Coordinated ordering decisions for products with

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Computers & Industrial Engineering 54 (2008) 602–612 www.elsevier.com/locate/dsw

Coordinated ordering decisions for products with short lifecycle and variable selling price Ping-Hui Hsu a, Hui-Ming Teng

b,c

, Yung-Tsan Jou c, Hui Ming Wee

c,*

a b

Department of Business Administration, De Lin Institute of Technology, Tu-Cheng, Taipei, Taiwan 236, ROC Department of Business Administration, Chihlee Institute of Technology, Panchiao, Taipei, Taiwan 22005, ROC c Department of Industrial Engineering, Chung Yuan Christian University, Chungli, Taiwan 32023, ROC Received 4 July 2006; received in revised form 22 January 2007; accepted 14 September 2007 Available online 21 September 2007

Abstract The supplier–buyer coordination is an important policy in the supply chain management. The buyer in the two-echelon inventory system with regular selling season has to face the uncertainty of customer demand, supplier’s delivery time and variable price change. At the same time, the supplier has to consider the inventory holding and delay cost. The objective of this study is to develop an integrated supply chain strategy for products with short lifecycle and variable selling price to entice cooperation. The strategy must provide a win–win situation for both the supplier and the buyer. A numerical case example, sensitivity analysis and compensation mechanism are given to illustrate the model. 2007 Elsevier Ltd. All rights reserved. Keywords: Short lifecycle; Price decrease; Coordination; Uncertainty

1. Introduction With the advances in technology and complex customer demand, the lifecycle of some products becomes shorter and shorter, especially mobile phones, digital cameras, and notebook computers. These products have common characteristics, such as rapid product substitution, uncertain market demand, and rapid price decrease. During the lifecycle of products, price might drop due to new products introduction. For example, in the case of a newspaper retailer (buyer) the price of a piece of a newspaper at 6 AM may be $0.5 but will decrease to $0.25 at 3 PM. The price of a notebook PC may be $1000 when it is first introduced to the market but may drop to $750 right after a new model is introduced to the market. The issue on selling price change can be classified as a single price change (Goyal, 1979; Markowski, 1990; Naddor, 1966) and a continuous price change (Erel, 1992; Buzacott, 1975; Moutaz & Sunjune, 2003). Price change significantly affects ordering policy. Pellegrini (1986) pointed out that return policy would encourage

*

Corresponding author. Tel.: +886 3 456 6142; fax: +886 3 456 3171/4499. E-mail address: [email protected] (H.M. Wee).

0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.09.010

P.-H. Hsu et al. / Computers & Industrial Engineering 54 (2008) 602–612

603

retailers to order more stocks. Lee, Padmanabhan, Taylor, and Whang (2000) showed that price protection could be an instrument for channel coordination regarding the price change. Due to the customer’s uncertain demand and the supplier’s uncertain delivery, the ordering policy for the retailer (buyer) has become more complex. In addition, decreasing selling prices as a result of new product introduction further complicates the ordering decisions. The buyer has to consider the uncertainty of the customer demand, future selling price drop, and unpredictability of the supplier delivery time to determine the order quantity in order to gain profits. In practice, if the buyer loses profit due to the supplier’s delay, the supplier should compensate the buyer for her loss. Uncertain supplier’s delivery time (lead-time) usually hurts the retailer’s business. If information sharing is implemented, the supplier can receive the delivery time information from the retailer, then the lead-time can be reduced and this will result in a win–win situation for both players. Because of the uncertainty of customer demand and supplier’s lead-time, some managerial experts realized the importance of supplier–buyer coordination. Information sharing is the key of coordination. Coordination consists of horizontal coordination (Ryu & Lee, 2003; Hsu & Wee, 2005; Andersen & Christensen, 2005) and vertical coordination (Buzzell & Ortmeyer, 1995; Weng, 1995; Fites, 1996; Weng & McClurg, 2003; Fiala, 2005). Supply chain coordination can increase system profit and result in a win–win situation through profit sharing mechanism. Mutual trust monitoring mechanism (Hsu & Wee, 2005) is necessary to ensure low risk collaboration. Under the uncertainty of customer demand and supplier’s lead-time, Weng and McClurg (2003) had shown that system profit would significantly increase if both sides decided on the order quantity through discussion. However, they assumed constant selling price. In this study, we extend Weng and McClurg’s model (2003) to consider uncertain demand, stochastic lead-time and a single price decrease. The discounted cash flow (DCF) is considered in deriving the optimal order strategy. The compensation mechanism for profit sharing is also implemented for a win–win situation. The following is the organization of this study. Section 1 is devoted to introducing the background and the purpose of the study. Section 2 describes the assumptions and notation. Section 3 shows the model development. In Section 4, a special case is used to illustrate the model. Section 5 presents numerical studies to verify the model. Section 6 discusses the compensation mechanism. Conclusion and further research are given in the last section.

2. Assumptions and notations In this study, a supply chains with a buyer and supplier is assumed. The buyer obtains the products from the supplier for sale to the customers. The buyer has to consider the uncertainty of customers’ demand and delivery time (lead-time) from the supplier’s perspective. Placing an optimal order before the selling period of the product is vital to the buyer. During the selling season, the selling price of the product will drop when a new product is introduced into the market. This study divides the selling season into two stages: stage 1 and stage 2, i.e., the selling price of stage i is pi, i = 1, 2 (p1 > p2). The customers’ demands, depending on the selling price, are assumed to be random variables, i.e., Xi, i = 1, 2. The supplier’s order complete time is assumed to be a random variable Y. The extension to multiple stages is straightforward. When the order is delivered on time, the probability density function (PDF) and the cumulative distribution function (CDF) of demand in stage 1 are denoted by fX 1 ðx1 jy; y 6 0Þ and F X 1 ðx1 jy; y 6 0Þ, respectively. When the order is delivered late, the PDF and CDF of demand in stage 1 are denoted by fX 1 ðx1 jy; y > 0Þ and F X 1 ðx1 jy; y > 0Þ, respectively. The PDF and CDF of demand in stage 2 are denoted by fX 2 ðx2 Þ and F X 2 ðx2 Þ, respectively. The delivery delay will not go beyond stage 2. The buyer makes a contract with the supplier to deliver a predetermined amount of product at the beginning of the selling season. Because of the uncertainty of customer demand and price decrease, the buyer has to place the optimal order quantity on the basis of the buyer’s expected profit. Despite unpredictable production and processing time, the supplier must set a stipulated delivery date. If the supplier finishes the order earlier than requested, the inventory holding cost will be held by the supplier until the due date. If the supplier is late

604


with the delivery of the product, lost sale will result. In this situation, the supplier must compensate the buyer through lower purchase price. The following notation is used throughout the analysis: Eb buyer’s maximum expected profit without coordination EJb buyer’s maximum expected profit with coordination supplier’s maximum expected profit without coordination Es supplier’s maximum expected profit with coordination EJs E expected system profit without coordination (E = Eb + Es) maximum expected system profit with coordination (EJ = EJb + EJs) EJ Exb the buyer’s extra profit the supplier’s extra profit Exs r buyer’s shortage cost per unit; represents lost of goodwill costs s buyer’s salvage value per unit in stage 2 Q order quantity by the buyer at the beginning of stage 1 random demand faced by the buyer for stage i, i = 1, 2 Xi Y supplier’s order completion time, random variable (Y 6 0 means delivery to the buyer will be on time; Y > 0 means delivery to the buyer is delayed byy periods), i.e., lead-time g(y) probability density function (PDF) of the supplier’s order completion time t supplier’s per unit production cost h supplier’s per unit holding cost per period pi selling price at the market per unit in stage i, i = 1, 2 (p1 > p2 > s) buyer’s wholesale purchase price per unit for stage 1 when the supplier delivers on time co a discount factor for costs in stage 2, 0 < a 6 1 b delay factor of the customer’s demand in stage 1 (b > 0) h the negotiation factor (h P 0)

3. Model development and analysis 3.1. Without coordination Without coordination, the buyer will decide on the order quantity that maximizes its expected profit. The supplier tries to meet the stipulated delivery date despite uncertainties. In our model, delivery time is assumed to be independent of the order quantity. Since the selling season contains two stages, we consider the discounted cash flow (DCF) to estimate the net present value (NPV) (Thompson, 1975). (i) When the order is delivered on time, i.e., y 6 0. We obtain the decision model using backward deducing. Towards the end of stage 1, with leftover stock q, and discount factor a, the buyer’s conditional expected profit in stage 2, EB2(q), is Z q h c i co o s ðq x2 Þ fX 2 ðx2 Þdx2 p2 EB2 ðqÞ ¼ x2 a a 0 Z 1 h i co þ p2 ð1Þ q rðx2 qÞ fX 2 ðx2 Þdx2 : a q In stage 1, if the buyer’s order quantity is Q then the buyer’s conditional expected profit, EB(Qjy, y 6 0), is Z Q EBðQjy; y 6 0Þ ¼ ½ðp1 co Þx1 þ aEB2 ðQ x1 ÞfX 1 ðx1 jy; y 0 Z 1 ½ðp1 co ÞQ rðx1 QÞ þ aEB2 ð0ÞfX 1 ðx1 jy; y 6 0Þdx1 : ð2Þ 6 0Þdx1 þ Q

Let Qb be the buyer’s optimal order quantity, the buyer’s maximum expected profit, Eb, is


605

Eb ¼ EBðQb jy; y 6 0Þ Z Qb x1 Z Qb co ðQb x1 Þ rðEðX 2 Þ Qb þ x1 Þ ðp2 s þ rÞ ðp1 co Þx1 þ a p2 F X 2 ðx2 Þdx2 ¼ a 0 0 Z 1 fX 1 ðx1 jy; y 6 0Þdx1 þ ½ðp1 co ÞQb rðx1 Qb Þ arEðX 2 ÞfX 1 ðx1 jy; y 6 0Þdx1 Qb

¼ r½EðX 1 jy; y 6 0Þ þ aEðX 2 Þ þ ½p1 ap2 þ rð1 aÞCðQb jy; y 6 0Þ þ ðp1 co þ rÞQb ½p1 ap2 þ rð1 aÞQb F X 1 ðQb jy; y 6 0Þ Z Qb x1 Z Qb aðp2 s þ rÞ F X 2 ðx2 Þdx2 fX 1 ðx1 jy; y 6 0Þdx1 ; þ 0

ð3Þ

0

RQ where CðQjy; y 6 0Þ ¼ 0 x1 fX 1 ðx1 jy; y 6 0Þdx1 . Due to the supplier’s earlier delivery, the inventory holding cost will be held by the supplier until the due date. The supplier’s expected profit without coordination when y 6 0 is Z 0 ðco t þ hyÞgðyÞdy: 1

(ii) When the order is delivered late for y unit time, i.e., y > 0. Since there is a potential loss for the buyer due to delivery delay from the supplier, the supplier must compensate the buyer through discounted wholesale price, which is under the ‘‘rule of financial equivalent’’. That is, the supplier would be required to make payment to the buyer such that the buyer would be in the same position he would have been (Whang, 1992). The discounted wholesale price c(y) that satisfies the contractual requirement EB(Qbjy, y > 0) = Eb, is cðyÞ ¼ frEðX 1 jy; y > 0Þ þ ½p1 ap2 þ rð1 aÞCðQb jy; y > 0Þ þ ðp1 þ rÞQb arEðX 2 Þ Z Qb Z Qb x1 aðp2 s þ rÞ F X 2 ðx2 Þdx2 ½p1 ap2 þ rð1 aÞQb F X 1 ðQb jy; y > 0Þ þ 0 0 fX 1 ðx1 jy; y > 0Þdx1 Eb Qb ; ð4Þ RQ where CðQjy; y > 0Þ ¼ 0 x1 fX 1 ðx1 jy; y > 0Þdx1 . That Ris, EB(Qbjy, y > 0) = EB(Qbjy, y 6 0), and the supplier’s expected profit without coordination when 1 y > 0 is 0 ðcðyÞ tÞgðyÞdy. Therefore, the supplier’s expected profit without coordination is Z 1 Z 0 ðcðyÞ tÞgðyÞdy þ ðco t þ hyÞgðyÞdy : ð5Þ Es ¼ Qb 0

1

The expected system profit without coordination, E, will be E = Eb + Es. 3.2. With coordination We further consider the situation with coordination. That is, both the buyer and the supplier jointly determine the buyer’s ordering quantities through information sharing to maximize the expected system profit. If the buyer’s order quantity is Q, then the buyer’s expected profit will be EB(Qjy, y 6 0); the discounted wholesale price c(y, Q) satisfies the contractual requirement that is EB(Qjy, y > 0) = EB(Qjy, y 6 0): cðy; QÞ ¼ rEðX 1 jy; y > 0Þ þ ½p1 ap2 þ rð1 aÞCðQjy; y > 0Þ þ ðp1 þ rÞQ arEðX 2 Þ Z Q Z Qx1 ½p1 ap2 þ rð1 aÞQF X 1 ðQjy; y > 0Þ þ aðp2 s þ rÞ F X 2 ðx2 Þdx2 0 0 fX 1 ðx1 jy; y > 0Þdx1 EBðQjy; y 6 0Þ Q: ð6Þ

606


The supplier’s expected profit is EJs ðQÞ ¼ Q system profit EJ(Q) is as follows:

hR 1 0

ðcðy; QÞ tÞgðyÞdy þ

Z EJ ðQÞ ¼ EJb ðQÞ þ EJs ðQÞ ¼ EBðQjy; y 6 0Þ þ Q

i ðc t þ hyÞgðyÞdy . The expected o 1

R0

1


Z

0

ðco t þ hyÞgðyÞdy :

ð7Þ

1

0

Let QJ be the optimal system order quantity with coordination; i.e., maximize EJ(Q), and EJ = EJ(QJ), EJb = EJb(QJ), EJs = EJs(QJ). We can easily show that maximum expected system profit with coordination, EJ, is better than the one without coordination, E (See Appendix A). 4. An illustrative case A special case is used to illustrate the model where the probability distribution of the random demand is exponential and the lead-time is uniform. The random demand faced by the buyer for stage i, i = 1, 2, is exponentially distributed with parameter ki. We define a period in which the expected demand is one unit per period. Given that the order is completed by the supplier at time y periods, the PDF of the realized demand for the buyer is fX i ðxi jy; y 6 0Þ ¼ ki eki xi ;

i ¼ 1; 2;

when the order is delivered on time. fX 1 ðx1 jy; y > 0Þ ¼ ðk1 þ byÞeðk1 þbyÞx1 ; when the order is delivered late. fX 2 ðx2 jy; y > 0Þ ¼ fX 2 ðx2 jy; y 6 0Þ: This means that the delivery delay will not go beyond stage 2. The supplier’s actual order completion date is uniformly distributed over the interval (b, b), where b represents the number of time periods in the planning season. We further derive the buyer’s optimal order quantity without coordination, Qb, that maximizes the buyer’s expected profit in the absence of supplier–buyer coordination. Proposition 1. If p1 > p2 > s r, for exponential demand distribution, the buyer’s optimal order quantity Qb satisfies the following expression: k1 aðp2 s þ rÞ k1 aðp2 s þ rÞ k2 Qb as þ p1 þ r ek1 Qb e þ as co ¼ 0; k1 6¼ k2 ; k2 k1 k2 k1 ð8Þ ½k1 aðp2 s þ rÞQb as þ p1 þ rek1 Qb þ as co ¼ 0;

k1 ¼ k2 :

Proof. Please refer to Appendix B. In addition, we find the optimal system order quantity, QJ, which maximizes the expected system profit in the presence of supplier–buyer coordination. Proposition 2. If p1 > p2 > s r, for exponential demand distribution, the optimal system order quantity QJ satisfies the following expression:

Z 1

d EBðQjy; y 6 0ÞjQ¼QJ þ ðp1 þ r tÞ ½p1 ap2 þ rð1 aÞF X 1 ðQJ jy; y > 0Þ dQ 0 0 Z Q Z Qx1 d aðp2 þ s rÞ F X 2 ðx2 ÞfX 1 ðx1 jy; y > 0Þdx2 dx1 jQ¼QJ gðyÞdy þ dQ 0 0 Z 0 ðc0 t þ hyÞgðyÞdy ¼ 0: ð9Þ þ

1

Z

1

gðyÞdy

1

Proof. Please refer to Appendix C.


607

Table 1 Parameter values Parameter

Value

Discount factor for costs in stage 2 (a) Delay factor of the customer’s demand in stage 1 (b) Unit cost to buyer (co) Upper bound on delivery time (b) Unit revenue to buyer in stage 1 (p1) Unit revenue to buyer in stage 2 (p2) Parameter of exponentially demand distribution in stage 1 (k1) Parameter of exponentially demand distribution in stage 2 (k2) Unit salvage value (s) Unit shortage cost (r) Unit production cost (t) Period holding cost factor (h)

0.92, 0.95, 0.98 1.105, 4.106, 6.106, 8.106 45, 55, 65 70, 75, 80 100 75, 80, 85 0.002 0.003 4.5 5 15 0.001

Qb 719 579 457 719 579 457 719 579 457

α =0.95 ,

% profit increase QJ 1399 20.30 % % 1399 36.79 % 1399 62.28 % 1394 20.10 % 1394 36.50 % 1394 61.88 % 1390 19.91 1390 36.24 % 1390 61.49 %

p 2 =75 ,

C0

b

Qb

QJ

45 55 65 45 55 65 45 55 65

70 70 70 75 75 75 80 80 80

729 588 466 729 588 466 729 588 466

1414 1414 1414 1409 1409 1409 1405 1405 1405

α =0.98 , C0 45 55 65 45 55 65 45 55 65

b 70 70 70 75 75 75 80 80 80

p 2 =75 ,

Qb 739 598 474 739 598 474 739 598 474

60 50

Co=45

40

Co=55

20 10 0

b=75

b=80

β =1 × 10-5

70 60 50 Co=45

40

Co=55 30

Co=65

20 10 0 b=70

b=75

α =0.98 , p 2=75 ,

β =1 × 10 - 5

QJ % profit increase % 1429 19.83 % 1429 35.73 % 1429 60.33 % 1424 19.64 1424 35.45 % 1424 59.95 % 1420 19.45 % 1420 35.19 % 1420 59.57 %

b=70

α =0.95 , p 2=75 ,

% profit increase % % % % % % % % %

Co=65

30

β =1 × 10 - 5 20.07 36.33 61.15 19.87 36.06 60.77 19.68 35.79 60.39

β =1 × 10-5

70

% profit increase

b 70 70 70 75 75 75 80 80 80

α =0.92 , p 2=75 ,

β =1 × 10 - 5

% profit increase

C0 45 55 65 45 55 65 45 55 65

p 2 =75 ,

b=80

β =1 × 10-5

70 60

% profit increase

α =0.92 ,

50 Co=45 40

Co=55 Co=65

30 20 10 0 b=70

b=75

b=80

Fig. 1. The effect of discount factor a for costs in stage 2 on the increase in system profits.

608


5. Numerical case studies In order to further investigate the effect of the coordination under uncertain demand and lead-time, different parameter values are assumed. In Theorem 1, the maximum expected system profit with coordination is always better than the one without coordination. In Table 1, the percentage of profit change for each variable is tabulated when the other parameters (k1, k2, s, r, t, p1, h) are fixed. The parameters, a = 0.95, p2 = 75, b = 1 · 105, are used as the standard values. Figs. 1–3 show the optimal order quantity with and without coordination. In Fig. 1, the delay factor of the customer’s demand in period 1, b, is fixed, but the discount factor for cost in stage 2, a, is a variable. When a increases, both Qb and QJ increase but the percentage of profit increase drop. In such situation the effect of coordination is insignificant. In Fig. 2, both a and b are fixed but p2 is a variable. When p2 increases, both Qb and QJ increase accordingly but the percentage of profit increase tends to drop. This means the effect of coordination will decrease as p2 increases to approach p1. In this situation the effect of coordination is less significant. In Fig. 3, both a and p2 are fixed but b is a variable. When b increases Qb remains unchanged but QJ decreases and the percentage profit increases. Due to the contractual requirement, it will not affect the buyer’s profit, and Qb remains unchanged. In summary, when the supplier delivers on time, the buyer’s wholesale purchase price per unit, co, increases. The value of Qb decreases while QJ is unchanged, and the percent profit increases. We use

Qb 746 605 482 746 605 482 746 605 482

α =0.95 , C0 45 55 65 45 55 65 45 55 65

b 70 70 70 75 75 75 80 80 80

p 2 =85 , Qb 763 622 498 763 622 498 763 622 498

α =0.95 , C0 45 55 65 45 55 65 45 55 65

b 70 70 70 75 75 75 80 80 80

QJ % profit increase 1432 19.46 % % 1432 35.03 1432 58.76 % % 1428 19.27 % 1428 34.77 % 1428 58.38 1424 19.09 % 1424 34.51 % 1424 58.01 %

Qb 780 638 514 780 638 514 780 638 514

QJ 1450 1450 1450 1446 1446 1446 1442 1442 1442

p 2 =90 ,

60 Co=45 Co=55 Co=65

40 30 20

0 b=70

b=75

α =0.95 , p 2=85 ,

b=80

β =1 × 10-5

60 50 40 Co=45 Co=55 Co=65

30 20 10 0

b=70

b=75

α =0.95 , p 2=90 ,

β =1 × 10 - 5

QJ % profit increase % 1468 18.25 1468 32.67 % 1468 54.26 % % 1464 18.08 1464 32.42 % % 1464 53.91 1459 17.90 % 1459 32.18 % % 1459 53.57

50

10

β =1 × 10 - 5 %profit increase % 18.86 33.77 % 56.46 % 18.67 % % 33.51 % 56.10 18.50 % 33.26 % 55.75 %

β =1 × 10-5

70

% profit increase

b 70 70 70 75 75 75 80 80 80

α =0.95 , p 2=85 ,

β =1 × 10 - 5

% profit increase

C0 45 55 65 45 55 65 45 55 65

p 2 =80 ,

b=80

β =1 × 10-5

60 50

% profit increase

α =0.95 ,

40

co=45 co=55 co=65

30 20 10 0 b=70

b=75

b=80

Fig. 2. The effect of selling price p2 at the market per unit in stage 2 on the increase in system profits.


Qb 729 588 466 729 588 466 729 588 466

α =0.95 , C0 45 55 65 45 55 65 45 55 65

b 70 70 70 75 75 75 80 80 80

Qb 729 588 466 729 588 466 729 588 466

α =0.95 , C0 45 55 65 45 55 65 45 55 65

b 70 70 70 75 75 75 80 80 80

QJ % profit increase 1451 21.77 % % 1451 38.76 1451 64.59 % 1449 21.66 % 1449 38.60 % 1449 64.38 % % 1446 21.56 1446 38.46 % 1446 64.17 %

QJ % profit increase % 1438 21.15 1438 37.88 % % 1438 63.35 % 1434 21.01 1434 37.68 % 1434 63.07 % 1431 20.87 % 1431 37.48 % 1431 62.80 %

QJ % profit increase 1425 20.59 % 1425 37.08 % 1425 62.21 % 1421 20.42 % 1421 36.84 % 1421 61.87 % 1417 20.25 % 1417 36.60 % 1417 61.54 %

50 co=45 co=55 co=65

40 30 20 10 0 b=70

b=75

α =0.95 , p 2=75 , 70

b=80

β =6 × 10-6

60 50 co=45 co=55 co=65

40 30 20 10 0

β =8 × 10 - 6

p 2 =75 , Qb 729 588 466 729 588 466 729 588 466

60

β =6 × 10 - 6

p 2 =75 ,

β =4 × 10-6

70

% profit increase

b 70 70 70 75 75 75 80 80 80

α =0.95 , p 2=75 ,

% profit increase

C0 45 55 65 45 55 65 45 55 65

β =4 × 10 - 6

p 2 =75 ,

b=70

b=75

α =0.95 , p 2=75 , 70

b=80

β =8 × 10-6

60

% profit increase

α =0.95 ,

609

50 co=45

40

co=55

30

co=65

20 10 0 b=70

b=75

b=80

Fig. 3. The effect of delay factor b of the customer’s demand in stage 1 on the increase in system profits.

Table 2 Detailed values of the first three items in Fig. 1 a = 0.98, p2 = 75, b = 1 · 105 Co 45

b

70

Qb

QJ

739

1429

Eb Es

17854 21049

598 55

70

Eb Es

65

70

Eb Es

6368 40250

64.33 91.22

7922 54540

171.79 135.52

22212 68830

480.21 196.23

1429 11191 23157

474

EJ(1429) = EJb + EJs = 6368 + 40250 = 46618. EJ(1429) = EJb + EJs = 7922 + 54540 = 46618. EJ(1429) = EJb + EJs = 22212 + 68830 = 46618.

EJb EJs

% Profit increase

EJb EJs 1429

5842 23235

EJb EJs

610


the first three items in list 3 of Fig. 1 as an example (see Table 2) to see why QJ remains unchanged. With b = 70, co ranges from 45 to 65, QJ remains unchanged as 1429, and the maximum expected system profit fixes at $46618 with EJb ranges from $6368 to $–22212 (loss), EJs ranges from $40250 to $68830. This means a higher co is not good for the buyer, but the maximum expected system profit remains the same. This is because ocoo EJ ðQ; co Þ ¼ 0:25 1010 Q. Consequently, ocoo EJ ðQ; co Þ ! 0, for any 0 < Q < 2000, the influence of EJ to co is limited. When the upper bound on delivery time, b, increases, Qb remains the same but QJ drops, so the percentage of profit increases. This is because the contractual requirement will influence the supplier’s profit but not the buyer’s profit. This results in decreasing QJ. The phenomenon is due to increased uncertainty of lead-time. The result is the same as when the selling price is constant (Weng & McClurg, 2003). To improve this condition, the manager should aim at shortening the lead-time. 6. Compensation mechanism Although there is an increase in the maximum expected system profit with coordination, the gain is always unilateral. The distribution contract of the maximum expected system profit with coordination is assumed to compensate the buyer’s loss (Eb–EJb), The remaining value K = (EJs–Es)–(Eb–EJb) = EJ–E, (the distribution ratio of Exb = hExs, where h is the negotiation factor; Exb) is the buyer’s extra profit, and Exs is the supplier’s extra profit. hK K hK Then Exb ¼ hþ1 , Exs ¼ hþ1 , the buyer’s actually expected profit after distribution contract is S Jb ¼ Eb þ hþ1 , K the supplier’s actually expectedprofit after distribution contract is S Js ¼ Es þ hþ1. The amount supplier should K compensate the buyer is EJs Es þ hþ1 . Table 3 shows the change in profit for variable negotiation factor and other fixed parameter. For the parameters in the table, the profit gain with coordination will be k = EJ–E = $7715 as compared to the case without coordination. When h = 0.5 the buyer’s extra profit is $2572. The buyer’s actually expected profit after distribution contract is $20426 (Buyer’s maximum expected profit without coordination, Eb, is $17854), The supplier’s extra profit is $5143. The supplier’s actually expected profit after contract is $26192 (Supplier’s maximum expected profit without coordination, Es, is $21049).The supplier has to compensate $14058 to the buyer. 7. Conclusion The study develops a supply chain model for products with short life cycle and variable selling price. A coordination policy is discussed for comparison. Coordination between the supplier and the buyer through information sharing can effectively improve the expected system profit. Both sides can reach a better profit through a compensation mechanism. The buyer needs to evaluate accurately the customer’s demand for profit distribution. On the other hand, the supplier needs to evaluate the value of cooperating with the buyer and reducing the lead-time.

Table 3 Sensitivity analysis for negotiation factor a = 0.98, p2 = 75, b = 1 · 105, Co = 45, b = 70, Qb = 739, QJ = 1429, Eb = 17854, EJb = 6368, Es = 21049, EJs = 40250 h

Exb

SJb

Exs

SJs

Supplier’s compensation to the buyer

0 0.5 1 10 100

0 2572 3857 7014 7639

17854 20426 21712 24868 25493

7715 5143 3857 701 76

28764 26192 24907 21750 21125

11486 14058 15344 18500 19125


611

When we apply this model to real business case study, two issues deserve our attention. Firstly, correct information sharing relies on trust and accurate prediction. Trust is the corner stone of market economies. In this study, we assume that both parties agree with the buyer’s distribution of demand and the supplier’s distribution of lead-time. Secondly, success in maintaining the contract and compensation mechanism needs cooperation to achieve a win–win strategy. For the discount factor, a, we use the average discount rate when the lifecycle is short; otherwise continuous discount rate is used. As to the sensitivity analysis, when p2 increases, the percentage profit increase will drop. This means the worse the business condition, the better the effect of the coordination. This also helps the manager to understand the importance of cooperation. Appendix A. Proof of EJ > E From Eq. (7), EJ ðQÞ ¼ EJb ðQÞ þ EJs ðQÞ ¼ EBðQjy; y 6 0Þ þ Q

Z

1


Z

0

Then EJ(QJ) = EJ is the global optimal.

0

ðco t þ hyÞgðyÞdy :

1

h

Appendix B. Proof of proposition 1 (i) When k1 5 k2 d d EBðQjy; y 6 0Þ ¼ aðp2 þ s rÞ dQ dQ

Z

Q

Z

Qx1

F X 2 ðx2 ÞfX 1 ðx1 jy; y 6 0Þdx2 dx1 þ ðp1 co þ rÞ k1 aðp2 s þ rÞ þ ðap2 p1 þ ar rÞ F X 1 ðQjy; y 6 0Þ ¼ as þ p1 þ r ek1 Q k2 k1 k1 aðp2 s þ rÞ k2 Q e þ as co : ð10Þ k2 k1

Solve

d EBðQjy; y dQ

0

0

6 0Þ ¼ 0, Q > 0, with respect to Q, we have Qb.

d2 k2 k1 EBðQjy; y 6 0Þ ¼ aðp2 s þ rÞ ðek1 Q ek2 Q Þ ½p1 ap2 þ rð1 aÞk1 ek1 Q : 2 k2 k1 dQ

ð11Þ

With a(p2 s + r) > 0, if k2 < k1 then k2 k1 < 0, and ek1 Q ek2 Q < 0, "Q > 0, the first term of Eq. (11) will be negative; if k2 > k1 then k2 k1 > 0, and ek1 Q ek2 Q > 0, "Q > 0, the first term of Eq. (11) will be negative too. d2 When p1 ap2 > 0, r(1 a) > 0, k1 ek1 Q > 0, the second term will be negative; we have dQ 2 EBðQjy; y 6 0Þ < 0, "Q > 0, then EB(Qjy, y 6 0) is a convex function. (ii) When k1 = k2 d EBðQjy; y 6 0Þ ¼ ½k1 aðp2 s þ rÞQ as þ p1 þ rek1 Q þ as co : dQ Set

d EBðQjy; y dQ

ð12Þ

6 0Þ ¼ 0, Qb is derived.

d2 EBðQjy; y 6 0Þ ¼ ½k1 aðp2 s þ rÞQ þ ðp1 ap2 Þ þ ð1 aÞrðk1 Þek1 Q : dQ2

ð13Þ

Since p2 s + r > 0, p1 ap2 > 0, 1 a > 0, therefore k1a(p2 s + r)Q + (p1 ap2) + (1 a)r > 0. In addid2 tion, when k1 < 0, one has dQ h 2 EBðQjy; y 6 0Þ < 0, then EB(Qjy, y 6 0) is a convex function.

612


Appendix C. Proof of proposition 2 d We derive QJ by solving dQ EJ ðQÞ ¼ 0, Q > 0 with respect to Q. 2 Z 1 Z 1 2 d d E ðQÞ ¼ 1 gðyÞdy EBðQjy; y 6 0Þ þ ½p1 ap2 þ rð1 aÞfX 1 ðQjy; y > 0Þ J dQ2 dQ2 0 0 Z Q Z Qx1 d2 2 aðp2 s þ rÞ F X 2 ðx2 ÞfX 1 ðx1 jy; y > 0Þdx2 dx1 gðyÞdy dQ 0 0 Z 1 1 d2 EBðQjy; y 6 0Þ ½p1 ap2 þ rð1 aÞðk1 þ byÞeðk1 þbyÞQ gðyÞdy ¼ 2 dQ2 0 Z 1 ðk1 þ byÞk2 ðk1 þbyÞQ ½e aðp2 s þ rÞ ek2 Q gðyÞdy: k2 k1 by 0 d2 dQ2 ðk1 þbyÞQ

Since

ð14Þ

EBðQjy; y 6 0Þ < 0, "Q > 0 (see the proof in Appendix B); p1 ap2 + r(1 a) > 0, ðk1 þ byÞ

1 þbyÞk2 > 0, "Q > 0; aðp2 s þ rÞ ðk ½eðk1 þbyÞQ ek2 Q > 0, "Q > 0 (the proof is analogous to Theorem 2 e k2 k1 by when substituting k1 + by for k1, here we neglect the situation that k1 + by = k2, because the integrand of last term 2 in (14) will be aðp2 s þ rÞðk1 þ byÞ Qeðk1 þbyÞQ gðyÞ > 0, it does not affect the integral); and g(y) > 0, " y > 0; then d2 E ðQÞ < 0, "Q > 0; therefore EJ (Q) is a convex function. h dQ2 J

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