Optimum profit model based on order quantity, product price, and process quality level

Expert Systems with Applications 38 (2011) 7886–7893

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Short Communication

Optimum profit model based on order quantity, product price, and process quality level Chung-Ho Chen ⇑, Chih-Lun Lu Institute of Industrial Management, Southern Taiwan University, 1 Nan-Tai Street, Yung-Kang City, Tainan 710, Taiwan

a r t i c l e

i n f o

Keywords: Order quantity Process mean Product price Taguchi’s symmetric quadratic quality loss function

a b s t r a c t The maximum expected profit model between the producers and the purchasers is an important objective for the supply chain system. The producer’s profit needs to consider the problem of sales revenue and manufacturing cost. The purchaser’s profit needs to consider the problem of order quantity and used cost of customer. How to get a trade-off between them should be an important topic. Chen and Liu (2007) presented the optimum profit model between the producers and the purchasers for the supply chain system with pure procurement policy. However, their model with simple manufacturing cost did not consider the used cost of customer. In this study, the modified Chen and Liu’s model will be addressed for determining the optimum product and process parameters. The authors propose a modified Chen and Liu’s model with indirectly measurable quality characteristic under the single stage screening procedure. The surrogate variable with high correlation with indirectly measurable quality characteristic will be directly measured. The used cost of customer can be obtained by adopting Taguchi’s quadratic quality loss function. The optimum purchaser’s order quantity, the producer’s product price, and the process quality level will be jointly determined by maximizing the expected profit between them. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The trade-off problem between the producers and the purchasers is an important problem for the manufacturing industry. The producers need to consider the sale price and the manufacturing cost of product for obtaining the maximum profit. The purchasers need to consider the order quantity and the customer’s used cost of product for obtaining the maximum profit. Hence, the supply chain system needs to solve ‘‘how to get a trade-off between them’’. Seifert, Thonemann, and Hausman (2004) explained the resulting procurement change and proposed the benefits of using spot markets from a supply chain perspective. They developed the mathematical model that determines the optimal order quantity to purchase via forward contracts and spot markets. Haksoz and Seshadri (2007) discussed about supply chain system with spot market and presented a literature review about recent work. Chen and Liu (2007) presented the optimum profit model between the producers and the purchasers for the supply chain system with pure procurement policy from the regular supplier and mixed procurement policy from the regular supplier and the spot market. Chen and Liu (2008) further proposed an optimal consignment policy considering a fixed fee and a per-unit commission. Their model determines a higher manufacturer’s profit than the traditional production system and coordinates the retailer to obtain a ⇑ Corresponding author. E-mail address: [email protected] (C.-H. Chen). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.12.046

large supply chain profit. Li and Liu (2008) considered the problem about the retailer determining his optimal order quantity and the manufacturer determining his optimal reserve capacity. Their model can make two sides of the supply chain increase profit. The selection of optimum process quality level setting and the process improvement are the important topics for canning/filling industry. The study of canning/filling problem started from Springer (1951). He considered the scrap cost in the production process and tried to determine the optimum process mean. Most of the measurable quality characteristic of product is considered for setting the optimal process quality level. However, some situations are not easy to directly measure the product characteristic of interest. One needs to obtain the optimal process parameters of product characteristic of interest by measuring the indirectly product characteristic. Lee, Hong, Kwon, and Kim (2000) considered the single-stage screening procedure for determining the screening limits of surrogate variable and the optimum process mean of performance variable. They assumed that the surrogate variable is high correlated with the performance variable and the product is classified as three grades. However, they did not consider the quality loss of product within specifications for the customer. Hence, their model underestimate the quality cost of product. Lee, Hong, and Elsayed (2001) and Lee and Elsayed (2002) further extended the modified models based on the two-stage screening procedure for determining the optimum process parameters. Taguchi (1986) proposed the concept of quadratic quality loss function and redefined the product quality as the loss of society

C.-H. Chen, C.-L. Lu / Expert Systems with Applications 38 (2011) 7886–7893

for the sold product. Taguchi’s (1986) quadratic quality loss function has successfully applied in some topics of quality control. It connects with the methodology of process control and quality improvement. Recently, Chen (2006), Chen and Kao (2008), Chen and Khoo (2008) and Chen and Lai (2007a, 2007b) have addressed about Taguchi’s quality loss function applied in the economic specification limits setting, optimum process quality level setting, and economic manufacturing quantity model. Chen and Liu’s (2007) model presented the optimum profit model between the producers and the purchasers for the supply chain system. However, their model with simple manufacturing cost did not consider the used cost of customers in pure procurement policy. Hence, the modified Chen and Liu’s (2007) model will be addressed for determining the optimum product and process parameters. In this study, the author will propose a modified Chen and Liu’s (2007) model with indirectly measurable quality characteristic under the single stage screening procedure. The surrogate variable with high correlation with indirectly measurable quality characteristic will be directly measured. The used cost of customer can be obtained by adopting Taguchi’s quadratic quality loss function. The optimum purchaser’s order quantity and the producer’s product price and process quality less will be jointly obtained by maximizing the expected profit of them. The motivation behind this work stems from the fact that the neglect of the quality loss within the specification limits should have the overestimated expected profit per item. 2. Literature review – Chen and Liu’s (2007) pure procurement model The Chen and Liu’s (2007) pure procurement model is actually based on the standard news-vendor model without spot markets. They consider a single period supplier–buyer relationship in which a regular supplier produces short-life cycle products and a buyer orders products from the regular supplier and then sells to the end customer. Assumptions: 1. A buyer purchases a finished product from a regular supplier and resells it at a price, R, to the end customer. 2. The regular supplier produces each unit at a cost, C. 3. The regular supplier and the buyer enter into a contract at a wholesale price, W. 4. The regular supplier sets the wholesale price to maximize his expected profit while offering the buyer a specific order quantity, Q. 5. When realized demand exceeds procurement quantity, unmet demand is lost; therefore, demand uncertainty exposes the buyer to risks associated with mismatches between the procurement quantity and demand. 6. The procurement lead time is long relative to the selling season, so that the buyer cannot observe demand before placing the order. 7. The consumer demand, X, is   an uniform distribution, i.e., X  U lx  ðrx =2Þ; lx þ ðrx =2Þ , where lx is the mean of X and rx is the standard deviation of X. The buyer’s profit is given by

pRPS ¼



RX  WQ þ SðQ  XÞ; X < Q ; RQ  WQ ;

X P Q;

ð1Þ

where R is the sales price per unit; S is the salvage value per unit; C is the regular supplier’s production cost per unit; Q is the quantity procured by the buyer from the regular supplier; X is the stochastic demand; W is wholesale price per unit, paid by the buyer to the regular supplier.

7887

The buyer’s expected profit can be expressed as

  E pRPS ¼

Z

Q

½Rx  WQ þ SðQ  xÞf ðxÞdx

lx ðrx =2Þ Z lx þðrx =2Þ

þ

½ðR  WÞQ f ðxÞdx;

ð2Þ

Q

where f(x) is the probability distribution of X. Let the partial derivative of the buyer’s expected profit function dEðpR Þ with respect to Q be zero, i.e., dQPS ¼ 0. We have the optimal order quantity

Q ¼

1 r  ðR  WÞrX lx  x þ : 2 2 RS

ð3Þ

The regular supplier maximizes his expected profit and determines the wholesale price per unit based on the buyer’s order quantity Q⁄. Hence, the regular supplier’s expected profit is expressed as

  E pSPS ¼ ðW  CÞQ  :

ð4Þ

Let the partial derivative of the regular supplier’s expected profdEðpS Þ it function with respect to W be zero, i.e., dWPS ¼ 0. The optimal W value of Eq. (4) yields

W ¼

ðR  SÞlx ðR þ S þ 2C Þ : þ 4 2r x

ð5Þ

Substituting Eq. (5) into Eq. (3), the optimal Q value can rewritten as

Q ¼

1 r  ðR  CÞrX l  x þ : 2 x 2 2ðR  SÞ

ð6Þ

From Eqs. (2), (4)–(6), the buyer’s and supplier’s expected profit can be expressed as

h i   2 rX 2  R  ðR  SÞ ðQ Þ  lX  2 E pPS ¼ ; 2r X

  ðR  SÞlX R þ S  2C  E pSPS ¼ Q : þ 4 2r X

ð7Þ ð8Þ

3. Modified Chen and Liu’s (2007) pure procurement model Assume that the supplier consider the problem of determining the optimum process quality level of the quality characteristic of interest under single-stage screening. The inspection is performed directly on the surrogate variable that is high correlated with the quality characteristic of interest. Let Y be the quality characteristic of interest. L is the lower specification limit of Y. U is the upper specification limit of Y. Y is the nor  mal quality characteristic of product, Y  N ly ; r2y , where lY is the unknown mean of Y and rY is the known standard deviation h of Y. Let Z be a variable that is positively correlated with Y; Z  N k1 þ k2 ly ; k22 r2y þ r2 , where k1, k2, and r2 are the known constants. L and U are also the lower and upper screening limits on the decision variable Z, respectively. An accepted item with Y < L or Y > U incurs the penalty cost d which includes the cost of identifying and handling the non-conforming item, and the service and replacement cost. If Z < L or Z > U, the item is scrapped and sold at a discounted priceSp. The joint distribution of (Y, Z) is a bi-variate normal distribution. Chen and Liu’s (2007) pure procurement model with constant manufacturing cost per unit for the supplier is too simple. In fact, the manufacturing cost per unit should include the constant and variable production costs. The variable production cost is proportional to the value of quality characteristic. Chen and Liu’s (2007) model also did not consider the used cost of customers. The neglect of the quality loss within the specification limits should have the overestimated expected profit per item for the buyer. Hence, the authors propose the following modified Chen and Liu’s (2007) model.

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C.-H. Chen, C.-L. Lu / Expert Systems with Applications 38 (2011) 7886–7893

where

The buyer’s profit is given by

pRPS ¼



RX  WQ þ SðQ  XÞ  X  LossðYÞ; X < Q ; 1 < Y < 1; RQ  WQ  Q  LossðYÞ; X P Q ; 1 < Y < 1;

A1 ¼

L  ly ; ry r pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

deviation of Y; L is the lower specification of product; U is the upper specification limit of product; Loss(Y) is the quality loss per unit, LossðYÞ ¼ kðY  y0 Þ2k is the quality loss coefficient; y0 is the target value of product. From Appendix A, the buyer’s expected profit is h

     rx i 1 rx 2 E pRPS ¼ QðS  WÞ Q  lx þ þ ðR  SÞ Q 2  lx  2 2 2

Z  1 1 rx 2 1 2   Q  lx  LossðyÞf ðyÞdy rx 2rx 2 1   i rx 1 1h r  lx þ x  Q þ ½R  WQ lx þ  Q 2 rx rx 2 Z 1

LossðyÞf ðyÞdy;

Q

ð10Þ

ð11Þ

U() is the cumulative distribution function of standard normal random variable /() is the probability density function of standard normal random variable. The supplier’s profit is given by

p

U  ly ; ry r pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

B2 ¼

k2 r2y þr2

½ðR  SÞ  Alx  12 ½ðS þ R  2WÞ  Arx ; ðS  RÞ þ A S R  A rx 2 þ 2 þ t  2 þ lx ½ðR  SÞ  A ; W ¼ 2rx

Q ¼

k 2 ry

r

:

ð14Þ ð15Þ

where

b þ cly þ i1 !

t¼

Uk1 k2 ly

Lk1 k2 ly

!

ffi  U pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 "

k2 ry þr

Lk1 k2 ly

!

k2 ry þr

Uk1 k2 ly

!#

ffi þ 1  U pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Sp U pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 k2 ry þr



Uk1 k2 ly

!

k2 ry þr

Lk1 k2 ly

k2 ry þr

1

S PS

A2 ¼

!

ffi  U pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2

where

1  2

: LossðyÞf ðyÞdy ¼ k r2y þ ly  y0

;

From Appendix A, the optimum order quantity Q⁄ for the buyer and the optimum selling price W⁄ for the supplier are as follows:

1

Z

r

k2 r2y þr2

ð9Þ where Y is the normal quality characteristic of product, Y    N ly ; r2y ; lY is the unknown mean of Y; rY is the known standard

k2 ry

B1 ¼

8 L < z < U; L < y < U; < W  b  cy  i1 ; ¼ W  b  cy  i1  d; L < z < U; y < Lory > U; : z < L or z > U; 1 < y < 1; Sp  b  cy  i1 ;

ð12Þ where W is the selling price per unit for the conformance product; Sp is discounted price per unit for the non-conformance product scrapped; b is the constant production cost per unit; c is the variable production cost per unit; i1 is the inspection cost per unit; d is the penalty cost when the non-conforming item is determined as the conforming one. From Appendix A, the supplier’s expected profit is 8 2 0 1 0 13 > < U  k  k l L  k  k l 1 2 1 2 6 B B yC y C7 EðpP Þ ¼ W 4U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 > : k22 r2y þ r2 k22 r2y þ r2 2 0 1 0 13

BU  k1  k2 ly C7 6 BL  k1  k2 ly C þ Sp 4U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ 1  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 2 2 2 k2 ry þ r k22 r2y þ r2 2 0 1 U  k  k l A1 B1 C 1 2 y 6 B  d4W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 k22 r2y þ r2 1 þ B1 1 þ B21 0 13 L  k  k l A B 1 2 y 1 1 B C7 W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5 k22 r2y þ r2 1 þ B21 1 þ B21 82 0 1 0 13 > < 6 BU  k1  k2 ly C BL  k1  k2 ly C7  d 4U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 > : k22 r2y þ r2 k22 r2y þ r2 0 1 U  k  k l A2 B2 C 1 2 y B  W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA k22 r2y þ r2 1 þ B22 1 þ B22 9 0 19 > > = A2 B2 C= BL  k1  k2 ly þW@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA  b  cly  i1 > > ; k22 r2y þ r2 1 þ B22 1 þ B22 ; Q ; ð13Þ  RU R1 gðy; zÞdy dz L 1

k2 ry þr

! 8" > Uk1 k2 ly A1 B1 > ffi ; pffiffiffiffiffiffiffiffi2 ; pffiffiffiffiffiffiffiffi2 > W pffiffiffiffiffiffiffiffiffiffiffiffiffi > < 1þB1 1þB1 k22 r2y þr2 ! ! þd > > Uk1 k2 ly Lk1 k2 ly > >U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : k22 r2y þr2 k22 r2y þr2 !# 9 > Lk1 k2 ly A1 B1 > ffi ; pffiffiffiffiffiffiffiffi2 ; pffiffiffiffiffiffiffiffi2 > W pffiffiffiffiffiffiffiffiffiffiffiffiffi > = 1þB1 1þB1 k22 r2y þr2 ! !  > > Uk1 k2 ly Lk1 k2 ly > ffi U p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > U pffiffiffiffiffiffiffiffiffiffiffiffiffi ; k2 r2 þr2 k2 r2 þr2 2 y

2 y

! 8 > Uk1 k2 ly A2 B2 > pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; pffiffiffiffiffiffiffiffi2 ; pffiffiffiffiffiffiffiffi2 > > < W 1þB2 1þB2 k22 r2y þr2 ! ! þdþd  > > Uk1 k2 ly Lk1 k2 ly > > U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : k22 r2y þr2 k22 r2y þr2 ! 9 > Lk1 k2 ly A2 B2 > ffi ; pffiffiffiffiffiffiffiffi > W pffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffi > = 1þB22 1þB22 k22 r2y þr2 ! ! ; þ > > Uk1 k2 ly Lk1 k2 ly > ffi U p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > U pffiffiffiffiffiffiffiffiffiffiffiffiffi ; k2 r2 þr2 k2 r2 þr2 2 y

A¼k



r2y þ ly y0

ð16Þ

2 y

2

:

ð17Þ

The trade-off model between Eqs. (10) and (13) needs to obtain the optimal order quantity (Q⁄), the optimal wholesale price (W⁄), and   the optimal process quality level ly with the maximum expected profits for the buyer and the supplier. Substituting Eqs. (14) and (15) into Eq. (13), there is only one unknown decision variable ly. Let L < ly < U. One can adopt direct search method for obtaining the optimal ly with the maximum supplier’s expected profit for the given order quantity and wholesale price. Substituting this optimal ly into Eqs. (14) and (15), one obtains the optimal solution with the maximum expected profits for the buyer and the supplier. 4. Numerical example and sensitivity analysis Assume that some parameters are as follows: R = 40, S = 12,

lx = 125, rx = 250, y0 = 5, ry = 0.5, b = 0.1, i1 = 0.02, k = 1.1, c = 0.08, Sp = 0.1, d = 0.05, k1 = 2, k2 = 0.8, r = 0.5, L = 3, and U = 7. By solving Eq. (13), one obtains the optimal process quality level

ly ¼ 4:81. Hence, the optimal order quantity Q⁄ = 176.83 and the

7889

C.-H. Chen, C.-L. Lu / Expert Systems with Applications 38 (2011) 7886–7893 Table 1 The effect of parameters for optimal solution. Sp

ly

W⁄

Q⁄

32 36 40 44 48

4.74 4.78 4.81 4.82 4.83

16.08 18.09 20.10 22.11 24.11

198.08 185.64 176.83 170.28 165.20

S

ly

W⁄

Q⁄

9.6 10.8 12 13.2 14.4

4.83 4.82 4.81 4.79 4.76

20.11 20.11 20.10 20.10 20.09

162.71 169.47 176.83 184.88 193.71

lx

ly

W⁄

Q⁄

100 112.5 125 137.5 150

4.77 4.79 4.81 4.82 4.83

18.71 19.41 20.10 20.80 21.49

164.39 170.61 176.83 183.07 189.31

rx

ly

W⁄

Q⁄

200 225 250 275 300

4.83 4.82 4.81 4.79 4.78

21.84 20.87 20.10 19.47 18.94

153.95 165.39 176.83 188.30 199.75

y0

ly

W⁄

Q⁄

4 4.5 5 5.5 6

3.80 4.29 4.81 5.32 5.73

20.06 20.08 20.10 20.12 20.11

177.20 177.02 176.83 176.66 176.61

ry

ly

W⁄

Q⁄

0.4 0.45 0.5 0.55 0.6

4.82 4.81 4.81 4.80 4.79

20.15 20.13 20.10 20.07 20.04

176.65 176.75 176.83 176.94 177.06

b

ly

W⁄

Q⁄

0.08 0.09 0.1 0.11 0.12

4.81 4.81 4.81 4.80 4.80

20.09 20.10 20.10 20.11 20.11

176.93 176.88 176.83 176.80 176.75

i1

ly

W⁄

Q⁄

0.016 0.018 0.02 0.022 0.024

4.81 4.81 4.81 4.81 4.80

20.10 20.10 20.10 20.10 20.10

176.85 176.84 176.83 176.82 176.83

k

ly

W⁄

Q⁄

0.88 0.99 1.1 1.21 1.32

4.77 4.79 4.81 4.82 4.83

20.12 20.11 20.10 20.09 20.08

176.77 176.80 176.83 176.88 176.92

c

ly

W⁄

Q⁄

0.064 0.072 0.08 0.088 0.096

4.84 4.82 4.81 4.79 4.77

20.07 20.09 20.10 20.12 20.13

177.16 177.00 176.83 176.68 176.53

Sp

ly

W⁄

0.08 0.09 0.1

4.80 4.80 4.81

20.10 20.10 20.10

  E pSPS 3084.04 3263.15 3462.86 3675.31 3896.19   E pSPS 3186.65 3319.01 3462.86 3619.80 3791.71   E pSPS 2990.67 3222.43 3462.86 3711.98 3969.77   E pSPS 3281.53 3366.17 3462.86 3568.32 3680.35   E pSPS 3476.67 3469.23 3462.86 3456.43 3449.27   E pSPS 3468.74 3465.97 3462.86 3459.43 3455.66   E pSPS 3466.53 3464.70 3462.86 3461.03 3459.20   E pSPS 3463.60 3463.23 3462.86 3462.50 3462.13   E pSPS 3466.23 3464.52 3462.86 3461.24 3459.65   E pSPS

Q⁄

3477.04 3469.93 3462.86 3455.84 3448.84   E pSPS

176.84 176.84 176.83

3462.73 3462.80 3462.86

Per A 10.940 5.767 – 6.135 12.513 Per A 7.976 4.154 – 4.532 9.496 Per A 13.636 6.943 – 7.194 14.638 Per A 5.237 2.792 – 3.045 6.281 Per A 0.399 0.184 – 0.186 0.393 Per A 0.170 0.090 – 0.099 0.208 Per A 0.106 0.053 – 0.053 0.106 Per A 0.021 0.011 – 0.011 0.021 Per A 0.097 0.048 – 0.047 0.093 Per A 0.410 0.204 – 0.203 0.405

  E pRPS 1542.22 1632.32 1731.89 1836.85 1947.74   E pRPS 1592.98 1658.70 1731.89 1809.34 1895.41   E pRPS 1460.35 1601.72 1731.89 1846.88 1950.74   E pRPS 1597.29 1674.20 1731.89 1775.91 1811.55   E pRPS 1738.48 1734.43 1731.89 1728.82 1725.50   E pRPS 1734.83 1732.56 1731.89 1730.11 1727.67   E pRPS 1733.66 1731.89 1731.89 1729.63 1729.63   E pRPS 1731.89 1731.89 1731.89 1731.89 1731.40   E pRPS 1733.86 1732.82 1731.89 1730.58 1729.31   E pRPS

Per B

E(pT)

10.952 5.749 – 6.060 12.463

4626.26 4895.47 5194.75 5512.16 5843.93

Per B

E(pT)

8.021 4.226 – 4.472 9.442

4779.63 4977.71 5194.75 5429.14 5687.12

Per B

E(pT)

15.679 7.516 – 6.640 12.636

4451.02 4824.15 5194.75 5558.86 5920.51

Per B

E(pT)

7.772 3.331 – 2.542 4.600

4878.82 5040.37 5194.75 5344.23 5491.90

Per B

E(pT)

0.381 0.147 – 0.177 0.369

5215.15 5203.66 5194.75 5185.25 5174.77

Per B

E(pT)

0.170 0.039 – 0.103 0.244

5203.57 5198.53 5194.75 5189.54 5183.33

Per B

E(pT)

0.102 0.000 – 0.130 0.130

5200.19 5196.59 5194.75 5190.66 5188.83

Per B

E(pT)

0.000 0.000 – 0.000 0.028

5195.49 5195.12 5194.75 5194.39 5193.53

Per B

E(pT)

0.114 0.054 – 0.076 0.149

5200.09 5197.34 5194.75 5191.82 5188.96

Per B

E(pT)

0.383 0.129 – 0.262 0.428

5215.56 5204.06 5194.75 5183.19 5173.32

Per A

1738.52 1734.13 1731.89 1727.35 1724.48   E pRPS

Per B

E(pT)

0.004 0.002 –

1731.40 1731.40 1731.89

0.028 0.028 –

5194.13 5194.2 5194.75 (continued on next page)

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C.-H. Chen, C.-L. Lu / Expert Systems with Applications 38 (2011) 7886–7893

Table 1 (continued) Sp

ly

W⁄

Q⁄

0.11 0.12

4.81 4.81

20.10 20.10

176.83 176.84

d

ly

W⁄

Q⁄

0.04 0.045 0.05 0.055 0.06

4.81 4.81 4.81 4.81 4.81

20.10 20.10 20.10 20.10 20.10

176.83 176.83 176.83 176.83 176.83

k1

ly

W⁄

Q⁄

1.6 1.8 2 2.2 2.4

4.86 4.84 4.81 4.76 4.70

20.11 20.11 20.10 20.09 20.08

176.83 176.83 176.83 176.85 176.86

k2

ly

W⁄

Q⁄

0.64 0.72 0.8 0.88 0.96

4.87 4.86 4.81 4.69 4.56

20.11 20.11 20.10 20.08 20.04

176.84 176.84 176.83 176.88 176.97

r

ly

W⁄

0.4 0.45 0.5 0.55 0.6

4.82 4.81 4.81 4.80 4.79

20.10 20.10 20.10 20.10 20.10

Notes: Per A ¼

EðpSPS Þ3462:86 3462:86

 100%; Per B ¼

EðpRPS Þ1731:89 1731:89

  E pSPS 3462.93 3462.99   E pSPS 3462.86 3462.86 3462.86 3462.86 3462.86   E pSPS 3465.10 3464.31 3462.86 3460.44 3456.69   E pSPS

Q⁄

3465.71 3465.23 3462.86 3456.48 3446.52   E pSPS

176.85 176.85 176.83 176.83 176.82

3464.02 3463.49 3462.86 3462.16 3461.40



pRPS



Per A

E

0.002 0.004

1731.89 1731.89   E pRPS

Per A 0.000 0.000 – 0.000 0.000 Per A 0.065 0.042 – 0.070 0.178 Per A 0.082 0.068 – 0.184 0.472

1731.89 1731.89 1731.89 1731.89 1731.89   E pRPS 1732.20 1731.44 1731.89 1730.96 1728.65   E pRPS

Per B

E(pT)

0.000 0.000

5194.82 5194.88

Per B

E(pT)

0.000 0.000 – 0.000 0.000

5194.75 5194.75 5194.75 5194.75 5194.75

Per B

E(pT)

0.018 0.026 – 0.054 0.187

5197.3 5195.75 5194.75 5191.40 5185.34

Per B

E(pT)

0.038 0.018 – 0.289 0.531

5198.25 5197.43 5194.75 5183.37 5169.22

Per A

1732.54 1732.20 1731.89 1726.89 1722.70   E pRPS

Per B

E(pT)

0.034 0.018 – 0.020 0.042

1732.36 1731.89 1731.89 1731.40 1730.89

0.027 0.000 – 0.028 0.058

5196.38 5195.38 5194.75 5193.56 5192.29

 100%.

optimal wholesale price W⁄ = 20.10 with the supplier’s expected     profit E pSPS ¼ 3462:86, the buyer’s expected profit E pRPS ¼ 1731:89 and the society’s expected profit EðpT Þ ¼ 5194:75. Table 1 lists ±20% change for parameter value and presents the effect on the process quality level, the wholesale price, the order quantity, the supplier’s expected profit, the buyer’s expected profit, and the society’s expected profit. If the change percentage of the supplier’s and buyer’s expected profits of product are larger than 10%, then the parameter has the major effect on the expected profit. From Table 1, one has the following conclusions: 1. The process quality level and the wholesale price increase as the sales price per unit, R, increases. The order quantity decreases as R increases. The sales price per unit has a major effect on the supplier’s and buyer’s expected profits. 2. The process quality level decreases, the wholesale price is constant, and the order quantity increases as the salvage value per unit, S, increases. The salvage value per unit has a slight effect on the supplier’s and buyer’s expected profits. 3. The process quality level increases, the wholesale price varies, and the order quantity increases as the mean of the demand of customer, lx, increases. The mean of the demand of customer has a major effect on the supplier’s and buyer’s expected profits. 4. The process quality level decreases, the wholesale price decreases, and the order quantity increases as the standard deviation of the demand of customer, rx, increases. The standard deviation of the demand of customer has a slight effect on the supplier’s and buyer’s expected profits. 5. The process quality level increases, the wholesale price varies slight, and the order quantity varies slight as the target value of product, y0, increases. The target value of product has a slight effect on the supplier’s and buyer’s expected profits.

6. The process quality level is constant, the wholesale price decreases, and the order quantity increases as the standard deviation of quality characteristic of product, ry, increases. The standard deviation of quality characteristic of product has aslight effect on the supplier’s and buyer’s expected profits. 7. The process quality level is constant, the wholesale price increases, and the order quantity decreases as the constant production cost per unit, b, increases. The constant production cost per unit has a slight effect on the supplier’s and buyer’s expected profits. 8. The process quality level, the wholesale price, and the order quantity are constants as the inspection cost per unit, i1, increases. The inspection cost per unit has a slight effect on the supplier’s and buyer’s expected profits. 9. The process quality level increases, the wholesale price decreases, and the order quantity increases as the quality loss coefficient, k, increases. The quality loss coefficient has a slight effect on the supplier’s and buyer’s expected profits. 10. The process quality level decreases, the wholesale price increases, and the order quantity decreases as the variable production cost per unit, c, increases. The variable production cost per unit has a slight effect on the supplier’s and buyer’s expected profits. 11. The process quality level, the wholesale price, and the order quantity are constants as the discounted price per unit for the non-conformance product scrapped, Sp, increases. The discounted price per unit for the non-conformance product scrapped has a slight effect on the supplier’s and buyer’s expected profits. 12. The process quality level, the wholesale price, and the order quantity are constants as the penalty cost, d, increases. The penalty cost has a slight effect on the supplier’s and buyer’s expected profits.

7891

C.-H. Chen, C.-L. Lu / Expert Systems with Applications 38 (2011) 7886–7893

13. The process quality level decreases, the wholesale price decreases, and the order quantity increases as the constant k1, increases. It has a slight effect on the supplier’s and buyer’s expected profits. 14. The process quality level decreases, the wholesale price decreases, and the order quantity varies as the constant k2, increases. It has a slight effect on the supplier’s and buyer’s expected profits. 15. The process quality level decreases, the wholesale price is constant, and the order quantity decreases as the constant r, increases. It has a slight effect on the supplier’s and buyer’s expected profits.

Z

4:

U

Z

L

L

gðy; zÞdydz ¼

Appendix A. The derivation of optimal wholesale price and order quantity

¼

p

¼

L

þ

Z

Z Z

L

1

Wgðy; zÞdy dz þ

1 1

U



k2 r2y

L

h

ðk1 þk2 ly Þþ

1:

U

U

Z

Z

Z

L

1 1

Sp gðy; zÞdy dz 

Z

L

1

U

Z

L

2:

Z

L

1

3:

Z U

1

Z

/ðz1 Þdz1 dz;

ðA6Þ

k2 ry þr

pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i 2

k2 r2y þr2 z2 þðly r2 k1 k2 r2y Þ

L

k2 r2y

k2 r2y ðk1 þk2 ly Þþðly r2 k1 k2 r2y Þ k22 r2y þr2

pffiffiffi B

¼

Let A1 ¼

)A1 ¼

ð

Þð ffiffi

lr r rry

lr r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L ¼

l

k2 r2 y

and B1 ¼

2 2 2 y k2 y þ k22 2y þ 2

ð r rÞ r rr

r

pffiffiffiffiffiffiffiffiffiffiffiffi

k2 r2 þr2 2 y k2 r2 þr2 y 2p

ffiffi B

L  ly

¼

y pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

k22 r2y þr2

z2 :

Þ

B

k22 y 2y þ y 2 k22 2y þ 2

k2 r2y þr2

pffiffiffi B



2 2 k2 r2 y k1 þk2 ly þ ly r k1 k2 ry k2 r2 þr2 y 2p

L

pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

k22 r2y þr2

rr

y pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

k22 r2y þr2

.

;

k22 r2y þr2

pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

k2 2y þ 2 k22 2y þ 2

pffiffiffi B

2 2

k2 r y pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

¼

k2 r2y þr2

ryr pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 k2 r2y þr2

¼

k 2 ry

r

U

Z

:

Hence, Eq. (A6) can be rewritten as r þr 2 y Lk1 k2 ly

pffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2

Sp gðy; zÞdy dz

r r r r

k

2

/ðz2 ÞUðA1  B1 z2 Þdz2 ¼

Uk1 k2 ly Z p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 k

r þr

2 y

/ðz2 ÞUðA1  B1 z2 Þdz2

1

ry þr

1 L



dgðy; zÞdy dz

Lk1 k2 ly Z p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 k

r þr

2 y

/ðz2 ÞUðA1  B1 z2 Þdz2 ;

1

1

dgðy; zÞdy dz  b  cly  i1 :

Z

ðA2Þ

1

gðy; zÞdy dz ¼

L

Z

1

U

0

k

Eq. (A6) can be further rewritten as

Z L

ðA3Þ

0

Z

! a b p p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; k; : /ðzÞUða þ bzÞdz ¼ W * 2 2 1 1þb 1þb Z

U

1 L  k  k l 1 2 yC B gðy; zÞdy dz ¼ f ðzÞdz ¼ U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A; 1 1 k22 r2y þ r2 1

1

r2 þr2 2 y

1

1

Z

y k

1

k2 r2 þr2

k

1

0 1 U  k  k l 1 2 yC B gðy; zÞdy dz ¼ f ðzÞdz ¼ U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 2 1 L k 2 ry þ r2 0 1 BL  k1  k2 ly C  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A; k22 r2y þ r2

Z

mðzÞ

2 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 r2

p2 ffiffiffiy B

Uk1 k2 ly Z p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2

To obtain the supplier’s expected profit per unit, Eqs. (A3)–(A9) are needed as follows:

Z

U

2 k2 r2 y ðzk1 Þþly r k2 r2 þr2 2 y

where Z1 denotes the standard normal random variable. k r2 zþðl r2 k k r2 Þ k r2 ðzk Þþl r2 r2 r2 Let A ¼ 2 yk2 r2 1þr2 y ¼ 2 y k2 ry2 þr2 1 2 y and B ¼ k2 ry2 þr2 . 2 y 2 y 2 y   R LA RU RU pffi pffiffi dz. Let mðzÞ 1B /ðz1 Þdz1 dz ¼ L mðzÞU LA We have L B  zk1 k2 ly 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ z2 ¼ ¼ þ k l . We have dz dz, and z ¼ k 2 1 2 y 2 2 2 2 2 2

B1 ¼

Hence, the supplier’s expected profit per unit is

E1

Z L

ðA1Þ

S PS

pðyjzÞmðzÞdydz

1 L

k2 r2y

8 L < z < U; L < y < U; > < W  b  cy  i1 ; ¼ W  b  cy  i1  d; L < z < U; y < L or y > U; > : Sp  b  cy  i1 ; z < L or z > U; 1 < y < 1:

U

L

and

The supplier’s profit is given by

Z

L

1

L



Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k22 r2y þ r2 z2 . Hence

In this paper, the authors have presented a modified Chen and Liu’s (2007) pure procurement model with quality loss of product for indirectly measurable quality characteristic. The optimum process quality level, the wholesale price, and the order quantity are simultaneously determined in the modified Chen and Liu’s (2007) model. From the above numerical results, one has the following conclusions: the sales price per unit and the mean of the demand of customer have a major effect on the supplier’s and the buyer’s expected profits. The extension to modified Chen and Liu’s (2007) model with two-stage screening procedure may be left for further study.



U

k2 ry þr

5. Conclusions

pSPS

Z

ðA4Þ 1

BU  k1  k2 ly C f ðzÞdz ¼ 1  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A; k22 r2y þ r2 ðA5Þ

U

1 U  k  k l A B 1 2 y 1 1 B C gðy; zÞdy dz ¼ W@qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1 k22 r2y þ r2 1 þ B21 1 þ B21 0 1 L  k 1  k 2 ly B1 C B A1  W@qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA k22 r2y þ r2 1 þ B21 1 þ B21 0 1 U  k  k l A B 1 2 y B C 1 1 ¼ W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA k22 r2y þ r2 1 þ B21 1 þ B21 0 1 A1 B1 C BL  k1  k2 ly  W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; k22 r2y þ r2 1 þ B21 1 þ B21

Z

0

L

ðA7Þ

7892

Z

C.-H. Chen, C.-L. Lu / Expert Systems with Applications 38 (2011) 7886–7893

Z

U

L

1

gðz; yÞdy dz ¼

U

Z

U

Z

L

¼

2 0

1

pðyjzÞmðzÞdy dz

U

Z

U

mðzÞ

E1

Z U

L

k2 r2 ðzk1 Þþly r2 y k2 r2 þr2 2 y

;

1

rffiffiffiffiffiffiffiffiffiffiffiffi r2y r2



pSPS

ðA8Þ

/ðz1 Þdz1 dz

k2 r2 þr2 2 y

where Z1 denotes the standard normal random variable. Let A ¼

k2 r2y ðzk1 Þþly r2 k22 r2y þr2

k2 r2y zþðly r2 k1 k2 r2y Þ

¼

k22 r2y þr2

r2 r2

and B ¼ k2 ry2 þr2 . 2

y

h  i R1 RU pffiffi dz. We have L mðzÞ UA /ðz1 Þdz1 dz ¼ L mðzÞ 1  U UA pffi B B   zk1 k2 ly 1 ffi. We have dz2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dz, and z ¼ k1 þ k2 ly þ Let z2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 RU

k2 ry þr

k2 ry þr

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k22 r2y þ r2 z2 . Hence

U

k2 r

2 y

h

ðk1 þk2 ly Þþ



1 0 13 U  k  k l L  k  k l 1 2 yC 1 2 y C7 6 B B ¼ W 4U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 k22 r2y þ r2 k22 r2y þ r2 2 0 1 0 13 L  k  k l U  k  k l 1 2 1 2 6 B B yC y C7 þ Sp 4U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ 1  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 k22 r2y þ r2 k22 r2y þ r2 2 0 1 A1 B1 C 6 BU  k 1  k 2 ly  d4W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA k22 r2y þ r2 1 þ B21 1 þ B21 0 13 A1 B1 C7 BL  k1  k2 ly W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5 k22 r2y þ r2 1 þ B21 1 þ B21 82 0 1 0 13 > < U  k  k l L  k  k l 1 2 yC 1 2 y C7 6 B B  d 4U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 > 2 2 2 2 2 : k r þr k r þ r2 2

pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

k2 2y þ 2 z2 k1

r

r

þly r

ry r pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

ðk22 r2y ly þly r2 Þþk2 r2y

U

pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

k2 r2y þr2 z2

k22 2y þ 2

r r ry r pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k22 r2y þr2

¼

¼

U  ly k2 ry z :  ry r pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r 2 2

2

Ul

ðA10Þ

k

r

k r þr 2 y

Lly ry r

Uly ry r

k2 ry

r þr 2 y Lk1 k2 ly k

pffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 k

k r þr 2 y

k

¼

k r þr 2 y

Lk1 k2 ly p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 k

2

/ðz2 Þdz2 

ry þr

k

r þr

2 y

Lk1 k2 ly p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 k

2

/ðz2 ÞUðA2  B2 z2 Þdz2 ;

!

k

* 1

a b /ðzÞUða þ bzÞdz ¼ W pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 1þb 1þb

L

U

Z U



0 1 A2 B2 C BL  k1  k2 ly þ W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; k22 r2y þ r2 1 þ B22 1 þ B22 ðA9Þ

Z

 þ

From Eqs. (A3)–(A9), Eq. A2 can be rewritten as

Q

½Rx  WQ þ SðQ  xÞf ðx; yÞdx dy

lx rx 2

Z

Z

Q

lx rx 2

Z

Z

1

1

xLossðyÞf ðx; yÞdy dx

1 lx þrx 2

1 Q Z lx þrx Z 2

 Q (Z

½R  WQf ðx; yÞdy dx 1

QLossðyÞf ðx; yÞdy dx

1

Q

¼

lx rx

½ðR  SÞx þ Q ðS  WÞf ðxÞdx

)Z

1

LossðyÞf ðyÞdy

1

2



Z  1 rx 2 1 Q 2  lx  LossðyÞf ðyÞdy 2rx 2 1 (Z lx þrx ) Z

1 2 ½R  WQf ðxÞdx  LossðyÞf ðyÞdy þ



Q

r þr

2 y

Z

1

1

where A2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi and B2 ¼ r . 2 2 2 k

X P Q ; 1 < Y < 1:

Hence, the buyer’s expected profit is

  E pRPS ¼

0 1 A2 B2 C BU  k1  k2 ly  W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA k22 r2y þ r2 1 þ B22 1 þ B22

k2 ry

ðA11Þ

RQ  WQ  Q  LossðYÞ;

1 0 13 U  k  k l L  k  k l 1 2 1 2 6 B B yC y C7 gðz; yÞdy dz ¼ 4U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A  U@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 k22 r2y þ r2 k22 r2y þ r2

Uly ry r

items in order to

RX  WQ þ SðQ  XÞ  X  LossðYÞ; X < Q ; 1 < Y < 1:

2 0

1

gðy;zÞdy dz

ðA12Þ

Eq. (A6) can be further rewritten as

Z

Q

The buyer’s profit is given by

r y þr

pRPS ¼ Z

1

    Q E pSPS ¼ R U R 1  E1 pSPS : gðy; zÞdy dz L 1

r þr

2 y

Uk1 k2 ly Z p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2

r þr

2 y

The supplier needs to produce R U R 1

satisfy the buyer’s order quantity. Hence, the supplier’s expected profit is

/ðz2 Þ½1  UðA2  B2 z2 Þdz2

Uk1 k2 ly Z p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2

k2 ry

where A1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; B1 ¼ r ; A2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi, and B2 ¼ r . 2 2 2 2 2 2 L

Uk1 k2 ly Z p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2

2

k2 r2y þr2

y 2 y Let A2 ¼ pffiffiffiffiffiffiffiffiffiffiffi ry r ffi and B2 ¼ r . Hence, Eq. (A8) can be rewritten 2 2 2

as

y

1 U  k  k l A2 B2 C 1 2 y B  W@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 k22 r2y þ r2 1 þ B2 1 þ B22 0 19 > = L  k  k l A B 1 2 2 2 B C y þW@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiA  b  cly  i1 ; > k22 r2y þ r2 1 þ B2 1 þ B2 ;

2

k22 r2y þr2

k2 r2y þr2

2

y

0

i



1h

rx

lx þ

rx 2

i Z Q Q

1 1

1

LossðyÞf ðyÞdy:

ðA13Þ

7893

C.-H. Chen, C.-L. Lu / Expert Systems with Applications 38 (2011) 7886–7893

Eq. (A14) can be rewritten as

E



pRPS



h



   rx  i 1 rx 2 ¼ Q ðS  WÞ Q  lx þ þ ðR  SÞ Q 2  lx  2 2 2

Z  1 1 rx 2 1 2 Q  lx  LossðyÞf ðyÞdy   rx 2rx 2 1  1 i rx 1h r þ ½R  WQ lx þ  Q  lx þ x  Q  Q 2 rx rx 2 Z 1 LossðyÞf ðyÞdy;  1

 2

. where 1 LossðyÞf ðyÞdy ¼ k r2y þ ly  y0

ðA14Þ

R1

R1 Let A ¼ 1 LossðyÞf ðyÞdy. We need to obtain the maximum expected profit for the buyer. Let the partial derivative of Eq. (A15) dEðpR Þ with respect to Q be zero, i.e., dQPS ¼ 0. We have

h  i rx  ) ðS  WÞ Q  lx þ þ QðS  WÞ þ Q ðR  SÞ  QA 2 h   i rx þ ðR  WÞ lx þ  Q  ðR  WÞQ 2 h  i rx  lx þ  Q  Q A ¼ 0: 2 ½ðR  SÞ  Alx  12 ½ðS þ R  2WÞ  Arx : ðS  RÞ þ A

Let

b þ cly þ i1 !

t¼

!

Uk1 k2 ly

Lk1 k2 ly ffi U p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2

"

k2 ry þr

Lk1 k2 ly

!

k2 r y þ r

Uk1 k2 ly

!#

ffi þ 1  U pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Sp U pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 k2 r y þ r



k2 r y þ r

!

Uk1 k2 ly Lk1 k2 ly ffi U p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 k2 ry þr

!

k2 r y þ r

! 8" > Uk1 k2 ly A1 B1 > ffi ; pffiffiffiffiffiffiffiffi2 ; pffiffiffiffiffiffiffiffi2 > W pffiffiffiffiffiffiffiffiffiffiffiffiffi > < 1þB1 1þB1 k22 r2y þr2 ! ! þd > > Uk1 k2 ly Lk1 k2 ly >U p > ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U : k2 r2 þr2 k2 r2 þr2 2 y

2 y

!# 9 > > ffi ; pffiffiffiffiffiffiffiffi2 ; pffiffiffiffiffiffiffiffi2 > W pffiffiffiffiffiffiffiffiffiffiffiffiffi > = 1þB1 1þB1 k22 r2y þr2 ! !  > > Uk1 k2 ly Lk1 k2 ly > ffi U p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > U pffiffiffiffiffiffiffiffiffiffiffiffiffi ; k2 r2 þr2 k2 r2 þr2 Lk1 k2 ly

A1

B1

2 y

2 y

2 y

2 y

! 8 > Uk1 k2 ly A2 B2 > pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; pffiffiffiffiffiffiffiffi2 ; pffiffiffiffiffiffiffiffi2 > > < W 1þB2 1þB2 k22 r2y þr2 ! ! þdþd  > > Uk1 k2 ly Lk1 k2 ly > U p > ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U : k22 r2y þr2 k22 r2y þr2 ! 9 > Lk1 k2 ly A2 B2 > ffi ; pffiffiffiffiffiffiffiffi > ; pffiffiffiffiffiffiffiffi W pffiffiffiffiffiffiffiffiffiffiffiffiffi > = 1þB22 1þB22 k22 r2y þr2 ! ! : þ > > Uk1 k2 ly Lk1 k2 ly > ffi U p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > U pffiffiffiffiffiffiffiffiffiffiffiffiffi ; k2 r2 þr2 k2 r2 þr2

  E pSPS ¼ ðW  tÞ 

½ðR  SÞ  Alx  12 ½ðS þ R  2WÞ  Arx : ðS  RÞ þ A

ðA17Þ

We also need to obtain the maximum expected profit for the supplier. Let the partial derivative of Eq. (A18) with respect to W be zero, i.e.,

dEðpSPS Þ dW

Q þ ðW  t Þ

rx ¼ 0: ðS  RÞ þ A

¼ 0. We have

That, is ½ðR  SÞ  Alx þ 12 ½ðS þ R  2wÞ  Arx  ðW  tÞrx ¼ 0.

1 ) s½ðR  SÞ  Alx þ ½ðS þ R þ 2tÞ  Arx  2rx W ¼ 0: 2

ðA18Þ

From Eq. (A19), the optimal wholesale price for the supplier is

)W  ¼





rx 2S þ R2 þ t  A2 þ lx ½ðR  SÞ  A : 2rx

ðA19Þ

ðA15Þ

From Eq. (A16), the optimal order quantity for the buyer is

Q ¼

Substituting Eq. (A17) into Eq. (A12), we have the supplier’s expected profit is

ðA16Þ

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