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Journal of China University of Science and Technology Vol.49- 2011. 08 Abad 之供應商暫時降價存貨模式的最適封閉解 Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers Temporary Price Reduction 摘要 當供應商公告在某一時間點其供貨價格將有所調整時,零售商會依自身的庫 存量來決定下次訂貨量的多寡。一般當供應商突然採取暫時性的降價促銷時,零 售商會大量採購以節省採購成本, Abad [7] 探討了供應商將商品售價暫時性降價 後零售商採購的兩種模式。本篇文章中將探討 Abad [7] 所討論的兩種模式,針對 其尋找最適訂貨量及額外獲利提出一正確且快速的修正方法;文章中所提出的方 法為一封閉解的型式,尤其當有整數運算子存在數學模型中,可免除數值搜尋時 的費時與產生的誤差,可廣泛運用到現有的存貨模型中。 Abstract Supplier announces some or all of cost parameters may change after a decided time. According to the inventory is depleted at the time of the last opportunity to purchase before some or all of the cost parameters may change, Abad [7] presented the inventory system when supplier offers temporary price reduction as two models. In this article, author deals with alternative approaches to present simple solutions in order to decide the two Abad’s [7] models: (1) the discount is applicable to units resold during the promotion, and (2) the discount is applicable to units purchased during the promotion. Author derives the closed-form solutions to find the additional profit and the number of lots purchased by the reseller during the promotion. Also author uses the same data of Abad [7] to demonstrate the closed-form solutions can obtain better results than Abad’s iteration method. Keywords: Temporary price reduction; Discounted price, trade promotion I. Introduction The economic order quantity (EOQ) model is popular in supply chain management. The traditional EOQ inventory model supposes that the inventory parameters are 1 Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers Temporary Price Reduction constant during the sale period. Schwarz [1] discussed the finite horizon EOQ model, the costs of the model were static and the optimal ordering number can be found during the finite horizon. In real life, there are many reasons for suppliers offer a temporarily price discount to retailers. The retailers may engage in purchasing additional stock at reduced price and sale at regular price later. Lev and Weiss [2] considered the case where the cost parameters may change, and the horizon may be finite as well as infinite. However, the lower and upper bounds they used do not guarantee boundary conditions are met. Tersine [3] proposed a temporary price discount model, the optimal EOQ policy is obtained by maximizing the difference between regular EOQ cost and special ordering quantity cost during the sale period. Martin [4] revealed that Tersine’s [3] representation of average inventory in the total cost is flawed, and suggested the true representation of average inventory. But Martin [4] sacrificed the closed-form solution in solving objective function, instead of using search methods to find special order quantity and maximum gain. Wee and Yu [5] assumed that the items deteriorated exponentially with time and temporary price discount purchase occurred at the regular and non-regular replenishment time. Abad [6] provided a procedure to determine the optimal response of reseller when the supplier offers a temporary reduction in price. he assumed that the trade promotion is offered by the supplier only at one point in time and the market demand for the product is elastic with respect to the selling price. The reseller may engage in purchasing additional stock at the reduced price offered by the supplier for later sale at the regular selling price. Abad [7] extended the analysis to the case where the promotion lasted a finite time interval and he considered two cases. Nevertheless, there exist some flaws on Abad’s [7] computational procedure. Sarker and Kindi [8] proposed five different cases of the discount sale scenarios in order to maximize the annual gain of the special ordering quantity. Kovalev and Ng [9] shown a discrete version of the classic EOQ problem, they assumed that time and the product are continuously divisible and demand occurs at a constant rate. Cárdenas-Barrón [10] pointed out there are some technical and mathematical expression errors in Sarker and Kindi [8] and presented the closed form solutions for the optimal total gain cost. Li [11] presented a solution method which modified Kovalev and Ng’s [9] search method to find the optimal number of orders. Cárdenas-Barrón et al. [12] proposed economic lot size model that the supplier offers a temporary discount and specified a minimum quantity of additional units to purchase. García-Laguna et al. [13] illustrated a method to obtain the solution of the classic EOQ and economic production quantity models 2 Journal of China University of Science and Technology Vol.49- 2011. 08 when the lot size must be an integer quantity. Their approach obtained a rule to discriminate between the situation in which the optimal solution is unique and when there are two optimal solutions. Chang et al. [14] used closed-form solutions to solve Martin [4] and Wee and Yu [5] EOQ models with a temporary price discount. Other authors also considered similar issues have been performed by Khouja and Park [15], Wee et al. [16], Cárdenas-Barrón [17], etc. The purpose of this article is to present two algorithms that are simpler than Abad’s [7] computing procedure. Either in any one of Abad’s [7] two cases, we propose closed-form solutions to represent the computing procedure. Only follow the steps we suggested, you can quickly and correctly find the number of lots and the incremental profit. II. Notation We will follow the same notation used in Abad [7]. the reseller’s regular selling price to customer p0 p the reseller’s discounted selling price to customer, p p0 D( p0 ) the annual customer demand associated with the selling price p 0 , D0 D( p0 ) D ( p ) the annual customer demand associated with the selling price p , D D( p ) C v r t0 W0 the fixed ordering receiving and placement cost for the reseller the per unit price charged by the supplier to the reseller the annual holding cost fraction for the reseller the cycle time in regular policy, t 0 2C / rvD0 the reseller’s profit rate under regular policy, W0 ( p0 v 2CRv / D0 ) D0 the temporary price reduction offered by the manufacturer to the reseller ($/unit) the duration over which the promotion is offered by the manufacture (years) the number of lots purchased by the reseller during the promotion m Q the lot size by the reseller for each cycle, Q DT / m ( p, m) the incremental profit over time span T d T the duration in which the selling price from reseller to customer is p the duration in which the selling price from reseller to customer is p 0 ( p, m, , ) the incremental profit over time span T . III. Model formulation 3 Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers Temporary Price Reduction 3.1 Reseller’s response when the discount is applicable on units resold during the supplier’s promotion The inventory pattern for Abad’s [7] first model is shown in figure 1. When reseller sets regular selling price p 0 , the reseller’s profit rate under regular policy is W0 , p D / D v Crv / 2 D where can be solved by and p0 W0 ( p0 v 2CRv / D0 ) D0 . There are relations between annual demand D ( p ) and the selling price p which the reseller sets. The higher selling price p , the lower annual demand D ( p ) is. The supplier offered temporary price reduction d to reseller during the interval [0, T ] . The supplier can monitor the reselling price and insure that discount is applicable only on ‘sell-through’ units. During the interval [0, T ] , the product discounted selling price is p . There are m identical cycles with cycle time T / m . The last cycle ends at time T where inventory is zero. The reseller’s problem is to maximize the additional profit during time interval [0, T ] . The objective function is hDT 2 max ( p, m) ( p v d ) DT mC TW0 2m (1) where h r (v d ) . We define Hessian matrix as below 2 2 (2 D 2 DD )T CD pm p 2 D D H 2 2 CD 2C 2 pm m D m Because Hessian matrix is a semi-negative definite matrix, ( p, m) would exist maximum value in Eq.(1). Let p (m ) be the solution of 0 . i.e., p (m ) is the p solution of p where D D hT vd D 2m (2) dD . Hence, Eq. (1) can be represented by dp z (m) [ p(m), m] [ p(m) v d ]DT hDT 2 mC TW0 2m (3) dz (m) DhT 2 d 2 z (m) DhT 2 C 0 . We can say that z (m) is a , dm 2m 2 dm 2 4m 3 concave function of m . It means that there exists integer m such that z (m) has maximum value. By z (m 1) z (m) 0 , we can find Because 4 Journal of China University of Science and Technology Vol.49- 2011. 08 1 1 DhT 2 m 4 2C 2 (4) Taking m into Eq. (2), p D hT vd D 1 1 DhT 2 2 4 2C 2 (5) We can find the value of p (m ) from the above equation. The complete algorithm for solving ( m, p ) is shown below: Algorithm 1 Step 1. Compute reseller’s regular selling price p 0 by p where D D Crv v D 2D dD , and D0 D( p0 ) dp Step 2. Compute reseller’s profit rate W0 by W0 ( p0 v 2C R v ) D0 D0 Step 3. Compute reseller’s discounted selling price p by p D hT vd D 1 1 DhT 2 2 4 2C 2 where D D( p) , h r (v d ) . Step 4. Compute m by 1 1 D h T2 m 2 4 2C Step 5. The additional profit ( p, m) during interval [0, T ] is hDT 2 ( p, m) ( p v d ) DT mC TW0 2m 3.2 Reseller’s response when the discount is applicable on units purchased during 5 Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers Temporary Price Reduction the supplier’s promotion The inventory pattern for Abad’s [7] second model is shown in figure 2. The supplier offered temporary price reduction d to reseller during the interval [0, T ] . The supplier cannot monitor the reselling price. The reseller can purchase a large lot before the end of the promotion and sell a portion of the large lot later at the regular price. There are m identical cycles with cycle time T / m . The last cycle ends at time T where inventory is zero. At T the reseller purchased a large lot. During the interval [0, T ] , the product discounted selling price is p . During the interval (T , T ] , the product regular selling price is p 0 . The reseller’s problem is to maximize the additional profit during the interval [0, T ] . The objective function is max (m, p, , ) ( p v d ) D(T ) ( p 0 v d ) D0 (m 1)C h( DT 2 D 2 D0 2 D0 ) (T )W0 2m 2 2 (6) subject to 0 , 0 , where h r (v d ) . Because the model during the interval [T , T ] in fig.2 is like the model during the interval [0, z ] suggested by Abad [6] , shown in fig.3. We can use the procedure suggested by Abad [6] to find p , D , , . And the value m in Eq. (6) is obtained by Eq. (4). The complete algorithm for solving (m, p, , ) is shown below: Algorithm 2 Step 1. Compute reseller’s regular selling price p 0 by p where D D Crv v D 2D dD , and D0 D( p0 ) dp Step 2. Compute reseller’s profit rate W0 by W0 ( p0 v 2CRv ) D0 D0 Step 3. Compute reseller’s discounted selling price p by G ( p) G ( p0 ) 2G ( p) D 0 D D0 6 Journal of China University of Science and Technology Vol.49- 2011. 08 where G ( p) ( p v d ) D , G ( p) dG and D D( p) . dp Step 4. Compute by [ p (v d )]D [ p0 (v d )]D0 r (v d )( D D0 ) Step 5. Compute by d 2C r v/ D0 r (v d ) Step 6. Compute m by 1 1 D h T2 m 4 2C 2 where h r (v d ) . Step 7. The additional profit (m, p, , ) during time interval [0, T ] is (m, p, , ) ( p v d ) D(T ) ( p 0 v d ) D0 (m 1)C h( DT 2 D 2 D0 2 D0 ) (T )W0 2m 2 2 IV. Numerical example We use the same data of Abad [7] to show that we can quickly and correctly find the additional profits ( p, m) and (m, p, , ) . Table 1 and Table 2 are the results of comparing our method to Abad’s [7] method. Using Abad’s [7] data, v $8 / unit , d $8 / unit , C $80 / order , r 0.5$ / $ / yr , T 0.25yr and demand D( p) 10000000 p 3 . For first case, the discount is applicable on units resold during the supplier’s promotion, we can follow steps in Algorithm 1 and obtain p0 12.26$ / unit , W0 21254$ / yr , p 11.025$ / unit , D 7462 unit/yr , m 3 and the additional profit is ( p, m) 6084.85$ . For second case, the discount is applicable on units purchased during the supplier’s promotion, we can follow steps in Algorithm 2 and obtain p0 12.26$ / unit , W0 21254$ / yr , p 11.29$ / unit , D 6956 unit/yr , 0.177 yr , 0.141 yr , and the additional profit is m3 (m, p, , ) 2294.69$ . Either in case 1 or case2, we can quickly and correctly find the number of lots m and the incremental profit. Our result in case 2, the incremental profit (m, p, , ) 2294.69$ is larger than Abad’s incremental profit 7 Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers Temporary Price Reduction (m, p, , ) 2289.6$ . V. Conclusion The main purpose of this paper is to propose easy-to-apply methods in order to determine Abad’s [7] temporary price reduction over an interval offered by the supplier. The two algorithms we proposed not only solve the tediously numerical iteration calculation but also find the exact incremental profit. It can also apply to the cases when supplier temporarily reduce sale price during the sale period. Numerical examples show that the two algorithms proposed in this paper is accurate and rapid. Table 1. Comparing our method to Abad’s [7] method for case 1 p m ( p, m) p0 W0 D Abad’s method 12.26 Our method 12.26 21254 11.025 7462 3 6084.85 21254 11.025 7462 3 6084.85 Table 2. Comparing our method to Abad’s [7] method for case 2 p m (m, p, , ) p0 W0 D Abad’s method 12.26 21254 11.10 7312 0.156 0.162 3 2289.6 Our method 21254 11.29 6956 0.177 0.141 3 2294.69 12.26 8 Journal of China University of Science and Technology Vol.49- 2011. 08 Inventory level Q time 0 T m 2T m … T t0 T Fig. 1. Inventory pattern over time. Inventory level Q 0 T m 2T m … T T T time Fig. 2. Inventory pattern over time. inventory level zD D0 zD z z Fig 3. Adad’s [6] model suggested in 1997. 9 time Closed-Form Optimal Solutions for Abad’s Inventory Models When Supplier Offers Temporary Price Reduction References [1] Schwarz, L.B. (1972). Economic order quantities for products with finite demand horizon. AIIE Transactions, 4, 234-236. [2] Lev, B. and Weiss, H. J. (1990). Inventory models with cost changes. Operations Research, 38(1), 53-63. [3] Tersine, R.J. (1994). Principles of Inventory and Materials Management, 4th ed. Prentice-Hall, Englewood Cliffs, NJ. [4] Martin, G.E. (1994). 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