Optimal lot sizing for deteriorating items with expiration date

Transcription

1 Optimal lot sizing for deteriorating items with expiration date Ping-Hui Hsu 1 Hui Ming Wee Hui-Ming Teng 3 1, Department of Industrial Engineering Chung Yuan Christian University Chungli Taiwan R.O.C. 1 De Lin Institute of Technology Tu-Cheng City, Taipei County Taiwan 36 R.O.C. 3 Chihlee Institute of Technology Taipei Taiwan R.O.C. Abstract Inventory models with deteriorating items, season pattern demand, expiration, and bacordering, though common in practice; have received little attention from researchers. The objective of this study is to develop an inventory model of season pattern demand with expiration date when the product in stoc deteriorates with time. An algorithm is presented for determining an optimal replenishment cycle, a shortage period and an order quantity such that the total profit per unit time is maximized. The model for price discount is developed to compare with the model without price discount. umerical examples are given to illustrate the model. Keywords : Inventory, deteriorating, expiration, bacordering. weehm@hotmail.com Journal of Information & Optimization Sciences Vol. 7 (), o., pp c Taru Publications 5-667/6 $. +.5

2 7 P. H. HSU, H. M. WEE AD H. M. TEG 1. Introduction We attempt to discuss the common deteriorating items inventory problem such as fresh seafood, battery, volatile chemicals and semiconductor chips. The product will deteriorate with time resulting in decreasing utility or price from the original one. Moreover, customers demand decline when the product is close to its expiration date. Faced with the loss of profit caused by the deterioration and declining demand, an appropriate lot size and business policy to maximize the profit are developed. Many researchers have postulated the inventory models with a timedependent demand. Hariga [5] studied the effects of inflation and the time-value of money on an inventory model with time-dependent demand rate and shortages. Barbosa and Friedman [1] developed an inventory model with a deterministic demand pattern, a downward trend and shortages. Petruzzi and Dada [7] considered an inventory problem to determine the order quantity and sale price. Khouja [6] dealt with an inventory problem with the condition that multiple discounts is used to sell excess inventory. Urban and Baer [1] developed a deterministic inventory problem in which demand is a multivariate function of αp ε t y 1 i 1 ε, where p and i represent price and inventory level respectively. You [1] studied the optimal replenishment policy for product with season pattern demand and price. Ghare and Schrader [4] developed an EOQ model by assuming a constant rate of deterioration. Philip, Shah and Dave [8, 9, ] had assumed instantaneous replenishment rate with different assumptions on the patterns of deterioration. Eilon and Mallaya [3] analyzed the pricing policy for item with fixed shelf life, and no deterioration before the expiration date was assumed. Wee [11] developed a model with quantity discount, pricing and partial bacordering policy and Weibull rate of distribution. To the best of our nowledge, the model for deteriorating items with both season pattern demand and expiration has not been developed before. In this study, an algorithm is developed to determine the replenishment and bacordering policy of deteriorating items following a constant rate of deterioration with expiration date and season pattern demand. The effect of price discount is presented for comparison.

3 OPTIMAL LOT SIZIG 73. Assumptions and notations In this section, the following notations are used throughout this paper. T length of a seasonal interval expiration date v a critical time at which inventory level reaches zero Q order quantity each replenishment, decision variable Q 1 sales amount without bacordering over replenishment cycle Q Bacordered quantity at the end of replenishment cycle constant deterioration rate of on-hand-stoc, > p sales price p j discounted sale price for the jth season (during time interval [( j 1)T, jt]) p b sales price when shortages occur θ(η) the probability that customers are willing to purchase the item under the condition that they receive their order after η units of time h unit inventory holding cost per unit time c purchasing cost per unit c ordering cost per replenishment cycle r penalty cost of a lost sale including loss of profit δ discount factor, δ µ processing cost including maing an inventory and deteriorated items per season I j (t) inventory level during the jth season without discount I 1 () maximum inventory level at the start of a cycle Id j (t) inventory level during the jth season with discount In developing the model, the following assumptions are made: 1. The sales price p and bacorder price p b are predetermined and set at prices such that P b = λp > c, where < λ < 1.. Depending upon the seasonal changes of the discount sales price, p j, is set as: p j = p δ( j 1) >, j = 1,,..., n, n, where δ is discount factor.

4 74 P. H. HSU, H. M. WEE AD H. M. TEG 3. Demand for the product follows a deterministic function of price and season such that: αw( j) d j (t, p) = p β, j = 1,,...,,, j >, where w( j) = j + 1 is conserved function, and the values of α and β are nown constant with α, β >. This means that the customers demand is smaller when it is nearer to the product expiration date. 4. Demand during the stocout period is partially lost due to impatient customers. 5. Baclogged demand is satisfied at the beginning of each replenishment. 6. Shortage time is less than the length of a seasonal interval T. 7. The probability of customers bacordered is assumed to be linearly decreasing with waiting time η and is assumed to be. θ(η) = 1 η/t, η < T. 8. The capacity of the warehouse is unlimited. 9. There is no replacement or repair of deteriorated items during a given cycle. 3. Modeling and analysis In this section, we consider deteriorating items with constant deterioration rate and expiration date. The season pattern demand is d j (t, p). The purpose of this study is to maximize the unit profit by determining (1) the replenishment cycle () the duration of the shortages, and (3) the order quantity Q. We illustrate our problem using two case scenarios. 3.1 Case without discount The inventory system during a given cycle is depicted in Figure 1. Suppose replenishment cycle is set at nt. We derive the model by bacward deduction. Let I n (t), n, be the inventory level during the n th season.

5 OPTIMAL LOT SIZIG 75 Figure 1 Inventory system for deteriorating items (revised from You, 5) The differential equation governing the transition of the system during the season interval is di n (t) dt = I n (t) αw(n) p β, with initial condition I n (v) =. Solving the equation gives I n (t) = αw(n) p β [e(v t) 1], t v. (1) Let I n 1 (t) be the inventory level during the (n 1) th season, then di n 1 (t) dt = I n 1 (t) αw(n 1) p β, with initial condition I n 1 (T) = I n (). It gives I n 1 (t) = α p β {w(n 1)[e(T t) 1] + w(n)[e v 1]e (T t) }, Similarly, the inventory level during the j th season is I j (t) = α { p β w( j)[e (T t) n 1 1] + i= j+1 t T. () w(i)[e T [(i j)t t] 1]e + w(n)[e v 1]e [(n j)t t] }, t T, j = 1,,..., n. (3) ext we deduce the objective function. The objective function includes the sales revenues R(n, v), the purchasing cost C(n, v), the lost sale cost L(n, v), the processing cost B(n, v) (including inventory and deteriorated

6 76 P. H. HSU, H. M. WEE AD H. M. TEG items), the inventory holding cost, H(n, v) and the ordering cost, c. When the replenishment cycle and shortage length are set at nt and T v units of time, respectively, F(n, v) is the unit time profit without discount, Q 1 be the sales amount without bacordering over replenishment cycle, and Q be the bac-ordered quantity at the end of replenishment cycle (calculating according to the demand of the first season),then F(n, v) = 1 nt [R(n, v) C(n, v) L(n, v) B(n, v) H(n, v) c ], where R(n, v) = pq 1 + λpq, Q 1 = = Q = n 1 T v d j (t, p)dt + d n (t, p)dt j=1 α(n 1)( + n)t p β + T v = α p β T d 1 (t, p)θ(t t)dt ( T v ). < v T, n, (4) α( + 1 n)v p β Because the order quantity at each replenishment Q = I 1 () + Q, we have C(n, v) = [I 1 () + Q ]c. Because the amount of shortage is and T L(n, v) = α ( ) T p β + v T v r, v d 1 (t, p)[1 θ(t t)]dt, then where B(n, v) = nµ, H(n, v) = H T j n H T j + Hn 1 T + Hv n, n 3, j=1 T = h I j (t)dt

7 OPTIMAL LOT SIZIG 77 { ( ) = αh j + 1 e T 1 p β T + et 1 1 e T ( ) e T (n j)e (n j)t ( j+1) et [1 e(n j 1)T ] +(n j 1)e (n j+1)t 1 e T (1 e T ) + n + } 1 (e v (n j)t 1 e T 1)e, j = 1,,..., n. {[ ] n H T j = αh e T 1 (n )( + 3 n) j=1 p β T + et 1 1 e t ([ ] (e T ) (n )( + 3 n) 1 e t et 1 e T ( ) e (n )T (n 1) (+1)(e (n )T 1) +(n )e T 1 e T (1 e T ) (n )et (1 e t ) + 1 (1 e T ) 1 (1 e T ) n[e (n 1)T e T ] 1 e T (n 1)[e nt e T ] 1 e T + n + } 1 (e v 1) e(n 1)T e T, n 3 ( ) e (n 1)T (n 1)e T + (n ) (1 e T ) ( e nt (n 1)e T +(n )e T (1 e T ) ) H T n 1 T = h I n 1 (t)dt = αh p β [ ( ) n + e T 1 T + n + ] 1 (e v 1) et 1,

8 78 P. H. HSU, H. M. WEE AD H. M. TEG ( ) v Hn v = h I n (t)dt = αh n + 1 e v 1 p β v, ( ) H(1, v) = αh e v 1 p β v, H(, v) = αh p β ( e T 1 + αh 1 p β Our problem can be formulated as: Max : F(n, v) Subject to : 1 n, < v T. T + ) 1 (ev 1) et 1 ( ) e v 1 v. According to Eq. (4) the profit function F(n, v) is a function of two variables n and v, which v is real number and n is a discrete variable. Theorem 1 shows the concavity of F(n, v). Theorem 1. F(n, v) is concave in v. Proof. Please refer to Appendix A. Since F(n, v) is a concave function for v when n is fixed, we can find optimal v by solving the equation F(n, v)/ v =. Let v(n) be the solution and v (n) = min{v(n), T}. Since the integer variable n, cannot be founded by analytic method, the following solution search procedure is used. Solution search procedure Step 1. Set n = 1, n =, v = and F = Step. While n do Step 3-5 Step 3. Solve the equation F(n, v)/ v = to obtain v(n) and v (n) Step 4. Calculate F(n, v (n)) using (4) Step 5. If F(n, v (n)) > F(n, v ), let F = F(n, v (n)), n = n and v = v Example 1. Let α = 5, = 1, p = 3, β =.5, =.1, T = 1, λ =.9, h =.5, c = 1, r = 5, µ =, and c = 1. If we apply these data to the procedure, the optimal decision shown in Table 1 is obtained and the results are as follows:

9 OPTIMAL LOT SIZIG 79 n = 3, v = 94.7, the optimal replenishment cycle is nt = 3, the length of shortage is 5.793, the optimal order quantity Q = I 1 () + Q = = , the sales amount Q 1 + Q = = , and the optimal unit profit F = $ Table 1 Results without discount (α = 5, = 1, p = 3, β =.5, =.1, T = 1, λ =.9, h =.5, c = 1, r = 5, µ =, c = 1 ) n v (n) F(n, v (n)) Q sales n v (n) F(n, v (n)) Q sales amount amount * Case with discount In the previous case, the sales price is fixed, while the seasonal demand decreases throughout the seasons. In this case, we adjust the sale price downward depending upon the seasonal changes. We assume the sale price during the j th season is p j = p δ( j 1), j = 1,,..., n, δ. Then, the demand for the product during the j th season is αw( j) d j (t, p j ) =, = 1,,..., n. p β j Let Id n (t) be the inventory level during the n th season, one has did n (t) dt = Id n (t) αw(n) p β n with initial condition Id n (v) =. Solving it we have Id n (t) = αw(n) p β n [e(v t) 1], t v. With similar way, we have Id n 1 (t) = αw(n 1) p β n 1 [e (T t) 1]+ αw(n) pn β (ev 1)e (T t), t T.,

10 8 P. H. HSU, H. M. WEE AD H. M. TEG Id j (t) = αw( j) p β j [e(t t) 1] + n 1 i= j+1 αw(i) p β i (et [(i j)t t] 1)e + αw(n) p β n (ev 1)e [(n j)t t], t T, j = 1,,..., n. Let Fd(n, v) be the unit time profit with discount, then Fd(n, v) = 1 nt [Rd(n, v) Cd(n, v) Ld(n, v) Bd(n, v) Hd(n, v) c ], < v T, n, (5) where Rd(n, v) = n 1 T p j j=1 n 1 αw( j)t = j=1 p β 1 j Cd(n, v) = [I 1 () + Q ]c { α(e T 1) p β 1 + v d j (t, p j )dt + p n d n (t, p n )dt + λp 1 Q + αw(n)v p β 1 n n 1 αw(i) i= + λα p β 1 1 T ( T v = p β i (et 1)e [(i 1)T] + αw(n) pn β (ev 1)e [(n 1)T] + α ( T p β 1 T v Ld(n, v) = L(n, v) = α ( ) T p β + v T v r 1 Bd(n, v) = B(n, v) = nµ Hd(n, v) = = n Hd T j + Hdn 1 T + Hdv n j=1 n h j=1 T ) ) } c T v Id j (t)dt + h Id n 1 (t)dt + h Id n (t)dt. Our problem can be formulated as: Max : Fd(n, v) Subject to : 1 n, < v T. Theorem. Fd(n, v) is concave in v.

11 OPTIMAL LOT SIZIG 81 Proof. Please refer to Appendix B. Let vd(n) be the solution when Fd(n, v)/ v = and vd (n) = min{vd(n), T}, then the optimal values of n and vd can be solved by a heuristic search procedure. Example. Let α = 5, = 1, p = 3, β =.5, =.1, T = 1, λ =.9, h =.5, c = 1, r = 5, µ =, c = 1, δ = 1. The optimal decision derived using the search procedure is shown in Table. The results are as follows: n = 3, vd = 96.18, the optimal replenishment cycle is nt = 3, the length of shortage is 3.818, the optimal order quantity Q = Id 1 () + Q = = , the sales amount is 97.38, and the optimal unit time profit F = $ Table Results with discount (α = 5, = 1, p = 3, β =.5, =.1, T = 1, λ =.9, h =.5, c = 1, r = 5, µ =, c = 1, δ = 1 ) n v (n) F(n, v (n)) Q sales n v (n) F(n, v (n)) Q sales amount amount * Comparing the examples 1 and, the unit time profit with discount increases by 3.5%. Both inventory behaviors are shown in Figure. Figure Inventory behavior of with and without discount

12 8 P. H. HSU, H. M. WEE AD H. M. TEG 4. Sensitivity analysis Sensitivity analysis with parameters δ and β changes are carried out in this section. Table 3 shows the change in n, vd (n) and Fd (n, vd (n)) for variable δ and other fixed parameter. Table 4 shows the change in n, vd (n) and Fd (n, vd (n)) for variable β. Table 3 lists the discount factor δ at.5, 1,,4, and 8 with other variables unchanged. When δ = 4 then n = 3, F = , it shows that as δ increases, the customers demand increases. However if the price cut is too large, then the unit profit will decrease. When δ higher, then n drops. Table 4 shows the sensitivity analysis study when β equals to.65, 1.5,.5,.7, and.9. When β increases, the n also increases. Therefore demand is more sensitive to the higher sale price, hence discount can influence the profit. Table 3 Sensitivity analysis for parameter δ (α = 5, = 1, p = 3, β =.5, =.1, T = 1, λ =.9, h =.5, c = 1, r = 5, µ =, c = 1 ) δ n vd (n) Fd(n, v (n)) Q Table 4 Sensitivity analysis for parameter β (α = 5, = 1, p = 3, =.1, T = 1, λ =.9, h =.5, c = 1, r = 5, µ =, c = 1, δ = 1 ) β n vd (n) Fd(n, v (n)) Q T OTE: When β =.65, αw(1)/p β Tdt = 5 1/3.65 = 5967, which has higher demand and higher unit time profit

13 OPTIMAL LOT SIZIG Conclusion This study focuses on how to determine the optimal lot sizing policy and business strategy for deteriorating items under season pattern demand with expiration date. Because the demand is sensitive to the sale price, reducing the sale price can stimulate customers demand, compensates the loss of deteriorated items, and achieves an optimal policy. This can be used as a reference for the decision-maers. Future research can be done to consider recycle deteriorated items. Appendix A Proof of Theorem 1. αλp R(n, v) = v p β T, C(n, v) = c v [ α p β αr L(n, v) = v p β T >, B(n, v) =, v v c =, αh + 1 n H(n, v) = v p β + 1 n e (n 1)T e v e (n 1)T e T e v α ] p β, T And + αh + 1 n p β + αh p β αh H(1, v) = v p β ev >, e T 1 e v + 1 n e v >, n 3. αh 1 H(, v) = v p β e T 1 v + αh 1 p β ev >.

14 84 P. H. HSU, H. M. WEE AD H. M. TEG Then [ 1 F(n, v) = R(n, v) C(n, v) L(n, v) v nt v v v ] B(n, v) H(n, v) v v v c = 1 nt [ α(c λp) p β T v αc p β L(n, v) H(n, v) v + 1 n e (n 1)T e v ] ote λp > c. Thus, we have F(n, v)/ v <, this completes the proof.. Appendix B Proof of Theorem. αλp Rd(n, v) = v p β 1 T, [ Cd(n, v) = c v α p β n αr Ld(n, v) = v p β 1 T >, Bd(n, v) =, v v c =, + 1 n e (n 1)T e v α p β 1 T ], And αh + 1 n Hd(n, v) = v pn β + αh + 1 n pn β + αh p β n αh Hd(1, v) = v p β 1 ev >, e (n 1)T e T e v e T 1 e v + 1 n e v >, n 3.

15 OPTIMAL LOT SIZIG 85 Then αh 1 Hd(, v) = v p β e T 1 v + αh 1 p β ev >. [ 1 Fd(n, v) = Rd(n, v) Cd(n, v) Ld(n, v) v nt v v v ] Bd(n, v) Hd(n, v) v [ = 1 α(c λp) nt p β 1 T αc pn β v v v c + 1 n e (n 1)T e v ] Ld(n, v) Hd(n, v) v ote λp > c. Thus, we have Fd(n, v)/ v <, this completes the proof.. References [1] L. C. Barbosa and M. Friedman, Deterministic inventory lot size models a general root law, Management Sci., Vol. 4 (1978), pp [] U. Dave, On a discrete-in-time order-level inventory model for deteriorating items, Operational Research Quarterly, Vol. 3 (1979), pp [3] S. Eilon and R. V. Mallaya, Issuing and pricing policy of semiperishables, in Proceedings of 4th International Conference on Operational Research, Wiley-Interscience, ew Yor, [4] P. M. Ghare and G. F. Schrader, A model for exponentially decaying inventory, Journal of Industrial Engineering, Vol. 14 (1963), pp [5] M. A. Hariga, Effects of inflation and time-value of money on an inventory model with time-dependent demand rate and shortages, Eur. J. Oper. Res., Vol. 81 (1995), pp [6] M. J. Khouja, Optimal ordering, discounting, and pricing in the single-period problem, Internat. J. Prod. Econ., Vol. 65 (), pp [7]. C. Petruzzi and M. Dada, Pricing and the newsvendor problem a review with extensions, Oper. Res., Vol. 47 (1999), pp

16 86 P. H. HSU, H. M. WEE AD H. M. TEG [8] G. C. Philip, A generalized EOQ model for items with Weibull distribution deterioration, AIIE Transaction, Vol. 6 (1974), pp [9] Y. K. Shah, An order-level lot-size inventory model for deteriorating items, AIIE Transactions, Vol. 9 (1976), pp [1] T. L. Urban and R. C. Baer, Optimal ordering and pricing policies in a single-period environment with multivariate demand and mardowns, Eur. J. Oper. Res., Vol. 13 (1997), pp [11] H. M. Wee, Deteriorating inventory model with quantity discount, pricing and partial bacordering, Int. J. Production Economics, Vol. 59 (1999), pp [1] P. S. You, Optimal replenishment policy for product with season pattern demand, Operations Research Letters, Vol. 33 (1) (5), pp Received March, 5