Change(%) t 1 TC 1 ( t 1 )($) t 2 TC 2 ( t 2 )($) Change in cost (%) θ p q c 1 T r -25% % -10% % 10% % 25% % -25% % -10% % 10% % 25% % -25% % -10% % 10% % 25% % -25% % -10% % 10% % 25% % -25% % -10% % 10% % 25% % -25% % -10% % 10% % The sensitivity analyses for the different parameters of the proposed inventory system are presented in table 2. From the sensitivity analysis, it is clear that for increasing value of the deterioration parameter, it is clearly seen that t 1 and t 2 decreases while TC 1 ( t 1 ) and TC 2 ( t 2 ) increases. Both TC 1 ( t 1 )and TC 2 ( t 2 ) have low sensitivity, and t 1 and t 2 are moderately sensitive to change in. Table-2 135

10 shows that both total costs are highly sensitive to change of shelling price p but t 1 and t 2 are moderately sensitive. Both TC 1 ( t 1 ) and TC 2 ( t 2 ) have low sensitivity for the change of inflation rate q and optimal total costs decreases while t 1 and t 2 decreases with the increases value of theinflation rate. From table-2, TC 1 ( t 1 ) decreases TC 2 ( t 2 ) increases with the increases value of the purchase cost per unit item (c 1 ), which is expected as per the proposed model. TC 1 ( t 1 ) and TC 2 ( t 2 ) decreases while t 1 and t 2 increases with the increases of replenishment cycle ( ), and all t 1 and t 2 TC 1 ( t 1 ) and TC 2 ( t 2 )are highly sensitivity to change in It is found that t 1 and t 2 decreases while TC 1 ( t 1 )decreasesand TC 2 t 2 increases respectively, with the increase in the value of the reliability. However,t 1, t 2, TC 1 ( t 1 )and TC 2 ( t 2 )are moderately sensitive to change of reliability Now we give some graphs for analysis for the effect on optimal total cost of change of inventory parameters. We have taken the cases of delay payment as described in numerical example in the previous section. Figure 2. Effect of optimal total cost for rate of deterioration 136

11 Figure 3. Effect of optimal total cost for change of selling price From Figure 2, it is observed that when the rate of deterioration ( ) increases the percentage of total cost increases in every case of and the vice versa. Because when the deterioration rate increases by some percentage then the total cost of the inventory system increases. The pattern of the Figure 2 depicts that for more delay payment, the total cost is increased rapidly. From Figure 3, it is clear that if the selling price per unit time (p) increases then percentage of total cost increases in every case of and vice versa. Due to the fact that when the selling price per unit time increases by some percentage then the total cost of the inventory system increases. 137

12 Figure 4. Effect of optimal total cost for change of inflation. Figure 5. Effect of optimal total cost for change of purchase cost of item. Figure 4 depicts that if the inflation (q) decreases the total cost decreases in every cases of delay of payment and vice versa. Figure 4 shows clearly that the graph is linearly decreasing. Figure 5 shows that if the purchase cost of item (c 1 ) decreases the total cost of inventory system decreases in every case of and vice versa. Figure 6. Effect of optimal total cost for change of replenishment time. 138

13 Figure 7. Effect of optimal total cost for change of reliability. It is clear from Figure 6 that the total cost has increased when the replenishment time (T) has decreased in all four cases of M and vice versa. Figure 6 shows that the pattern of the graph is almost linearly decreasing. Figure 7 helps to conclude that if the reliability (r) decreases the percentage of total cost decreases in all cases of M and vice versa. Form the above figure we see that the graph is linearly decreasing. Above tables and figures shows that the analytical study of the inventory model is numerically exhibited of the truth of existence of the proposed inventory system. VII. Conclusion This paper has presented a deterministic inventory model allowing shortage and reliability of the product for realistic situation of demand. The proposed inventory model consists of some naturalistic feature such as deterioration, which is a common phenomenon of goods, also the price and amount of the stock. This paper successfully introduced a very important features of the delay in payment, in support of that a real-life example of proposed inventory system is given and sensitivity analysis is also performed. In sensitivity analysis, it is seen that the proposed model is moderately sensitive with the change in price. Also, the model is highly sensitive with the change in total time. The model is less sensitive for the change of inflation i.e. the inflation not so much effect the total cost of the system and which is sometime helpful for the retailer and the change of reliability is some effected in our proposed model. Lastly the paper also concludes at what time the stock should become zero so that total cost is minimum. 139

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