An Entropic Order Quantity (EnOQ) Model. with Post Deterioration Cash Discounts

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1 Int. J. Contemp. Math. Sciences, Vol. 6,, no. 9, An Entropic Order Quantity (EnOQ Model with Post Deterioration Cash Discounts M. Pattnaik Dept. of Business Administration, Utkal University Bhubaneswar-754, India Abstract This paper deals with an entropic order quantity (EnOQ model under cash discounts over a finite planning time. Its main aim lies in the need for an entropic cost of the cycle time is a key feature of specific perishable product like fruits, vegetables, food stuffs, fishes etc. To handle this multiplicity of objectives in a pragmatic approach, entropic ordering quantity model with cash discounts during post deterioration of perishable items to optimize its payoff is proposed. This paper explores the economy of investing in economics of lot sizing in EnOQ and EOQ models. Furthermore, numerical experiments are conducted to evaluate the difference between the EnOQ and EOQ models separately. The results indicate that this can become a good model and can be replicated by researchers in neighbourhood of its possible extensions. Mathematics Subject Classification: 9B5 Keywords: Two component demand, Post deterioration, Cash discounts. Introduction This paper deals with an entropic order quantity model with cash discounts under post deterioration of the perishable products. The predominant criterion in traditional inventory models is minimization of long-run average cost per unit time. The costs considered are usually fixed and variable ordering cost, holding cost. Costs associated with disorder in a system tied up in inventory are accounted for by including an entropy cost in the total costs. Entropy is frequently defined as the amount of disorder in a system. Jaber [] proposed an analogy between the behaviour of production system and the behaviour of physical system.

2 93 M. Pattnaik In deriving the economic order quantity formula, we usually omit the case of discounting on selling price. But in real world, it exists and is quite flexible in nature. On the other hand, in order to motivate customers to order more quantities, usually suppliers offer cash discounts on selling prices at post deterioration rate. Dave [] developed an inventory model under continuous discount pricing. Khouja [8] studied an inventory problem under the condition that multiple discounts can be used to sell excess inventory. Shah [] mentioned that discount is considered temporarily for exponentially decaying inventory model. This paper represents the issue in details. Recently, a number of research articles appeared which deal with the EOQ problem under constant demand. In last two decades the variability of inventory level dependent demand rate on the analysis of inventory system was described by researchers like Goswami [], Tripathy [3] and Pal [4]. They described the demand rate as the power function of on hand inventory. Researchers such as Mahata [4], Rafaat [3], Skouri [5], Pattnaik [9], Vujosevic [], Pal [4], Panda [5] and Goyal [6] discussed the EOQ model assuming time value of money, demand rate, deterioration rate, shortages and so on a constant or probabilistic number or an exponential function. Product perishable is an important aspect of inventory control. Deterioration in general, may be considered as the result of various effects on stock, some of which are damage, decay, decreasing usefulness and many more. Decaying products are of two types. Product which deteriorate from the very beginning and the products which start to deteriorate after a certain time. Lot of articles are available in inventory literature considering deterioration. Interested readers may consult the survey paper of Rafaat [3], Weatherford [7], Pattnaik [9], Tripathy [3], Panda [5] and Goyal [6]. The purpose of this paper is to investigate the effect of the approximation made by using the average payoff when determining the optimal values of the policy variables. A policy iteration algorithm is designed with the help Deb [6] and optimum solution is obtained through LINGO software. Numerical experiments are carried out to analyse the magnitude of the approximation error. The remainder of this paper is organised as follows. In Section assumptions and notations are provided for the development of the model. Section 3 describes the model formulation. In section 4, an illustrative numerical experiment is given to illustrate the procedure of solving the model. Finally in section 5 I make a summary, concluding remarks and provide some suggestions for future research.

3 Entropic order quantity model 933 Table- Major Characteristics of Inventory Models on selected researches. Author(s and published Year Mahata et al. (6 Panda et al. (9 Jaber et al. (8 Vujosevic et al. (996 Chung et al. (7 Skouri et al. (7 Present paper ( Structure of the Model Deterioration Inventory Model Based on Discount allowed Demand Backlogging allowed Fuzzy Yes (constant EOQ No Constant No Crisp Yes (constant EOQ Yes Stock Yes (partial dependent Crisp Yes (on hand inventory EnOQ No Unit selling price Fuzzy No EOQ No Constant No Crisp Yes (exponential EOQ No Selling price No Yes (partial Crisp Yes (Weibull EOQ No Ramp Yes (partial Crisp Yes (Heaviside EnOQ Yes Stock dependent No. Notations and Assumptions.. Notations C : set up cost c : per unit purchase cost of the product s : constant selling price of the product per unit (s>c h : holding cost per unit per unit time r : discount offer per unit after deterioration. Q : order level for post deterioration cash discounts. : cycle length for post deterioration cash discounts. T Assumptions. Replenishment rate is infinite. The deterioration rate θ = θ H (t-, ( < θ constant Where is the time measured from the instant arrivals of a fresh replenishment indicating that the deterioration of the items begins after a time from the instant of the arrival in stock. H (t - is the well known Heaviside s function., t H (t - =, otherwise

4 934 M. Pattnaik 3. Demand depends on the on hand inventory up to time from time of fresh replenishment beyond with it is constant and defined as follows. a + bi( t, t < R (I (t = a, t Where a> is the initial demand rate independent of stock level and condition of inventory. b> is the stock sensitive demand parameter I(t is the instantaneous inventory level at time t. r is the percentage discount after an unit selling price during 4 r, ( deterioration. α = ( n n ( n R the set of real numbers is the effect of discounting selling price on demand during deterioration. α is determined from priori knowledge of the seller such that the demand rate is influenced with the reduction state of selling price.. Mathematical Model with post deterioration cash discounts on unit selling price At the beginning of the replenishment cycle the inventory level raises to Q. As time progresses it decreases due to instantaneous stock dependent demand up to the time. After deterioration starts and the inventory level decreases for deterioration and constant demand. Ultimately inventory reaches zero level at T. As deterioration starts from, r, ( r is the percentage discount offer on unit selling price during deterioration is provided to enhance the demand of decreased quality items. This discount is continued for the rest of the replenishment cycle. Then the behaviour of inventory level is governed by the following system of linear differential equations. d I (t/dt = -a b I (t [ α a + θ I ( t ] With the initial boundary condition I ( = Q, t I (T =, t T = t ( t T ( Solving the equations, a bt bt I( t = [ e ] + Q e, t (3 b aα θ ( T t = [ e, ] t T (4 θ Now, at the point t = we have from equations (3 and (4

5 Entropic order quantity model 935 θ ( T ( e aα a b a Q = + e b (5 θ b Holding cost of inventories in the cycle is, HC = h[ T ( t dt I( t dt] I + o Purchase cost in the cycle is given by PC = cq. Setup cost in the cycle is given by OC =C. dσ Entropy generation rate must satisfy S = where, σ(t is the total dt entropy generated by time t and S is the rate at which entropy is generated. The entropy cost is computed by dividing the total commodity flow in a cycle of duration T i. ( t T i = The total entropy generated over time T i is ( σ T i Sdt, where s is constant selling price of the product per unit (s>c. Entropy cost per cycle is D( Qi With det erioration EC (T i = + (i= σ ( σ T where D ( = R( I ( t ( dt Q i with deterioration = Q R ( ( I( t σ = Sdt = dt = D( s s σ ( T ( I( t T T R a a dt = dt = s s s = sq EC = s + a ( T i ( T Total sales revenue in the order cycle can be found as T SR = s ( a + bi( t dt + α ( r a dt Thus total profit per unit time of the system is π ( r, T = [ SR PC HC EC OC] R S = ( I ( t T On integration and simplification of the relevant costs, the total profit per unit time becomes s

6 936 M. Pattnaik sa + s π = T + s Q ( r α a( T a + b b ( e sq a s a ( T cq C a + Q b h aα e + θ a e + b b θ ( T θ b ( T The post deterioration discount on selling price is to be given in such a way that the discounted selling price is not less that the unit cost of the product, i.e. s r c >. ( Applying this constraint on unit total profit function I have the following maximization problem. Maximize π ( r., T c Subject to r < r, T ( s 5. Numerical Example The parameter values are a=8, b=.3, h=.6, s=., C =., c=4., θ=.3, =., n=. are listed in Table. For entropic order quality model with post deterioration cash discounts, the post deterioration discount on unit selling price is 4.38% and discount starts at time. and continued to.839. Profit per unit time is The optimal order quantity is The unit profit and order quantity for EOQ model with post deterioration cash discounts are and with.9% and -3.98% changes respectively. The cycle length is.57 and.5 % less than that for EnOQ model with post deterioration cash discounts. Based on the results in Table, I also note that the relative difference can be high when the ordering policy is optimal in the discounted selling price under EnOQ and EOQ post deterioration models. Table Optimal values of the decision variables, order quantity and profit per unit time of different models. Nature Discount of Post deterioration cash discounts Decision variables EnOQ EoQ % change R T Q Π

7 Entropic order quantity model Conclusion I have presented an entropic order quantity model for perishable items under two component demand in which the criterion is to optimise the expected total discounted finite horizon payoff. To compute the optimal values of the policy parameters a simple and quite efficient policy model was designed. Finally, in numerical experiments the solution from the discounted model evaluated and compared to the solution of other traditional EOQ policy. However, I saw few performance differences among a set of different inventory policies in the existing literature. Although there are minor variations that do not appear significant in practical terms, at least when solving the single level, incapacitated version of the lot sizing problem. From my analysis it is demonstrated that the retailer s profit is highly influenced by offering post discount on selling price. The results of this study give managerial insights to decision maker developing an optimal replenishment decision for deteriorating product. Compensation mechanism should also be included to induce collaboration between retailer and dealer in a meaningful supply chain. In general, for normal parameter values the relative payoff differences seem to be fairly small. The optimal solution of the suggested entropic order quantity model has a higher total payoff as compared with optimal solution for the traditional EOQ policy. Conventional wisdom suggests that workflow collaboration in an entropic model in a varying deteriorating product in market place are promising mechanism and achieving a cost effective replenishment policy. The approach proposed in the paper based on EnOQ model seems to be a pragmatic way to approximate the optimum payoff of the unknown group of parameters in inventory management problems. The assumptions underlying the approach are not strong and the information obtained seems worthwhile. Investigating optimal policies when demand are generated by other process and designing models that allow for several orders outstanding at a time would also be challenging tasks for further developments. Its use may restrict the model s applicability in the real world. Future direction may be aimed at considering more general deterioration rate or demand rate. Uses of other demand side revenue boosting variables such as promotional efforts are potential areas of future research. The proposed paper reveals itself as a pragmatic alternative to other approaches based on two component demand function with very sound theoretical underpinnings but with few possibilities of actually being put into practice. The results indicate that this can become a good model and can be replicated by researchers in neighbourhood of its possible extensions. As regards future research, one other line of development would be to allow shortage and partial backlogging in the discounted model.

8 938 M. Pattnaik References [] A. Goswami, and K. S. Choudhury, An EOQ model for deteriorating items with linear time dependent demand rate and shortages under inflation and time discounting, Journal of Operational Research Society, 46(6, (995, 77. [] D. S. Dave, K. E. Fitzapatrick and J. R. Baker, An advertising inclusive production lot size model under continuous discount pricing, Computational Industrial Engineering, 3, (995, [3] E. Raafat, Survey of Literature on continuously deteriorating inventory model, Journal of Operational Research Society, 4, (99, [4] G. C. Mahata and A. Goswami, Production lot size model with fuzzy production rate and fuzzy demand rate for deteriorating item under permissible delay in payments, Journal of Operations Research Society of India, 43, (6, [5] K. Skouri, I. Konstantaras, S. PapaChristos and I. Ganas, Inventory models with ramp type demand rate, partial backlogging and weibull deterioration rate, European Journal of Operational Research, (7. [6] K. Deb, Optimization for engineering design, Prentice-Hall of India. New Delhi, (. [7] L. R. Weatherford and S. E. Bodily, A taxonomy and research overview of perishable asset revenue management: yield management, overbooking and pricing, Operations Research, 4, (99, [8] M. Khouja, Optimal ordering, discounting and pricing in the single period problem, International Journal of Production Economics, 65, (, - 6. [9] M. Pattnaik, An entropic order quantity (EnOQ model under instant deterioration of perishable items with price discounts, International Mathematical Forum, 5(5, (, [] M. Vujosevic, D. Petrovic, and R. Petrovic, EOQ formula when inventory cost is fuzzy, International Journal of Production Economics, 45 (996, [] M. Y. Jaber, M. Bonney, M. A. Rosen and I. Moualek, Entropic order quantity (EnOQ model for deteriorating items, Applied mathematical modelling, (8. [] N. H. Shah and Y.K. Shah, An EOQ model for exponentially decaying inventory under temporary price discounts, cahiers du CERO, 35, (993, 7-3. [3] P. K. Tripathy and M. Pattnaik, An entropic order quantity model with fuzzy holding cost and fuzzy disposal cost for perishable items under two component demand and discounted selling price, Pakistan Journal of Statistics and Operations Research, 4(, (8, 93-.

9 Entropic order quantity model 939 [4] S. Pal, K. Goswami and K. S. Chaudhuri, A deterministic inventory model for deteriorating items with stock dependent demand rate, Journal of Production Economics, 3, (993, [5] S. Panda, S. Saha and M. Basu, An EOQ model for perishable products with discounted selling price and stock dependent demand, CEJOR, 7, (9, [6] S. K. Goyal and B. C. Giri, Recent trends in modelling of deteriorating inventory, European Journal of Operational Research, 34, (, -6. Received: September,