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Optimal input quantity decisions considering commitment order contracts under yield uncertainty
International Journal of Production Economics 216 (2019) 398–412
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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe
Optimal input quantity decisions considering commitment order contracts under yield uncertainty
T
Jianhu Caia,*, Xiaoqing Hua, Feiying Jianga, Qing Zhoub, Xiaoyang Zhanga, Liyuan Xuana a b
School of Management, Zhejiang University of Technology, Hangzhou, 310023, PR China School of Management, Hangzhou Dianzi Uninversity, Hangzhou, 310018, PR China
ARTICLE INFO
ABSTRACT
Keywords: Supply chain management Input quantity decision Yield uncertainty Commitment order contracts
We examine a two-echelon supply chain with one supplier and one retailer under yield and demand uncertainty. A basic model is developed, in which the supply chain is assumed to operate under Vendor-Managed Inventory (VMI). Commitment order contracts are then proposed, and three possible situations are investigated: ① The wholesale price of the commitment order equals the wholesale price of the regular order; ② the wholesale price of the commitment order is lower than the wholesale price of the regular order; ③ the wholesale price of the commitment order is higher than the wholesale price of the regular order. Given the retailer's commitment order quantities, we obtain the supplier's optimal input quantities under different conditions. The impact of the retailer's commitment order quantity on the supplier's input quantity decision is also examined. Furthermore, the retailer's optimal commitment order quantities are discussed. We demonstrate the detailed boundary conditions for all possible optimal solutions of both members. Our analysis reveals that commitment order contracts are effective to stimulate the input quantity of the supplier. However, there still exist several regions in which the supplier's input quantity is independent of the commitment order quantity. In the special situation, we find that the supplier's input quantity decreases with the commitment order quantity. The numerical example shows that, with special sets of wholesale prices, the commitment order contracts can be used to improve the performance of the supply chain facing different random yield environments.
1. Introduction Yield uncertainties usually occur in an imperfect production system in which actual output quantity is less than the input quantity. Yield uncertainty also refers to the phenomenon in which the buyer receives less quantity than was ordered from the supplier. Yield uncertainty may occur due to instabilities in any step of the supply chain (SC), such as production, inventory and transportation. Instability is common in the production systems of agriculture, the chemical industry and the electronics industry (Keren, 2009). In most situations, yield uncertainty means that the actual yield is less than or equal to the input quantity. In batch production processes, e.g., agriculture, chemicals or pharmaceuticals, the actual yield may be more than the amount expected (Snyder and Shen, 2011). In this paper, we focus on the yield uncertainty related to special production systems. Yano and Lee (1995) describe different ways of modeling yield uncertainty and discuss some of their advantages and disadvantages. These methods are still effective for modeling yield uncertainty and are used by many researchers. This study is motivated by the case of Zhejiang Nali Clothing Co.,
*
Ltd., in China. In spring 2017, Nali orders 930 m of worsted fabric from the supplier, Jiangsu Danmao Textile Co., Ltd. Because of the complicated and uncertain production, defective products are inevitable. Although Danmao anticipates possible yield loss and inputs enough materials, the rate of imperfect items (30%) is still higher than expected. Ultimately, Nali only receives 680 m of worsted fabric from Danmao. Since the production cycle of worsted fabric is 65 days, it is very difficult for Danmao to replenish the products in time. This makes Nali not able to respond quickly to the market demand, and causes profit losses of both Nali and Danmao. Nali forms the assumption that more worsted fabric will be produced if a more reasonable order is placed. However, if a higher order is placed, Nali may receive more worsted fabric than needed. And Danmao may face inventory risk if the input quantity is too high. Both firms are eager to introduce an effective cooperative mechanism to reduce the impacts of yield uncertainty. Similar management problems are common in many other enterprises in Zhejiang Province. Another enterprise, Jinlong Machinery & Electronics Co., Ltd., produces mini motors (the diameter of the main product is 4–7 mm). It is difficult for Jinlong to ensure a stable output
Corresponding author. E-mail address: [email protected] (J. Cai).
https://doi.org/10.1016/j.ijpe.2019.06.021 Received 4 July 2018; Received in revised form 6 April 2019; Accepted 27 June 2019 Available online 04 July 2019 0925-5273/ © 2019 Elsevier B.V. All rights reserved.
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
for a given input, because the production process is very complicated. However, the company must determine the input quantity under yield and demand uncertainty in most situations to provide a relatively stable supply and meet the market demand. Hence, it is necessary for Jinlong and its buyers to introduce a simple and effective contract scheme to improve the performance of the SC. In the case of Nali and Danmao, some work has been done by the firms to construct tight cooperative relationships. First, Nali and Danmao are trying to establish efficient channels to exchange information in time. Second, Danmao tries its best to support the flexible production of Nali. In some special situations, without receiving the order of Nali, Danmao promptly makes an input quantity decision by analyzing yield and demand information. This mode is very close to the Vendor-Managed Inventory (VMI) mode. Third, Nali changes the traditional order decision into the commitment order decision by sharing demand information with Danmao. In this paper, we first develop a VMI SC model, and examine the SC members’ optimal decisions under yield and demand uncertainty. We then propose the commitment contracts under which the retailer commits a minimum order quantity before the selling season. Such commitment contracts are termed “commitment order contracts”. Three contracts are proposed according to the difference between the wholesale price of the commitment order and the wholesale price of the regular order: ① The wholesale price of the commitment order equals the wholesale price of the regular order (e.g., in the trade between Nali and Danmao, Nali can pre-order the products from Danmao and the wholesale price of products ordered in advance often keeps the same with the wholesale price of the regular order); ② the wholesale price of the commitment order is lower than the wholesale price of the regular order; for example, in the production of holiday celebration products (e.g., moon cakes, pumpkin pies, etc.), food contract manufacturers often place two orders from the packaging material suppliers. One is the advance order with a discount wholesale price, the other is the rush order with an original wholesale price. Both orders are delivered to the manufacturers at the beginning of the selling season (Chintapalli et al., 2017); ③ the wholesale price of the commitment order is higher than the wholesale price of the regular order; one example from the SC of Nali and Shaoxing Yonson Digital Printing Technology Co., Ltd. can illustrate this situation. In December 2018, Yonson produces a special fabric for Nali, because of uncertainty in production and market demand, the wholesale price they negotiate for the commitment order is relatively high. With the approach of the selling season, the market demand turns out to be positive, and Nali purchases more than the commitment order quantity with a lower wholesale price. Thus, the following important managerial questions must be addressed.
quantity. Hence, the retailer should choose proper commitment order quantity to encourage the input quantity of the supplier. (2) The distribution characteristics of the yield rate have a great influence on SC members' optimal decisions and the performance of the SC. It may affect both members' participation in the operation of the SC. Detailed boundary conditions are obtained for all possible optimal solutions of both members. The results in the paper can help enterprises make proper decisions when facing yield and demand uncertainty. (3) Under special conditions, commitment order contracts are efficient to stimulate SC members to cooperate with each other, thus improving the performance of the SC. Our work shows that the effectiveness of commitment order contracts is relevant to the exogenous wholesale prices and the yield rate. In this paper, the research structure is very simple, but finding the optimal decision is very difficult. The above management insights may help the managers of the enterprises make appropriate operational decisions to build tight relationship with each other and reduce the impacts of yield uncertainty, thus to improve the performance of the SC. The managers of these enterprises believe in great practical value of the results of such research structures. The rest of the paper is organized as follows. Section 2 reviews the related literature. In Section 3, we introduce the assumptions and develop a VMI SC model under yield and demand uncertainty. Section 4 proposes three commitment order contracts, and all members’ optimal decisions are obtained under different conditions. In Section 5, numerical analysis is conducted to illustrate and validate our approach, and the efficiencies of the commitment order contracts in improving the performance of the SC are examined and compared. Summary, conclusions and future research are discussed in Section 6. 2. Literature review 2.1. Yield uncertainty Yield uncertainty is a common phenomenon, many real cases with yield uncertainty are introduced to construct decision models. Kazaz (2004) studies random yield in the olive industry by defining the sales price and the purchasing cost as exogenous and increasing with decreasing yield. Cai et al. (2017) introduce the case of Mate 7, a smartphone launched by HUAWEI, to illustrate the yield uncertainty in the SC for electronic products. Yield uncertainty is also common in agriculture. For example, the output of soil crops is influenced by irrigation and N fertilization (Hao et al., 2017). Climate is an important factor for coffee-farming in Peru (Lechthaler and Vinogradova, 2017). Tang and Kouvelis (2014) find that semiconductor and agribusiness industries are seriously affected by random yields. He and Zhang (2010) present examples about the “Nintendo Wii” and the “textbook market” to illustrate the random yield SC. Gurnani and Gerchak (2007) think that random yield can also be found in assembly systems, and related examples include the mission of the Challenger and VCR and car assembly. Yin and Ma (2015) investigate a leisure food chain store, LPPZ, in China and find that a high degree of SC risk, including supply and demand uncertainty, hinders the achievement of a high service level. Similar to this paper, Li et al. (2017) use examples from the clothing industry, such as Meters/bonwe, Liz Claiborne and Benetton, to explain yield uncertainty. The flu vaccine industry in the United States has been described many times to illustrate yield uncertainty (Chick et al., 2008; Deo and Corbett, 2009; Arifoğlu et al., 2012; Cho and Tang, 2013). The impacts of yield uncertainty on SC performance have received widespread attention. Many researchers have developed different SC models considering yield uncertainty. Masih-Tehrani et al. (2011) construct an m-manufacturer, 1-retailer newsvendor system under supply uncertainty. Okyay et al. (2014) analyze three different types of supply uncertainty: Random yield, random capacity and both. They
(1) How do the commitment order contracts influence the supplier's optimal input quantity decisions? (2) How do the distribution characteristics of the yield rate influence the SC members' optimal decisions under the commitment order contracts? (3) Is it possible to improve the performance of the SC by introducing the commitment order contracts? In this study, we model the above three commitment order contracts as Stackelberg games in which the retailer acts as the leader. The wholesale price of the regular order and the wholesale price of the commitment order are assumed to be exogenous. Through the model development, we highlight our key findings and derive the following insights. (1) Under the commitment order contracts, the retailer's commitment order quantity has a positive impact on the supplier's input quantity. However, our paper also shows that the supplier's input quantity may be independent on the retailer's commitment order quantity, or even decreases with the retailer's commitment order 399
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present the example in which random demand, capacity and yield are directly dependent. Xu et al. (2015) develop a SC with one buyer and two different suppliers. They consider a special case in which one supplier is the buyer's regular long distance supply source but still faces uncertain yield. Fang and Shou (2015) study the Cournot competition between two SCs subject to supply uncertainty. Baruah et al. (2016) propose an optimal order revision policy for a buyer facing typical demand and supply uncertainty. They demonstrate the value of soft orders as well as the supplier's inventory position information for the buyer. Jabbarzadeh et al. (2017) present a production-distribution planning model that is robust to common supply interruptions and demand variations. Du et al. (2018) consider a SC composed of a lossaverse supplier with yield randomness and a loss-averse retailer with demand uncertainty. They obtain the SC members' optimal decisions and examine how loss aversion and yield variance contribute to the SC performance. In this paper, we propose a SC model considering yield and demand uncertainty. The supplier's yield is uncertain and the actual output quantity is a random fraction of the input quantity. Such stochastic proportional yield models are widely used in many literature (Yano and Lee, 1995; Guo et al., 2018). Different contracts are proposed to coordinate the SC under yield uncertainty. Xu (2010) considers a SC in which the supplier's production yield is random and the manufacturer's demand is stochastic. Then, the option contract is proposed to improve the performance of the SC. Hsieh and Wu (2008) consider a special SC with one Original Equipment Manufacturer (OEM), one manufacturer, and one distributor under supply and demand uncertainty. They propose effective coordination mechanisms through information sharing between the members. Yin and Ma (2015) consider the retailer's requirement of a service level of the product supply from the manufacturer with random yield. The bonus contracts can achieve Pareto improvements. Hu and Feng (2017) model a SC with revenue sharing contracts and service requirement under supply and demand uncertainty. He and Zhao (2012) propose optimal contract-design schemes for a three-tier SC with both supply and demand uncertainty. In their contract-design schemes, wholesale price contracts and a returns policy are united to coordinate the SC. Giri and Bardhan (2017) also discuss the coordination of a three-echelon SC under uncertain demand and random yield in production. Peng et al. (2018) develop a SC under the carbon emission capand-trade scheme, the manufacturer faces yield uncertainty. A contract of revenue-sharing with subsidy on emission reduction is introduced to coordinate the SC. Similar works regarding SC coordination under yield uncertainty can also be found in Güler and Keski˙N (2013) and Giri et al. (2016).
amount of quantity of newly developed products. Both parties attain maximum profits under the concept of the synergy effect. Xu (2011) studies a stochastic inventory model with a multi-period where the buyer commits to purchase the product, with at least a certain percentage of the purchased capacity during the planning horizon. The buyer benefits from these commitments. Most of the above works focus on the commitments made by the buyer to the supplier. The supplier can also make commitments to the buyer (Ghijsen et al., 2010). Based on a tight partnership, the supplier and the buyer can make mutual commitments to improve the performance of the SC (Durango-Cohen and Yano, 2006; Chen et al., 2017). In addition to quantity commitment, price commitment is also valuable. Liu et al. (2014) find that if brand manufacturers provide a retail price markup commitment strategy, retailers are willing to follow this strategy. Commitment contracts can also be used in the SC under yield uncertainty. He and Zhang (2010) propose a minimum commitment contract, specifying a commitment quantity that the supplier must supply. If the actual output quantity is less than the commitment quantity, the supplier has two choices: Pay a penalty to the retailer or go to a secondary market for procurement. By contrast, our paper focuses on the commitment order contracts, under which the retailer first commits a minimum order quantity before the selling season, the supplier then determines the input quantity. After yield and demand uncertainty is observed, the retailer determines whether to place a second order. 2.3. The distinctiveness of this research The key contribution of this study lies in introducing three different types of commitment order contracts to improve the performance of the SC under yield and demand uncertainty. Yield rate is a random variable following uniform distribution defined in a general region. In the basic model, the SC operates under VMI, in which the supplier takes charge of inventory management. Commitment order contracts are introduced to improve the performance of the SC. Stackelberg game models are then constructed, and optimal decisions are obtained. Specifically, we discuss different boundary conditions needed to obtain members’ optimal decisions. Most related works can be found in Cho and Tang (2013) and Tang and Kouvelis (2014). Cho and Tang (2013) assume that the production capacity is a constant and the actual output is subject to random production yield. The authors introduce a linear demand curve and show how advance selling strategies impact the SC's competition structures. In their model, the manufacturer acts as the game leader by first determining the wholesale prices. The retailer must decide the pre-book quantity or regular-order quantity. Tang and Kouvelis (2014) use a modified pay-back-revenue-sharing contract to coordinate the SC under uncertain yield and demand. In their model, they assume that the random yield rate is distributed on [0,1]. Boundary constraints thus do not require discussion. Neither study introduces a commitment order contract to change the competition structure of the SC. Table 1 further helps position this paper and identify the contributions. In short, this research contributes to the literature in the following three ways.
2.2. Commitment contract Kumar et al. (1995) demonstrate that relationships with greater total interdependence exhibit higher trust, stronger commitment, and lower conflict than relationships with lower interdependence. Nyaga et al. (2010) think that credible commitments may develop when incomplete contracting exists because neither party can completely rely on contractual mechanisms to manage the relationship. Trust and commitment lead to improved satisfaction and performance of the SC. Zhao et al. (2007) develop an analytical model to quantify the cost savings of an early-order commitment in a two-level SC in which demand is correlated serially. Lian and Deshmukh (2009) study a SC with demand forecast updating with the buyer able to receive discounts for committing to purchase in advance. These contracts are used by automobile and contract manufacturers and are common in the fuel oil and natural gas delivery markets. Xie et al. (2010) introduce the early-order commitment (EOC), wherein the retailer commits to purchase a fixedorder quantity from a manufacturer. They find that the EOC-based wholesale price-discount scheme can reduce SC costs. Motivated by a case in the cosmetic industry, Li et al. (2016) study a special quantityflexibility contract arrangement in which the retailer commits an
(1) Most of the existing literature assumes that the yield rate is a random variable distributed in the region [0,1]. In our paper, the yield rate is a random variable following a uniform distribution defined in a general region. Constructing the model based on such an assumption enriches the extant literature on SC models considering yield uncertainty. (2) Different from most researches which focus on introducing complicated contracts into the Retailer-Managed Inventory (RMI) SC under yield uncertainty, we develop a VMI SC model and introduce the commitment order contracts. The commitment order contracts proposed in this paper only include one parameter and can be 400
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Table 1 Summary of some relevant literature. Literature
Stochastic proportional yield
Contracts
Key findings
Cai et al. (2017)
The yield rate is a random variable distributed in (0,1].
Option contract; subsidy contract
Tang and Kouvelis (2014) He and Zhang (2010)
The yield rate is a random variable distributed in [0,1]. The yield rate is a random variable with the mean value µ .
Pay-back-revenue-sharing contract
Yin and Ma (2015)
The yield rate is a random variable distributed in [0,1]. The yield rate is a random variable distributed in [0,1].
Bonus contracts
The option contact is effective in coordinating the SC, it can be easily applied in a VMI SC; the subsidy contract can coordinate the SC, but it is not feasible for most situations under a VMI SC. A modified pay-back-revenue-sharing contract can coordinate the SC under yield and demand uncertainty. Different parties of the SC prefer different level of commitment quantity, and the reduction of random yield risk is useful in improving the performance of the SC. The bonus contracts can achieve Pareto improvements.
Li et al. (2017) Chick et al. (2008)
Commitment contract (the supplier commits a minimum supply quantity)
Commitment-option contracts
Effects of demand uncertainty, production yield variability, instant procurement price fluctuation, and the price parameters of option contracts are discussed. A variant of the cost sharing contract can provide incentives to both SC members so that the SC achieves global optimization. Three strategies are compared from both SC members' perspectives, respectively.
The yield rate is a random variable with the mean value µ . The yield rate is a random variable distributed in [ , ¯], and the mean ¯ value is 1. The yield rate is a random variable distributed in [0,1]. The yield rate is a random variable distributed in [L , U ], 0 L < U 1. The yield rate is a random variable distributed in [0,1].
Cost sharing contract
Peng et al. (2018)
The yield rate is a random variable with the mean value 1.
Revenue-sharing with subsidy contract
Güler and Keski˙N (2013)
The yield rate is a random variable
This paper
The yield rate is a random variable distributed in [ , ], 0 .
Buy-back contract; revenue sharing contract; quantity discount contract; quantity flexibility contract Commitment order contracts (the retailer commits a minimum order quantity before the selling season)
Cho and Tang (2013)
Xu (2010) Hu and Feng (2017) He and Zhao (2012)
distributed in [ l,
h] ,
0<
l
<
h
< 1.
Advance selling strategy; regular selling strategy; dynamic selling strategy Option contract
Both SC members can be better off under the option contracts.
Revenue sharing contract
Conditions to coordinate the SC are obtained under the revenue sharing contract. In a three-level SC, a returns policy used by the manufacturer and the retailer, combined with the wholesale price contract used by the raw-material supplier and the manufacturer, can perfectly coordinate the SC. The proposed contract can coordinate the low-carbon SC with yield uncertainty perfectly and the carbon emission reduction level can achieve the level under the centralized case. The randomness in the yield does not change the coordination ability of the contracts but affects the values of the contract parameters.
Wholesale price contract combined with a returns policy
conveniently introduced into the VMI SC connected by a wholesale price contract. (3) Detailed boundary conditions are obtained for all possible optimal solutions of both members. Diversified sensitivity analyses are conducted in the paper to find out management insights of the commitment order contracts. Thus, the models can be easily accepted by the enterprises, and the results can support their operations management.
The SC members' optimal decisions are obtained given different conditions under yield and demand uncertainty. It is possible to introduce the commitment order contracts to improve the performance of the SC given specific conditions.
then is obtained by the retailer. The retailer places its final order Min {D , xK } . Anticipating the retailer's optimal decision, the supplier determines the optimal input quantity. The supplier's expected profit can be expressed as s
(K ) = w 0
3. Basic model
xK 0
Df (x ) g (D ) dDdx +
+ xK
xKf (x ) g (D) dDdx (1)
cK It is easy to find that
Consider a two-echelon SC with one supplier and one retailer. Let p denote the retail price and w0 denote the wholesale price. The supplier's unit production cost is c . We assume c < w0 < p . The supplier's yield is uncertain. If the supplier's input quantity is K , then its actual output quantity is xK . Here, x is a stochastic variable with a uniform distribution. Let f (x ) and F (x ) denote the density function and the cux mulative distribution function of x , respectively. Assume 0 . Here, and are both constants, F ( ) = 0 and F ( ) = 1. The mean + value of x is µ , i.e., µ = 2 . Demand D is also uncertain. Let g (D) and G (D) denote the density function and the cumulative distribution function of D . In the next, a basic game model is proposed to describe a two-echelon SC operating under VMI. And the decision process can be described as follows.
s
(K ) is concave in K . Let K denote the input
quantity determined by solving
µw0
c
w0
s (K )
K
xf (x ) G (xK ) dx = 0
= 0 . We then have (2)
Let K * denote the supplier's optimal input quantity, we then have the following theorem. Theorem 1. If µ w , then the supplier does not input any quantity, 0 c i.e., K * = 0 . If µ > w , then the supplier's optimal input quantity is 0 K* = K . With the support of demand information sharing from the retailer, the supplier obtains its optimal input quantity when no commitment order contract is introduced. According to Theorem 1, under yield and demand uncertainty, the supplier must relate the wholesale price and the mean value of yield rate together to judge whether to take part in c the operation of the SC or not. If and only if µ > w , the supplier is 0 willing to choose a positive input quantity. Thus, the supplier’s c
① Under yield and demand uncertainty, the supplier determines an optimal input quantity. Then, the products are produced. ② Both members find actual output quantity. Demand information
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J. Cai, et al.
expected profit is s (K *) , and the retailer’s expected profit is r (K *) . The total expected profit of the SC then is T (K *) = s (K *) + r (K *) . Such an operation mode equals a VMI mode. And it reflects the supplier’s supply support for the retailer. In the next, our work shows that commitment order contracts can be used to stimulate the input quantity of the supplier.
Lemma 1. Given µ , c and w0 , the supplier's expected profit changes with K as follows. (i) If µ
(ii) If µ
Lemma 1 can be interpreted as follows. Firstly, given a relatively low mean value of yield rate, the more quantity the supplier inputs, the more losses it bears. Then the supplier does not choose a positive input quantity whatever a large commitment order quantity is placed by the retailer. Secondly, given a relatively high mean value of yield rate, the supplier can maximize its expected profit with an optimal input quantity. In order to obtain explicit solutions to maximize the supplier’s expected profits, we ignore the special situation with = in the paper. We then have the following theorem.
4. The Retailer's commitment order contracts Assume that the retailer decides to offer commitment order contracts before the input quantity is determined by the supplier. According to the commitment order contracts, the retailer commits to purchase at least a quantity with the wholesale price w1. Then, the supplier determines an optimal input quantity and produces the products. After yield and demand uncertainty is observed, the retailer determines whether to place a second order with the wholesale price w0 by comparing the actual output quantity and the actual demand. In reality, two wholesale prices w1 and w0 are always determined through the negotiation between the members at the beginning of the trade. During the negotiation, characteristics of industry, market and product may all influence the relative size of w1 and w0 . In the next, we consider different situations according to the following three conditions: ① w1 = w0 ; ② w1 < w0 ; ③ w1 > w0 .
Theorem 2. If µ w , the supplier does not input any quantity, i.e., 0 K a * = 0 . If µ > wc and 1( ) wc , the supplier's optimal input quantity is c
0
, i.e., K xK
(K ) = w 0 + w 0 +
+
(xK
xK
(ii) When , i.e., K can be expressed as sa (K )
c w0
and
Here,
1(
)=
= w0 K xK
+ xK
(xK
(D
K
cK
K
(3)
, the supplier's expected profit
) f (x ) g (D ) dDdx +
xKf (x ) g (D) dDdx + K
sa (K ) = (µw0
K
, i.e., K
the supplier's optimal input quantity is K a * = K .
xf (x ) 1
( ))dx .
G x
0
c
c , ① w0 a*
in the region
[0,
1
. We then have
1) ,
the supplier's
c
+ 0
f (x ) g (D ) dDdx
cK
0
K a2 ( ) . ① Anticipating K a * = K , which is independent of , the retailer’s optimal commitment order quantity is 0. Then, the retailer’s expected profit can be expressed as
(4) (iii) When expressed as
xf (x ) G (xK a2 ( )) dx = 0 .
(i) When µ w , anticipating K a * = 0 , the retailer does not make a 0 commitment order. No member takes part in the operation of the SC. c (ii) When µ > w , the supplier’s optimal input quantity is K or
) f (x ) g (D ) dDdx )
+ 0
(
K a2 ( )
optimal input quantity K is independent of ; ② in the region [ 1, + ) , the supplier's optimal input quantity K a * increases with . Theorem 3 shows that if the retailer’s commitment order quantity is relatively small ( < 1), the supplier’s input quantity decision is not influenced by the retailer’s commitment order quantity. Then, the supplier makes the optimal decision according to the demand information shared by the retailer and the yield information, i.e., just like the VMI mode. If the retailer’s commitment order quantity is relatively large ( 1), then the more quantity the retailer commits, the more quantity the supplier inputs. This conclusion is very close to the reality, c since a high mean value of yield rate ( µ > w ) indicates a stable output, 0 and a large commitment order indicates a prosperous market. Anticipating the supplier’s optimal input quantities, the retailer determines its optimal commitment order quantities. The following situations should be considered.
K
+ w0
c , w0
Theorem 3. When µ >
) f (x ) g (D ) dDdx
) f (x ) g (D) dDdx
) >
w0
= w . Combining with Eq. (2), we conclude K = 0 the following theorem. 1 ( 1)
, the supplier's expected profit can be
(D
1(
c
It is easy to find that 1 ( ) decreases with . We have 1(0) = µ and c lim 1 ( ) = 0 . Thus, when µ > w , a unique 1 exists and satisfies
The above game can be analyzed by using backward induction. We first determine the supplier's optimal input quantities given the retailer's commitment order quantities. Because w1 = w0 , w1 is replaced by w0 in all equations in this section. By noting the actual quantity Min {xK , Max {D , }} ordered, we can express the supplier's expected profit as functions of K in different regions as follows.
sa
If µ >
+
K
0
K a * = K a2 ( ) , where K a2 ( ) satisfies µw0
4.1. w1 = w0
(i) When expressed as
c , the supplier's expected profit sa (K ) decreases with K . w0 c > w , the supplier's expected profit sa (K ) is concave in K . 0
, the supplier's expected profit can be
ra
(5)
c) K
xK
(0) = +
The supplier determines the optimal input quantity K that maximizes its profit function sa (K ) in the region K [0, + ) . Although the supplier's expected profit functions vary in different regions of K , the supplier's expected profit is differentiable and continuous in the whole region. The following lemma summarizes that how the supplier's expected profits change with K under different conditions. a*
(p
0
+ xK
w0 ) Df (x ) g (D ) dDdx (p
w0 ) xK f (x ) g (D ) dDdx
(6)
② Anticipating K a * = K a2 ( ) , the retailer’s expected profit can be expressed as
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International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
ra
+ xK a2 ( )
( )=
xK a2 ( )
+ +
K a2 ( )
(p
xK a2 ( ) 0
w0 ) Df (x ) g (D ) dDdx
w0 xK a2 ( )) f (x ) g (D ) dDdx
(pD
(7)
(p 1 ( K
a2
Here,
( 2
a1 i ))
c)
a1 i
is determined by
a1 i)
w0 G (
1
F
a1 i
K a2 (
a1 i)
0
Ta (K
a*
c ) K a*
) = (µp
xK a *
p
0
1(
2
a*
sb
, i.e., K xK
(K ) = w 1 + w 0 +
+
(xK
xK
f (x ) G (D ) dDdx .
, the supplier's expected profit can be
= w0
xK
+ + w1
K
+ xK
K K
+ 0
(xK
(D
(D
) f (x ) g (D) dDdx
cK
1(
(8)
w0
, the supplier's expected profit
K
sb
(K ) = (µw1
2
w1 . w0
2
()
2 (K )
=
1(
1(
( )>
w1 ; w0
=1
x2
2
) >
c , w0
K
2
( )>
w1 w0
>
2
()
2
1 K
sb (K )
and and
c w0
) 2
()
2
()
K
and
obtain
2
( ) g(x )dx .
K
K = K0
satisfies
K0
( )=1
sb (K )
(B)
w1 . w0
x 2K 3g (xK ) dx ,
3
0;
K = K0
and
G( )
is obtained from
2
K b20 ( )
F
,
K b20 ( )
c , if 1 w1
G( )
xK
1
f (x ) Dg (D) dDdx <
then the supplier's optimal input quantity is K
xK b20 ( )
( )>
w1 w0
>
2
( ),
f (x ) Dg (D ) dDdx > (w0 then
the
supplier's
sb (K )
K
K = K0
w0 G ( ) optimal
b*
w1 w0
= K ; if and
>0
w1 ) input
quantity
is
= ) ; otherwise, the supplier does not input any quantity, i.e., c c K b * = 0 . When µ > w , if 1( ) > w , then the supplier's optimal input b*
K b20 (
0
quantity is K b * = K ; otherwise, the supplier's optimal input quantity is b* = K b20 ( ) , where K b20 ( ) satisfies the following equation.
) f (x ) g (D ) dDdx ) + 0
>
1(
we
1
G( )
(C)
1
K
c , w0
)
w1 w0
G( )
And
c , w0
)
1
) f (x ) g (D ) dDdx
xKf (x ) g (D) dDdx +
, i.e.,
and
( )=1
and
) f (x ) g (D) dDdx
f (x ) g (D ) dDdx
cK
w0
(9) (iii) When K expressed as
c w0
)
c , w0
)
Theorem 5. When µ
(ii) When , i.e., K can be expressed as sb (K )
(B)
the first derivative of sb (K ) given in Eq. (9). c Lemma 2 shows that given w1 < w0 , when µ > w , an optimal input 1 quantity exists and maximizes the supplier’s expected profit. When c µ , it is very complicated to analyze the supplier’s optimal input w1 quantity decision. First, we can infer that the supplier’s expected profit is closely related to the retailer’s commitment order decision. Second, there exist two special cases. In Case ①, two conditions influence the supplier’s optimal input quantity decision. If one of the conditions is satisfied, there exists one local maximum in the region, and the supplier’s optimal input quantity can be obtained by comparing the boundary value and the local maximum. In Case ②, three conditions influence the supplier’s optimal input quantity decision. Any one of the conditions is satisfied, the supplier’s expected profit decreases with K . Then the supplier cannot benefit from any input quantity. According to Lemma 2, we have the following theorem.
c
Similar to Section 4.1, the game can be analyzed by using backward induction. We first determine the supplier's optimal input quantities given the retailer's commitment order quantities. Then, the supplier's expected profit can be expressed as functions of K in different regions as follows. K
1(
Here
4.2. w1 < w0
(i) When expressed as
c ; w0
> 0.
K = K0
2 (K 0 )
c
1(
(A)
denote the retailer’s optimal commitment order quantity. When µ w , 0 a* =0 . When µ > wc , a * {0, a } . Then, the total expected profit of the SC is
) >
is concave in K .
② If one of the following conditions is satisfied, then the supplier's expected profit decreases with K .
=0
K a2 ( 2)) = p . Given w1 = w0 , we let
1(
(A) K
Theorem 4. Anticipating K a * = K a2 ( ) , the retailer can find at least one optimal commitment order quantity in the set a { 1} Z , where = { a1i } , (i = 1,2, ..., n) , and a1i is determined by the following [ 1, 2 ) . equation in the region a1 i)
c , then the supplier's expected profit w1 c , then there are two scenarios: w1
sb (K )
The retailer determines the commitment order quantity a that maximizes its expected profit given in Eq. (7). And according to the [ 1, + ) . Then, the proof of Theorem 3, a must fall into the region following theorem summarizes the results.
K a2 (
(i) If µ >
(ii) If µ ① If one of the following conditions is satisfied, then the supplier's expected profit first decreases then increases, and finally decreases with K .
w0 ) f (x ) g (D ) dDdx
(pD
0
K a2 ( ) K a2 ( )
+
w0 ) xK a2 ( ) f (x ) g (D) dDdx
(p
, the supplier's expected profit can be
= w1
(10)
c) K
K b20 ( )
K b20 ( )
F
F (x ) dx
K b20 ( )
F (x ) dx
K b20 ( )
K b20 ( )
F
K b20 ( )
K b20 ( )
xf (x ) G (xK b20 ( )) dx
+c
Theorem 5 shows that, given w1 < w0 , even the mean value of yield rate c is relatively low, i.e., µ , the supplier is still willing to choose a positive w
The supplier determines the optimal input quantity K that maximizes its profit function sb (K ) in the region K [0, + ) . Similarly, the supplier's expected profit is differentiable and continuous in the whole region. The following lemma summarizes that how the supplier's expected profits change with K under different conditions. b*
1
input quantity if the retailer commits a proper quantity. When µ > w , if the 1 retailer’s commitment order quantity is relatively small, the supplier’s optimal input quantity equals K , just like the VMI mode; if the retailer’s commitment order quantity is large enough, the supplier may choose its optimal input quantity with the commitment order quantity considered. According to Theorem 5, we first let
Lemma 2. Given µ , c , w0 and w1, the supplier's expected profit changes with K as follows. 403
c
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al. 3(
)=1
derivative of given by
) is
3(
1 ( 1)
the region
c w0
=
[0,
only if µ >
xK
1
G( )
3( )
and
=
1(
and
1)
K b * increases with ; if 1 4 , then the supplier's optimal input [ 1, 4], and increases quantity K b * decreases with in the region [ 4, + ). with in the region Theorem 6 is analogous to Theorem 3, given a relatively high mean c value of yield rate ( µ > w ), if the retailer’s commitment order quantity 1 is relatively small ( 1), the supplier makes its input quantity decision according to the VMI mode. If the retailer’s commitment order quantity is large enough, then the supplier’s input quantity decision is influenced by the retailer’s commitment order quantity. Different from the situation with w1 = w0 , given w1 < w0 , the supplier adds a threshold commitment order quantity ( 4 ) to judge the relationship between the retailer’s commitment order quantity and the actual market demand. Then,
f (x ) Dg (D ) dDdx . Then, the first order xK
1
f (x ) Dg (D ) dDdx . Because
2
) decreases with , we conclude 1(
in the region
c w0
)
c . w0
Additionally, we always have
w1 , w0
then a unique
1(
1 c )> w 0
1(
is
in
[ 1, + ) if and when µ
c w0
)
c , w0
under which the supplier’s optimal input quantity is K = 0 . Re( ) [0, 1) membering that K = 1 , we conclude 3 > 0 in the region c c when w < µ w . The following two situations should be discussed: ① If
lim 3(
1
3(
0
) <
) >
1
exists and satisfies
3
holds in the region
w1 w0 w1 w0
) < holds in the region corollary. 3(
[0,
Because
2
( )=1
let
4(
4)
} {
0<
4
[0,
c w0
) >
2
()
a
w1 . w0
w1 w0
c w0
c , w1
<µ
1(
1
unique
exists
4
in the region
f (x ) Dg (D ) dDdx
and
c w0
)
w1 , and w0 w1 , then w0
We then have the following
}.
K b20 ( )
Given
② if lim 3 ( )
Then, we conclude
w1 w0
xK b20 ( )
) = w0
) <
< 1,
w1 w0
G ( 4) = and
3(
=
① If the retailer’s actual commitment order quantity is larger than Max { 1, 4 } , then the supplier ensures that the market demand is really large. Thus, the more quantity the retailer commits, the more quantity the supplier inputs. Max { 1, 4 } , the supplier is very cautious in ② If Min { 1, 4} determining its input quantity and two scenarios may be considered. In the first scenario, i.e., 1 > 4 , the supplier’s optimal input [ 4 , 1]. In the second quantity is independent of in the region scenario, i.e., 1 4 , the supplier’s optimal input quantity de[ 1, 4]. It is a little counter-intuitive creases with in the region that the supplier’s optimal input quantity decreases with the commitment order quantity. In fact, it is reasonable since the supplier adds a threshold commitment order quantity ( 4 ). In the region [ 1, 4], the supplier may think that the retailer only pursues a low wholesale price through the commitment order quantity, not really for a forecasting of an optimistic market demand. Thus, in the [ 1, 4], the more quantity the retailer commits, the more region pessimistic market anticipation the supplier supposes, and the less quantity the supplier inputs. If and only if the retailer’s commitment order quantity is larger than 4 , the supplier may really ensure an optimistic market demand.
there exists at least one positive quantity in the set
<µ Corollary 1. When commitment order 1(
1) .
3 ( 3)
c , w1
c w0
{
3) ;
[0,
b*
2
2
( )>
and w1 w0
satisfies
in the region
[ 4 , + ) . At the same time, (w 0
w0 G ( )
( )>
w1 w0
>
2
w1 )
1
F
K b20 ( )
.
( ), there must be
> 1, and falls into the region [ 1, 4 ) . Under specific settings of the parameters { , , c, w1, w0, p} , we can find at least one commitment = { i } [ 1, 4 ) , , where order quantity in the set (i = 1,2, ..., n) , and i satisfies 4 ( i ) > 0 . Thus, the supplier chooses a positive input quantity given the retailer’s commitment order quantity in the set . Because the supplier’s expected profit function is piece-wise continuous and differentiable, we summarize the supplier’s optimal input quantity decisions given any closed interval of commitment order quantity. We then have the following lemma. 4
Anticipating the supplier’s optimal input quantities, the retailer determines its optimal commitment order quantities. The following situations should be considered.
Lemma 3. Given the retailer's commitment order quantities, we obtain the supplier's optimal input quantities as follows. ① When
c w0
c , w1
<µ
(A) if
3 ( 1)
>
w1 , w0
then in the region
(i) When µ
3], w1 , w0
[0,
( )<
sb (K )
w1 , w0
K
K = K0
> 0 and
4(
) > 0 ; (D) otherwise, the sup-
plier's optimal input quantity is K = 0 . c ② When µ w , the supplier's optimal input quantity is K b * = 0 . ③ When µ >
0
c , w1
b*
(A) in the region
[0,
1],
[ 1, + ) , the sup-
Because
1 ( 1)
satisfies
4
3 ( 3)
satisfies 1
=
w1 w0
G ( 4) =
if and only if
w1 , w0
3 ( 1)
1
>
satisfies
w1 , w0
=
c , w0
it is easy to find
But it is hard to determine the relative sizes of
4
and
1
c , ① w1 b*
in the region
[0,
1],
and 3
given µ >
By analyzing the supplier’s optimal input quantity given µ > then have the following theorem. Theorem 6. When µ >
rb
the supplier's optimal
input quantity is K b * = K ; (B) in the region plier's optimal input quantity is b * = K b20 ( ) .
c , w1
the supplier’s optimal input quantity K b * is in the
set {0, K , K b20 ( )} . Let b1 *denote the retailer’s optimal commitment order quantity that maximizes its expected profit, then, ① Anticipating K b * = 0 , the retailer does not make a commitment order. No member takes part in the operation of the SC, i.e., b1 * = 0. ② Anticipating K b * = K , the retailer’s expected profit can be expressed as
the supplier's optimal input quantity is K b * = K ; (B) if 3 ( 1) [0, 1], the supplier's optimal input quantity is then in the region K b * = K ; (C) if there exists a region [ 1, 4], then the supplier's b* = K b20 ( ) optimal input quantity is given 2
c , w1
( )=
(pD
0
xK
+
+ xK
w1 ) f (x ) g (D ) dDdx +
((p
w0) D + (w0
((p
w1) xK + (w0
w1) ) f (x ) g (D) dDdx w1) ) f (x ) g (D) dDdx
(11)
It is easy to find that rb ( ) is concave in , and Eq. (11) is maximized by 4 . According to Lemma 3, b1 * must fall into the region [0, Min { 1, 3}] to induce the supplier to choose a positive input w w quantity. Remind that 3 satisfies 3 ( 3) = w1 if and only if 3 ( 1) > w1 , 0 0 and there is 3 4 if 3 exists. Hence, one and only one optimal b1 * can be found by the retailer in the set commitment order quantity { 1, 3 , 4} .
3
4. c . w1
we
the supplier's
optimal input quantity K is independent of ; ② in the region [ 1, + ) , if 1 > 4 , then the supplier's optimal input quantity
③ Anticipating pressed as
404
b*
= K b20 ( ) , the retailer’s expected profit can be ex-
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
rb
xK b20 ( )
K b20 ( )
( )=
0
K b20 ( )
+
K b20 ( )
+
K b20 ( )
+
K b20 ( )
w1 ) xK b20 ( ) f (x ) g (D ) dDdx
+ xK b20 ( )
induction. We first determine the supplier's optimal input quantities given the retailer's commitment order quantities. Then, the supplier's expected profit can be expressed as sc ( ) = sb ( ) in the regions
K
w1 ) f (x ) g (D ) dDdx ((p
w 0 ) D + (w 0
w1 ) ) f (x ) g (D ) dDdx
(i) If µ > (ii) If µ
(12) Combining with Eq. (A.3), the first order derivative of respect to is rb
( )
= (p 1 ( K b20 ( ))
c)
K b20 ( )
+ (w 0
w0 G ( )
w1 ) 1
F
rb
( ) with
)) = Here, 1 ( falls into the region K b20 ( )
G (xK b20 (
xf (x )(1 [ 1, 4], then w1
K b20 ( )
))) dx . Because b1 * w0 + w0 G ( ) 0 and
0 . Recalling
1(
K b20 ( 1)) =
c , w0
(p 1 ( K b20 (
b1 i ))
c)
(ii) When µ >
K b20 (
b1 i)
b1 i
c , w1 b2 *
+ (w0
w0 G (
b1 i)
then in the region
w1 ) 1
F
b1 i b2 K 0 ( b1i )
Theorem 9. When µ >
=0
µ
① Anticipating K b * = K , the retailer’s expected profit is given in Eq. (11). According to Lemma 3, b2 * must fall into the region [0, 1], one and only one optimal commitment order quantity can be found by the retailer in the set { 1, 4} . ② Anticipating b * = b20 ( ) , the retailer’s expected profit is given in Eq. (12). [ 1, + ) , According to Lemma 3, b2 * must fall into the region the following theorem then summarizes the results.
[0,
1],
the supplier's
(
c c , w1 w 0
, the commitment order contract is always efficient to
c
(i) When µ , anticipating K c * = 0 , the retailer does not make a w1 commitment order. No member takes part in the operation of the SC. c (ii) When µ > w , the supplier’s optimal input quantity is K or
b2 0(
1
f (x ) G (D ) dDdx .
K c2 ( ). ① Anticipating K c * = K , which is independent of , the retailer’s optimal commitment order quantity is 0. Then, the retailer’s expected profit is rc (0) = ra (0) . ② Anticipating K c * = K c2 ( ) , the retailer’s expected profit is rc ( ) = rb ( ) , which is given in Eq. (12). The retailer determines the optimal commitment order quantity c that maximizes its expected profit. And according to the proof of Theorem 9, c must fall [Max {0, 1}, + ) . Then, the following theorem into the region summarizes the results.
Similar to Section 4.2, the game can be analyzed by using backward
Theorem 10. Anticipating K c * = K c2 ( ) , the retailer can find at least {0, 1} one optimal commitment order quantity in the set c , where = { c1i } ,(i = 1,2, ..., n) , and c1i is determined by the following [Max {0, 1}, 5]. equation in the region
(p 1 ( K b20 (
Here,
µ
in the region
stimulate the supplier to take part in the operation of the SC, and the supplier’s input quantity increases with the retailer’s commitment order quantity. Anticipating the supplier’s optimal input quantities, the retailer determines its optimal commitment order quantities. The following situations should be considered.
= ) , the retailer can find at least one Theorem 7. Anticipating optimal commitment order quantity in the set b2 * { 1} , where = { b2i } ,(i = 1,2, ..., n) , and b2i is determined by the following [ 1, 5]. equation in the region
b*
c , ① w0 c*
optimal input quantity K is independent of ; ② in the region [ 1, + ) , the supplier's optimal input quantity K c * increases with c c . When w < µ w , the supplier's optimal input quantity K c * always 1 0 increases with . Theorem 9 is analogous to Theorem 3, but there are still some difc c ferences. Given a relatively low mean value of yield rate ( w < µ w ), 1 0 the supplier still inputs a positive quantity. Hence, in the region
the supplier’s optimal input quantity is in the set
b*
c , then the supplier's expected profit is concave in K . w1 c , then the supplier's expected profit decreases with K . w1
theorem.
{K , )} . Let denote the retailer’s optimal commitment order quantity that maximizes its expected profit. Then, K b20 (
) respectively.
c , then the supplier does not input any quantity, w1 c c and 1( ) > w , then the supplier's optimal input w1 0 c c quantity is K c * = K . If µ > w and 1( ) w , then the supplier's 0 1 c* c2 c2 ( ) = b20 ( ) . optimal input quantity is K = K ( ) ; here, c Recall that 1 exists if and only if µ > w , we then have the following 0
[ 1, 4], 1( K b20 ( 1)) > always holds. Hence, similar to the conclusion in Theorem 4, we summarize as follows: Anticipating b* = K b20 ( ) , the retailer can find at least one optimal commitment order quantity in the set b1 * { 1} , where = { b1i },(i = 1,2, ..., n) , and b1i is determined by the following equation in the region [ 1, 4]. c p
,+
Theorem 8. If µ i.e., K c * = 0 . If µ >
< 0 (from the proof of Theorem 6), it is easy to find
b2 1 ( K 0 ( ))
and K
,
Lemma 4 is analogous to Lemma 1, given a relatively high mean c value of yield rate ( µ > w ), the supplier can maximize its expected 1 profit with an optimal input quantity; given a relatively low mean value of yield rate, the supplier does not input any quantity. But different from the situation with w1 = w0 , given a higher w1, the supplier is more likely to participate in the operation of the SC even when the mean value of yield rate is relatively low. We then have the following theorem.
(13)
K b20 (
,K
Lemma 4. Given µ , c , and w1, the supplier's expected profit changes with K as follows.
w1 ) ) f (x ) g (D) dDdx
w0 ) xK b20 ( ) + (w0
((p
0,
The supplier determines the optimal input quantity K c * that maximizes its profit function sc (K ) in the region K [0, + ) . Then, the following lemma summarizes that how the supplier's expected profits change with K under different conditions.
w1 ) xK b20 ( ) f (x ) g (D) dDdx
(pD
xK b20 ( )
+
(p
(p
0
K b20 ( )
w1 xK b20 ( )) f (x ) g (D ) dDdx
xK b20 ( )
+
+
(pD
b2 i ))
5
c)
K b20 (
b2 i)
b2 i
+ (w 0
is determined by
1(
w0 G (
b2 i)
b2 0 ( 5))
w1 ) 1
F
b2 i b2 K 0 ( b2i )
=0
c
= p . Given w1 < w0 , we let
denote the retailer’s optimal commitment order quantity. When c c , b* = b1 *. When µ > w , b* = b2 *. Then, the total expected w 1
profit of the SC is
1
Tb (K
b*
) = (µp
c ) K b*
p
xK b *
0
4.3. w1 > w0
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(p 1 ( K c2 (
c1 i ))
c)
K c2 ( c1 i
c1 i)
+ (w 0
w0 G (
c1 i)
w1 ) 1
F
c1 i
K c2 (
c1 i)
Table 3. Compared with the basic model, by introducing the commitment order contract with w1 = w0 , the SC members may obtain a Pareto improvement. Compared with the commitment order contract with w1 = w0 , the retailer can benefit from the commitment order contract with w1 < w0 . For the supplier, a lower wholesale price w1 of the commitment order may result in a lower expected profit. For example, given = 0.5 and = 0.8, the supplier's expected profit is 141.24 under the basic model, 175.56 under the commitment order contract with w1 = w0 , and 131.12 under the commitment order contract with w1 < w0 . However, under specific situations, a lower w1 can encourage the retailer to make a commitment, thus the supplier can benefit more. For example, given = 0.6 and = 1.0 , the supplier's expected profit is 495.90 under the basic model, 495.90 under the commitment order contract with w1 = w0 , and 501.19 under the commitment order contract with w1 < w0 . After introducing the commitment order contract with w1 < w0 , the total expected profit of the SC can also be improved. For example, given = 0.5 and = 0.9 , the total SC's expected profit is 1551.51 under the basic model, 1626.49 under the commitment order contract with w1 = w0 , and 1630.21 under the commitment order contract with w1 < w0 . We conclude that the SC may perform better under the commitment order contact with w1 < w0 . As to the commitment contract with w1 > w0 , it is a good way to stimulate the supplier to take part in the operation of the SC even the mean value of yield rate is very low. Take the chip industry as an example, because of a long R&D cycle and a low yield rate, few companies are willing to take part in the chip production in China. But the chip is an essential part of many electronic products. Then, the commitment order contract with a higher wholesale price w1 is an effective way to stimulate the supplier's input in the chip production. Our numerical analysis shows that given = 0.5 and = 0.7 , the supplier is still willing to participate in the operation of the SC. Thus, both the supplier and the retailer can obtain profits under a relatively low mean value of yield rate.
=0
Given w1 > w0 , we let c * denote the retailer’s optimal commitment c c , c *=0 . When µ > w , c * {0, c } . Then, order quantity. When µ w1 1 the total expected profit of the SC is c* xK c ) K c* p f (x ) G (D ) dDdx . Tc = (µp 0
5. Numerical analysis In this section, to gain further insights, we present a numerical study based on the following setting. We assume that demand D is normally distributed with mean = 50 and variance 2 = 52 , i.e., D~N (50, 52) . Let c = 30 , w0 = 50 , p = 80 and = 0.5, then the yield rate x is uniformly distributed in the region [0.5, ]. Specifically, in the situation with w1 < w0 , we let w1 = 49; in the situation with w1 > w0 , we let w1 = 51. Fig. 1 shows that given = 0.8, the supplier's expected profit is concave in K under the commitment order contracts when = 10 or = 60 . Then the results in Lemmas 1(ii), 2(i) and 4(i) are confirmed respectively. Special cases are shown in Fig. 1(c), given w1 < w0 and = 0.72 , the supplier's expected profit first decreases then increases, and finally decreases with K when = 10 or = 29, and the supplier's expected profit decreases with K when = 60 , as the conclusions illustrated in Lemma 2(ii). Furthermore, Fig. 2 shows the impact of the retailer's commitment order quantity on the supplier's input quantity decision, as the results illustrated in Theorem 6 and Theorem 9. Anticipating the supplier's decisions under different conditions, the retailer chooses the optimal commitment order quantities to maximize its expected profits, as shown in Table 2. In the next, we discuss the performance of three commitment order contracts by comparison. Table 2 shows the SC members’ optimal decisions given different and in four models (i.e., the basic model, and the commitment order contracts with w1 = w0 , w1 < w0 , w1 > w0 , respectively). The corresponding optimal expected profits are shown in
Fig. 1. The supplier's expected profits change with K under commitment order contracts.
406
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
Fig. 2. The supplier's optimal input quantities change with under commitment order contracts.
Table 2 The SC members’ optimal decisions given different The retailer's optimal commitment order quantity
= 0.5 Basic model
Commitment order contract with w1 = w 0 Commitment order contract with w1 < w 0 Commitment order contract with w1 > w 0
= = = = = = = = = = = = = = = =
0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0
– – – – – 55.42 54.88 53.79 – 56.14 55.74 54.83 54.56 54.70 – –
and
The supplier's optimal input quantity
= 0.6 – – – – 50.96 51.82 51.84 – 51.40 52.47 52.47 51.59 50.51 – – –
inventory management and must determines the optimal input quantity under yield and demand uncertainty. A real case involving Zhejiang Nali is used to illustrate the management problem in this paper. We find that the SC members are eager to introduce a simple contract to improve the performance of the SC facing yield and demand uncertainty. Specifically, they want to keep the regular wholesale price stable during the trading period. In this study, we introduce commitment order contracts to construct a cooperative relationship between the members. Under the commitment order contracts, the retailer promises to purchase at least a quantity with a special wholesale price. Three possible situations are investigated: ① The wholesale price of the commitment order equals the wholesale price of the regular order; ② the wholesale price of the commitment order is lower than the wholesale price of the regular order; ③ the wholesale price of the commitment order is higher than the wholesale price of the regular order. Under these contracts, Stackelberg game models between the supplier and the retailer are constructed, in which the retailer determines the commitment order quantity and the supplier determines the input quantity. By using backward induction, we obtain the supplier's optimal input quantities and the retailer's optimal commitment order quantities under different conditions. Sensitivity analysis for the retailer's commitment order quantity on the supplier's input quantity is also investigated. The result shows that, the retailer's commitment order quantity decision has a great influence on the supplier's input quantity decision. Specially, the supplier's input quantity may decrease with the retailer's commitment order quantity under specific conditions, thus a proper commitment order quantity should be chosen by the retailer to
in four models.
= 0.5 – 60.16 57.01 53.68 – 71.18 64.63 58.98 – 71.59 65.07 58.98 78.40 70.76 57.01 53.68
= 0.6 65.07 61.57 57.69 54.11 74.05 67.76 62.21 54.11 74.34 68.19 62.42 57.26 73.74 61.57 57.69 54.11
6. Discussion and conclusion In this paper, we consider a SC operating under yield and demand uncertainty. In the traditional mode, the supplier takes charge of Table 3 The SC members’ optimal expected profits given different
and
in four models.
The retailer's expected profit
Basic model
Commitment order contract with w1 = w 0
Commitment order contract with w1 < w0
Commitment order contract with w1 > w0
= = = = = = = = = = = = = = = =
0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0
= 0.5
= 0.6
– 1167.61 1184.88 1181.74 – 1254.66 1225.15 1203.75 – 1300.00 1268.23 1244.60 1245.06 1209.90 1184.88 1181.74
1263.60 1281.00 1278.74 1271.59 1336.35 1312.70 1288.95 1271.59 1383.90 1358.43 1322.50 1311.03 1289.19 1281.00 1278.74 1271.59
407
The supplier's expected profit
= 0.5
= 0.6
– 141.24 264.62 369.77 – 171.56 313.95 422.74 – 131.12 273.00 385.61 46.90 219.14 264.62 369.77
153.80 288.00 400.62 495.90 183.57 331.45 449.61 495.90 136.69 287.53 407.63 501.19 229.81 288.00 400.62 495.90
The SC's expected profit
= 0.5
= 0.6
– 1308.85 1449.50 1551.51 – 1430.22 153,910 1626.49 – 1431.12 1541.23 1630.21 1291.96 1429.02 1449.50 1551.51
1417.40 1569.00 1679.36 1767.49 1519.92 1644.15 1738.56 1767.49 1520.59 1645.96 1740.13 1812.22 1519.00 1569.00 1679.36 1767.49
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
stimulate the input quantity of the supplier. Our numerical analysis demonstrates that, with special sets of wholesale prices, the commitment order contracts can be used to improve the performance of the SC facing different random yield environments. Specially, when the supplier's yield rate is very low, the commitment order contract with w1 > w0 performs better, it can stimulate the supplier to participate in the operation of the SC. When the supplier's yield rate is relatively high, the commitment order contract with w1 < w0 performs better, it can encourage the retailer to make a commitment. When the yield rate is mid-high, the commitment order contract with w1 = w0 performs better. Hence, enterprises need to choose the appropriate commitment order contract according to the actual situation, thus they can effectively reduce the impacts from yield uncertainty and improve profits. The main contributions of the paper can be summarized as follows.
construct a tighter relationship between the retailer and the supplier based on commitment order contracts. The proposed contracts can be easily applied by the members through simple contract parameters. Based on this paper, given different sets of wholesale prices, the retailer and the supplier can rapidly determine their optimal commitment order decisions and input quantity decisions under yield and demand uncertainty. It is difficult to eliminate all possible stock shortages with yield and demand uncertainty. Most studies thus introduce replenishment tactics to improve the service level. Replenishment tactics can effectively improve the supply capability of the supplier. This paper does not consider replenishment tactics because of the following reasons: ① In most cases, replenishment tactics are difficult to implement during a selling season because of cost, time and other factors. ② Based on the model, traditional replenishment tactics actually can be absorbed, and corresponding optimal solutions can be conveniently obtained. ③ In future work, we want to introduce replenishment tactics that consider changes in cost, time and demand. There are several future directions for this research. Competition between multiple suppliers or retailers can be introduced into the model. Information among the SC members is asymmetric in most situations. When considering information asymmetry, the competition structures become more complex. Furthermore, a model with several trade periods would be very interesting.
(i) The yield rate is a random variable following uniform distribution defined in a general region. This definition very closely models practical problems related to yield uncertainty. (ii) The commitment order contracts proposed in this paper can be conveniently introduced into the VMI SC connected by a wholesale price contract. (iii) All optimal solutions are obtained for both members under different conditions. Hence, the models can be easily accepted by the enterprises to support their operations management. The retailer wants to improve the performance of procurement without incurring more inventory costs. The operation mode of VMI SC therefore is popular for releasing the retailer from the pressure of inventory management. However, it is difficult for the supplier to maintain a stable or sufficient quantity for the retailer under yield and demand uncertainty. In this paper, we propose an effective way to
Acknowledgments This research is supported by National Natural Science Foundation of China (71572184, 71874046, 71601169, 71702167, 71701184), Natural Science Foundation of Zhejiang Province (LY19G020011), Xinmiao Talent Project of Zhejiang Province (2019R403075).
Appendix Proof of Theorem 1. It is straightforward and details are omitted. □ Proof of Lemma 1. We consider the supplier's expected profits in different regions. (i) In the region K 1(
(
)=
1(
)
then
sa (K )
G x
sa (K )
the first order derivative of then µ
1(
decreases with K ; ② if
c and
sa (K )
K
K=
2
) . Because 1(
) >
, the first order derivative of
,
= µw0
K=
),
( ))dx ,
xf (x ) 1 c , w0
(ii) In the region K K
,+
c , w0
sa (K ) K2
then
sa (K )
sa (K )
< 0,
sa (K )
sa (K )
K
= w0 1 ( )
K=
K
= w0
c and
xf (x )(1
sa (K )
K
K
+
G (xK )) dx
c . Let
c < 0 , we have ① if
=
is concave in K .
given in Eq. (4) is
c , we have ① if µ
= w0 1 ( )
sa (K )
given in Eq. (3) is
c , w0
then
sa (K )
K sa (K )
= µw0
c
w0
K
xf (x ) G (xK ) dx . Because
decreases with K ; ② if µ >
c , w0
2
sa (K ) K2
< 0,
then we consider the
following two scenarios. (A) If
1(
c , w0
)
sa (K )
(iii) In the region K
is concave in K ; (B) if
1(
) >
c , w0
sa (K )
, according to Eq. (5), we have ① if µ
0,
increases with K . c , w0
then
sa (K )
decreases with K ; ② if µ >
c , w0
then
sa (K )
increases with K .
The supplier's expected profit is a piece-wise continuous and differentiable function of K . From the above discussions, we consider the following two situations. (i) If µ
(ii) If µ >
c , w0 c , w0
then
(A) When
1(
)
(B) When
1(
) >
sa (K )
decreases with K .
then we consider the following two scenarios. c , w0 c , w0
sa (K )
is concave in K and maximized in the region K
,
sa (K )
is concave in K and maximized in the region K
,+
Proof of Theorem 2. From the proof of Lemma 1, it is easy to find that. (i) If µ
c , w0
then the supplier's optimal input quantity is K a * = 0 . 408
.
). □
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
(ii) If µ >
and
c w0
µw0
c
1(
c , w0
)
by solving the first order derivative of
given in Eq. (4), a unique K a2 ( ) exists and satisfies
sa (K )
xf (x ) G (xK a2 ( )) dx = 0
w0
(A.1)
K a2 ( )
Then, K a * = K a2 ( ) . (iii) If µ > □
c w0
and
1(
c , w0
) >
by solving the first order derivative of
Proof of Theorem 3. When µ >
[0,
and
1)
1(
a unique
in the region
c w0
)
c , w0
independent of . In the region
exists and satisfies
1
given in Eq. (3), we obtain the supplier's optimal input quantity K a * = K .
sa (K )
1 ( 1)
c . w0
=
Because
1(
) decreases with , we have
[ 1, + ) . From Theorem 2, we conclude that in the region
[ 1, + ) , K
a*
K a2 ( )
= K ( ) . From Eq. (A.1), we find a2
=
1) ,
[0,
K
1(
) >
in the region
= K . Obviously, K is
a*
K a2 ( ) G ( ) 2G ( ) + (K a2 ( ))3
c w0
>0.□
x 2g (xK a2 ( )) dx K a2 ( )
Proof ra ( )
of
= (p 1 ( K
Here, Because 1(
K
Theorem
a2
a2
4.
( ))
c)
K a2 ( )) =
1(
K a2 ( )
> 0,
c . w0
( 1)) =
Combining
K a2 ( )
with
w0 G ( ) 1
F
Eq.
K a2 ( )) decreases with
Meanwhile, we have lim
1(
+
and
< 0 , we conclude
a
2;
=
first
order
in the region
K
a2
(B) when
must fall into the region
2),
[ 1,
ra ( )
lim
of
ra (
2
a2 1 ( K ( ))
K a2 ( )) is 1 ( 1)
=
c w0
exists and satisfies
a1 i ))
K a2 (
c)
a1 i
a1 i)
w0 G (
a1 i)
1
F
1(
K
a1 i
(
Eq.
(7)
is
x 2f (x ) g (xK a2 ( )) dx . c
K a2 ( 2)) = p . Obviously,
[ 1, + ) . Then, (A) when
2
c
0
is positive or negative, two
ra ( )
= 1
a1 i },
={
1.
>
[ 2 , + ) , p 1 ( K a2 ( ))
a
a2
in
K a2 ( )
=
< 0 , then = 1 ; (b) if there exists at least an extremum, let possible situations should be discussed: (a) If [ 1, 2 ) . solution set of determined by the following equation in the region
(p 1 ( K a2 (
given
)
and Eq. (A.1) together, we conclude
< 0 . Since we cannot determine whether
2
ra ( )
1(
[ 1, + ) . Considering
( )) = 0 . Given p > w0 , a unique a
derivative
.
K a2 ( )
The retailer's optimal commitment order quantity ra ( )
the
G (xK a2 ( ))) dx . The first order derivative of
xf (x )(1
1(
(A.1),
(i = 1,2, ..., n) denote the
=0
a1 i)
Considering (A) and (B) together, we conclude that at least one
a
can be found in the region
[ 1, + ) ; here
a
.□
{ 1}
Proof of Lemma 2. We consider the supplier's expected profits in different regions. (i) In the region K
,+
(ii) In the region K
,
sb (K )
K
= w0
K
F
w1 We 2
sb (K ) K2
2
K3
Here, 2
f
, the first order derivative of
()
F (x ) dx
K
sb (K )
K
( )1 K
let
2 (K )
( )=1
G( )
① When
2
( )>
K=
F
()
= µw0
G( )
>
2
sb (K )
c , w1
2
and
c
K w1 w0
x 2K 3g (xK ) dx
K
1 2
K
② When (A) If µ > (B) If µ
③ When
2
then
w1 , w0
1(
2
( ),
sb (K ) K2
) >
K=
()
w1 , w0
c.
= w0 1 ( )
c , w0
then
sb (K )
c w0
sb (K ) K2
is concave in K .
The
second
order
derivative
sb (K )
K
1(
)
x 2K 3g (xK ) dx ,
and
decreases
2 (K )
with
K.
is concave in K . There exists a unique K 0 satisfying 2 (K 0 ) =
0 and and
and
c w0
sb (K )
K
sb (K )
sb (K )
K
1(
K = K0
0,
0 and
c , w0
) 1(
)
sb (K )
K
sb (K )
is
We
obtain
2
sb (K )
c w0
and
w1 w0
and maximizing
( )=1 sb (K )
K
first increases then decreases with K ; (b) when sb (K )
K
K = K0
> 0,
decreases with K ; (c) when
sb (K ) 1(
G( )
and
. We consider the 1(
) >
c , w0
sb (K )
first decreases then increases, and finally
) >
c , w0
sb (K )
first decreases then increases
increases with K . We consider the following scenarios.
increases with K .
sb (K )
of
.
then we have the following conclusions. (a) When 2
) >
given in Eq. (9) is
then we have the following conclusions. (a) When
()
2 c , w1 c , w1
1(
( ) g(x )dx . Then, 3
decreases with K ; (b) when with K .
decreases with K ; ② if
sb (K )
(A.2) sb (K )
following scenarios. c (A) If µ > w , then we have the following conclusions. (a) When 1 increases with K . (B) If µ
then
c
K
=1
2
w1 w0
K
1
x2
c , w0
)
K
G( )
1
1(
xf (x ) G (xK ) dx
K
F (x ) dx
K
find
= w0
), from Eq. (8), we have ① if
1(
)
c , w0
sb (K )
decreases with K ; (b) when
decreases with K . We consider the following scenarios. 409
1(
) >
c , w0
sb (K )
is convex in K .
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
(A) If µ >
c , w1 c , w1
(B) If µ
then we have the following conclusions. (a) When then
1(
)
(iii) In the region K
and
c w0
sb (K )
c , w0
)
decreases with K .
c , w1
, from Eq. (10), we have ① if µ
0,
1(
then
is concave in K ; (b) when
sb (K )
decreases with K ; ② if µ >
sb (K )
c , w1
1(
) >
then
c , w0
sb (K )
sb (K )
increases with K .
increases with K .
From the above discussions, we consider the following three situations. (i) If µ >
c , w1
① When
1(
② When
1( c , w1
(ii) If µ ① When
1(
then we consider the following two scenarios.
) ) > )
K = K0
② When
1(
③ When 2
is concave in K and maximized in the region K
,
sb (K )
is concave in K and maximized in the region K
,+
c , w0
sb (K )
decreases with K in the regions K
> 0 hold, then c , w0
) >
1(
( )>
.
).
decreases with K in the region K
sb (K )
)
w1 w0
>
2
,+
,
0,
( ) and
sb (K )
K
,+
); in the region K
,
, if
2
( )>
w1 w0
0 ; (B)
K = K0
2
()
w1 ; w0
(C)
2
()
first decreases then increases with K in the region K
sb (K )
).
and one of the following conditions is satisfied,
c w0
and K
0,
>
2
( ) and
,
, and
first decreases then increases, and finally decreases with K .
sb (K )
is concave in K in the region K
sb (K )
(A)
sb (K )
then we consider the following three scenarios.
sb (K )
K
c , w0 c , w0
sb (K )
decreases with K .
w1 . w0
□
Proof of Theorem 5. From Lemma 2, it is easy to find that, (i) When µ
c , w1
① When i.e., II sb (KI )
1(
we consider the following three situations.
) >
= w0 (
(b) If 1 ② When
we discuss as follows. (A) The boundary value is
w1 w0
By comparing (a) If 1
c , w0
(1
and
sb (0)
1
G( )
sb (K
xKI
1
xK
f (x ) Dg (D) dDdx <
G( )
1
xK
f (x ) Dg (D) dDdx
c , w0
)
( )>
2
w1 w0
>
2
(K b20 ( )) = (w1
=(w1
K b20 ( )
w0 )
w0 + w0 G ( ))
1
F
K b20 ( )
K b20 ( )
K b20 ( )
= w1
F
F (x ) dx
By comparing (a) If w0 (b) If w0
2
sb (0)
xK b20 ( ) K b20 ( )
xK b20 ( ) K b20 ( )
③ When (A)
K b20 ( )
( )>
1( w1 w0
c w0
) >
2
and
F
K = K0
sb (0)
(K )
sb
and K b * = K .
(0) and K b * = 0 .
> 0 hold, we discuss as follows: (A) The boundary value is
c ) K b20 ( )
xK b20 ( )
K
) >
= 0 ; (B) the local
w0
xK b20 ( ) K b20 ( )
f (x ) G (D ) dDdx
= 0 given in Eq. (A.2), i.e.,
xf (x ) G (xK b20 ( )) dx
+c
K b20 ( )
(A.3)
)) , the supplier's optimal input quantity K w0 G ( )
w1 )
1
F
f (x ) Dg (D ) dDdx
w0 G ( )
w1 )
1
F
(w0
K b20 ( )
K b20 ( )
sb (K )
K = K0
0 ; (B)
2
()
w1 ; w0
(C)
2
()
w1 . w0
410
b*
can be obtained as follows.
, then , then
and one of the following conditions is satisfied, K b * = 0 . K
sb (0)
f (x ) Dg (D) dDdx
f (x ) Dg (D ) dDdx > (w0
( ) and
),
, i.e.,
K b20 ( )
b2 sb (K 0 (
sb
K b20 ( )
sb (K )
K b20 ( )
then
K
F (x ) dx
K b20 ( )
sb (K
F (x ) dx + (µw0
+ w0
K b20 ( )
then
sb (K )
,
Here, K b20 ( ) is determined by solving
w0
,+
f (x ) Dg (D ) dDdx ))
w1 , w0 w1 , w0
( ) and
maximum is obtained in the region K sb
= 0 ; (B) the local maximum is obtained in the region K
) , the supplier's optimal input quantity K b * can be obtained as follows.
G( )
1(
sb (0)
b2 sb (K 0 (
sb
)) >
(K b20 ( ))
sb (0)
sb
and K b * = K b20 ( ) .
(0) and K b * = 0 .
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
(ii) When µ > ① If
1(
) >
② If
1(
)
c , sb (K ) is concave in K . We consider the following two w1 c , the supplier's optimal input quantity K b * = K . w0 c , the supplier's optimal input quantity K b * = K b20 ( ) . □ w0
Proof of Corollary 1. When if
w1 . w0
lim 3 ( ) 1
{
1(
c w0
) >
Thus,
} {
c w0
<µ
) <
w1 w0
c , given 1( w1 c c <µ w , w0 1
when
3(
c , w0
) >
we have
there
exists
c , w1 K b20 ( )
[ 1, + ) , K b * = K b20 ( ) . From Eq. (A.3), we find = (w1
w0 + w0 G ( ))
2
+ w0 (K b20 ( ))
Given K b20 ( ) > , we conclude that (K b20 ( ))
[ 4 , + ) ; in the region
can obtain
3 K b20 ( )
3 K b20 ( )
=
(w1
4.
1
and
Hence, T ( 4 ) > 0 . Recalling
xf (x ) G (
4 ) dx
4],
[ 1,
K b20 ( 1)
1,
=
µ w1 . Because µ w
=µ
at
w1 w0
least
}
{ 0
one
<
3}
positive
if lim
3(
1
w1 ; w0
) >
commitment
{
3(
order
1],
w1 w0
) <
}
{ 0
quantity
in
< the
1}
set
K b * = K . Obviously, K is independent of ; ② in the region
w0 + w 0 G ( )) K b20 ( ) . M
K b20 ( )
x 2g (xK b20 ( )) dx > 0 and
because
K b20 ( ) 2
> 0 in the region
> K b20 ( )
>
4,
then w1
[ 1, + ) ; (B) if
4,
1
w0
w0 G ( ) > 0 .
then
K b20 ( )
0 in
G ( ) , we then
g (xK b20 ( )) dx = G ( K b20 ( ))
K b20 ( )
w0 + w0 G ( K b20 ( ))) 2 . Because K b20 ( ) > , we haveG ( K b20 ( 4)) > G ( 4 ) given
( ) 1
µ w1 is true given µ > w
>µ
1
x 2g (xK b20 ( )) dx
we then obtain G ( K b20 ( 1)) = G c w0
0
[0,
x 2g (xK b20 ( )) dx . Then, we discuss as follows: (A) If
w0 + w0 G ( K b20 ( ))) 2 . Let T ( ) = (w1
> (w 1
) <
① in the region
(K b20 ( ))3
the region
3(
}. □
Proof of Theorem 6. According to Lemma 3, when µ > Here,
{
situations.
0
( ). Meanwhile, we have we conclude G (x ) > G (
G x
c , w1
( )dx = µ
1
1
xf (x ) G x 4) .
c w0
1
Thus, we conclude
> G ( 4 ) . Hence, T ( 1) > 0 . [ 1, 4], at least one exists and satisfies < 0 . If Assumption I In the next, we first propose an assumption, i.e., Assumption I: In the region is true, then at least one exists and satisfies T ( ) < 0 . Because T ( 1) > 0 and T ( 4 ) > 0 , a region of (let denote this region) exists and satisfies
G(
K b20 ( 1))
T( )
T( )
< 0 . Since
K b20 ( )
= w0 g ( K b20 ( )) K b20 ( )
(a) If w0 g ( K b20 ( ))
< 0 , because w0 g ( K b20 ( )) > 0 , then
K b20 ( )
true, then
K b20 (
w0 + w0 G ( ))
K b20 ( )
in the region
<0
Proof of Theorem 7. According to the expression of
then
K b20 ( 5))
1(
rb ( )
[ 5, + ) , b2 *
=
1;
=
c . p
Thus,
< 0 and
b2 *
1(
K b20 (
=
5;
② If and
1
4,
1
4,
>
at least one
K b20 ( )
then
b2 1 ( K 0 ( ))
> 0 in the region
[ 1,
))
K b20 ( )
1
i.e., (p 1 ( K b20 (
4
<
[ 4 , + ) . Recalling
[ 1,
1(
Proof of Lemma 4. In the region K
,
b2 *
{ 1}
={ b2 i ))
to that of Lemma 1, and we omit the details. □
0 , i.e., K b20 ( ) decreases with
.
K b20 ( 1))
4],
[ 1,
=
and
1(
can be found in the region 2
sc (K ) K2
< 0 and
K b20 ( 5)) b2 *
))
c p
4,
>
c , w0
then w0
in the region
w0 G ( )
in the region
b2 i)
[ 1,
4];
=
c , p
w1 ) 1
K b20 ( )
F
5
[ 1, 5]. Then, (A) in the
denote the solution set of
w0 G (
w1 < 0 and
together with p > w0 , a unique
similar to that of Theorem 4: (a) If
0 in the region c w0
1
K b20 ( 1)) =
1(
K b20 (
i
b2 1 ( K 0 ( ))
, because w1 > w0 , we find
< 0 and
5], the proof is b2 i } ,(i = 1,2, ..., n ) K b2 ( b2 ) c ) 0 b2 i + (w 0
5 . Then, similar to the discussions in ①, at least one
Hence, we conclude that at least one
K b20 ( )
Thus
[ 5, + ) , and 1 (
can be found in the set { 1}
0 (from the proof of Theorem 6) and
0 in the region
[ 4 , + ) , i.e.,
b2 *
5],
4].
b2 1 ( K 0 ( ))
in the region
(B) in the region
[ 1,
On the other hand, in the region
given in Eq. (13), we discuss as follows: ① If
[ 1, + ) . Because c p
4 ]).
0.
(b) if there exists at least an extremum, let
the following equation in the region conclude that if
[ 1,
rb ( )
> 0 (from the proof of Theorem 6) in the region
exists and satisfies
(
0 . If Assumption I is true, then
)
Hence, Assumption I leads to the contradiction. Then [ 1, 4]. □
region
< 0.
< 0.
If Assumption I is true, then
K b20 ( )
K b20 ( )
.
w0 + w0 G ( K b20 ( ))) < 0 , then G ( K b20 ( )) < G ( 4 ) . Recall G ( K b20 ( 1)) > G ( 4 ) and G ( K b20 ( 4)) > G ( 4 ) . If G ( K b20 ( )) < G ( 4 ) is
(b) If 2 (w1
(w1
w0 + w0 G ( K b20 ( ))) , we consider the following two possible scenarios in the region
+ 2 (w 1
b2 i
K b20 ( b2i )
rb ( )
< 0,
determined by
= 0 . Hence, we
0 (from the proof of Theorem 6)
we conclude that
can be found in the set { 1}
5
only exists in the region .
[ 1, + ) . □ sc (K )
K
decreases with K . Then, the rest of the proof is similar
Proof of Theorem 8. It is similar to that of Theorem 2 and we omit the details. □ c Proof of Theorem 9. Given w1 > w0 , we have w >
holds. Then, we discuss as
region
[ 1, + ) ,
1 ( 1)
c c c c c c . When µ > w , a unique 1 exists and satisfies 1 ( 1) = w ; when w < µ w , 1( ) w always w1 0 0 1 0 0 c c c* [0, 1], 1 ( 1) w and K = K . Obviously, K is independent of . In the follows: (i) When µ > w , in the region 0 0 c c c < w and K c * = K c 20 ( ) = K b20 ( ) ; (ii) when w < µ w , K c * = K c 20 ( ) = K b20 ( ) . Here, K b20 ( ) increases with . □ 0 1 0 0
411
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Proof of Theorem 10. It is similar to that of Theorem 4 and we omit the details. □
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Contents lists available at ScienceDirect
International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe
Optimal input quantity decisions considering commitment order contracts under yield uncertainty
T
Jianhu Caia,*, Xiaoqing Hua, Feiying Jianga, Qing Zhoub, Xiaoyang Zhanga, Liyuan Xuana a b
School of Management, Zhejiang University of Technology, Hangzhou, 310023, PR China School of Management, Hangzhou Dianzi Uninversity, Hangzhou, 310018, PR China
ARTICLE INFO
ABSTRACT
Keywords: Supply chain management Input quantity decision Yield uncertainty Commitment order contracts
We examine a two-echelon supply chain with one supplier and one retailer under yield and demand uncertainty. A basic model is developed, in which the supply chain is assumed to operate under Vendor-Managed Inventory (VMI). Commitment order contracts are then proposed, and three possible situations are investigated: ① The wholesale price of the commitment order equals the wholesale price of the regular order; ② the wholesale price of the commitment order is lower than the wholesale price of the regular order; ③ the wholesale price of the commitment order is higher than the wholesale price of the regular order. Given the retailer's commitment order quantities, we obtain the supplier's optimal input quantities under different conditions. The impact of the retailer's commitment order quantity on the supplier's input quantity decision is also examined. Furthermore, the retailer's optimal commitment order quantities are discussed. We demonstrate the detailed boundary conditions for all possible optimal solutions of both members. Our analysis reveals that commitment order contracts are effective to stimulate the input quantity of the supplier. However, there still exist several regions in which the supplier's input quantity is independent of the commitment order quantity. In the special situation, we find that the supplier's input quantity decreases with the commitment order quantity. The numerical example shows that, with special sets of wholesale prices, the commitment order contracts can be used to improve the performance of the supply chain facing different random yield environments.
1. Introduction Yield uncertainties usually occur in an imperfect production system in which actual output quantity is less than the input quantity. Yield uncertainty also refers to the phenomenon in which the buyer receives less quantity than was ordered from the supplier. Yield uncertainty may occur due to instabilities in any step of the supply chain (SC), such as production, inventory and transportation. Instability is common in the production systems of agriculture, the chemical industry and the electronics industry (Keren, 2009). In most situations, yield uncertainty means that the actual yield is less than or equal to the input quantity. In batch production processes, e.g., agriculture, chemicals or pharmaceuticals, the actual yield may be more than the amount expected (Snyder and Shen, 2011). In this paper, we focus on the yield uncertainty related to special production systems. Yano and Lee (1995) describe different ways of modeling yield uncertainty and discuss some of their advantages and disadvantages. These methods are still effective for modeling yield uncertainty and are used by many researchers. This study is motivated by the case of Zhejiang Nali Clothing Co.,
*
Ltd., in China. In spring 2017, Nali orders 930 m of worsted fabric from the supplier, Jiangsu Danmao Textile Co., Ltd. Because of the complicated and uncertain production, defective products are inevitable. Although Danmao anticipates possible yield loss and inputs enough materials, the rate of imperfect items (30%) is still higher than expected. Ultimately, Nali only receives 680 m of worsted fabric from Danmao. Since the production cycle of worsted fabric is 65 days, it is very difficult for Danmao to replenish the products in time. This makes Nali not able to respond quickly to the market demand, and causes profit losses of both Nali and Danmao. Nali forms the assumption that more worsted fabric will be produced if a more reasonable order is placed. However, if a higher order is placed, Nali may receive more worsted fabric than needed. And Danmao may face inventory risk if the input quantity is too high. Both firms are eager to introduce an effective cooperative mechanism to reduce the impacts of yield uncertainty. Similar management problems are common in many other enterprises in Zhejiang Province. Another enterprise, Jinlong Machinery & Electronics Co., Ltd., produces mini motors (the diameter of the main product is 4–7 mm). It is difficult for Jinlong to ensure a stable output
Corresponding author. E-mail address: [email protected] (J. Cai).
https://doi.org/10.1016/j.ijpe.2019.06.021 Received 4 July 2018; Received in revised form 6 April 2019; Accepted 27 June 2019 Available online 04 July 2019 0925-5273/ © 2019 Elsevier B.V. All rights reserved.
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for a given input, because the production process is very complicated. However, the company must determine the input quantity under yield and demand uncertainty in most situations to provide a relatively stable supply and meet the market demand. Hence, it is necessary for Jinlong and its buyers to introduce a simple and effective contract scheme to improve the performance of the SC. In the case of Nali and Danmao, some work has been done by the firms to construct tight cooperative relationships. First, Nali and Danmao are trying to establish efficient channels to exchange information in time. Second, Danmao tries its best to support the flexible production of Nali. In some special situations, without receiving the order of Nali, Danmao promptly makes an input quantity decision by analyzing yield and demand information. This mode is very close to the Vendor-Managed Inventory (VMI) mode. Third, Nali changes the traditional order decision into the commitment order decision by sharing demand information with Danmao. In this paper, we first develop a VMI SC model, and examine the SC members’ optimal decisions under yield and demand uncertainty. We then propose the commitment contracts under which the retailer commits a minimum order quantity before the selling season. Such commitment contracts are termed “commitment order contracts”. Three contracts are proposed according to the difference between the wholesale price of the commitment order and the wholesale price of the regular order: ① The wholesale price of the commitment order equals the wholesale price of the regular order (e.g., in the trade between Nali and Danmao, Nali can pre-order the products from Danmao and the wholesale price of products ordered in advance often keeps the same with the wholesale price of the regular order); ② the wholesale price of the commitment order is lower than the wholesale price of the regular order; for example, in the production of holiday celebration products (e.g., moon cakes, pumpkin pies, etc.), food contract manufacturers often place two orders from the packaging material suppliers. One is the advance order with a discount wholesale price, the other is the rush order with an original wholesale price. Both orders are delivered to the manufacturers at the beginning of the selling season (Chintapalli et al., 2017); ③ the wholesale price of the commitment order is higher than the wholesale price of the regular order; one example from the SC of Nali and Shaoxing Yonson Digital Printing Technology Co., Ltd. can illustrate this situation. In December 2018, Yonson produces a special fabric for Nali, because of uncertainty in production and market demand, the wholesale price they negotiate for the commitment order is relatively high. With the approach of the selling season, the market demand turns out to be positive, and Nali purchases more than the commitment order quantity with a lower wholesale price. Thus, the following important managerial questions must be addressed.
quantity. Hence, the retailer should choose proper commitment order quantity to encourage the input quantity of the supplier. (2) The distribution characteristics of the yield rate have a great influence on SC members' optimal decisions and the performance of the SC. It may affect both members' participation in the operation of the SC. Detailed boundary conditions are obtained for all possible optimal solutions of both members. The results in the paper can help enterprises make proper decisions when facing yield and demand uncertainty. (3) Under special conditions, commitment order contracts are efficient to stimulate SC members to cooperate with each other, thus improving the performance of the SC. Our work shows that the effectiveness of commitment order contracts is relevant to the exogenous wholesale prices and the yield rate. In this paper, the research structure is very simple, but finding the optimal decision is very difficult. The above management insights may help the managers of the enterprises make appropriate operational decisions to build tight relationship with each other and reduce the impacts of yield uncertainty, thus to improve the performance of the SC. The managers of these enterprises believe in great practical value of the results of such research structures. The rest of the paper is organized as follows. Section 2 reviews the related literature. In Section 3, we introduce the assumptions and develop a VMI SC model under yield and demand uncertainty. Section 4 proposes three commitment order contracts, and all members’ optimal decisions are obtained under different conditions. In Section 5, numerical analysis is conducted to illustrate and validate our approach, and the efficiencies of the commitment order contracts in improving the performance of the SC are examined and compared. Summary, conclusions and future research are discussed in Section 6. 2. Literature review 2.1. Yield uncertainty Yield uncertainty is a common phenomenon, many real cases with yield uncertainty are introduced to construct decision models. Kazaz (2004) studies random yield in the olive industry by defining the sales price and the purchasing cost as exogenous and increasing with decreasing yield. Cai et al. (2017) introduce the case of Mate 7, a smartphone launched by HUAWEI, to illustrate the yield uncertainty in the SC for electronic products. Yield uncertainty is also common in agriculture. For example, the output of soil crops is influenced by irrigation and N fertilization (Hao et al., 2017). Climate is an important factor for coffee-farming in Peru (Lechthaler and Vinogradova, 2017). Tang and Kouvelis (2014) find that semiconductor and agribusiness industries are seriously affected by random yields. He and Zhang (2010) present examples about the “Nintendo Wii” and the “textbook market” to illustrate the random yield SC. Gurnani and Gerchak (2007) think that random yield can also be found in assembly systems, and related examples include the mission of the Challenger and VCR and car assembly. Yin and Ma (2015) investigate a leisure food chain store, LPPZ, in China and find that a high degree of SC risk, including supply and demand uncertainty, hinders the achievement of a high service level. Similar to this paper, Li et al. (2017) use examples from the clothing industry, such as Meters/bonwe, Liz Claiborne and Benetton, to explain yield uncertainty. The flu vaccine industry in the United States has been described many times to illustrate yield uncertainty (Chick et al., 2008; Deo and Corbett, 2009; Arifoğlu et al., 2012; Cho and Tang, 2013). The impacts of yield uncertainty on SC performance have received widespread attention. Many researchers have developed different SC models considering yield uncertainty. Masih-Tehrani et al. (2011) construct an m-manufacturer, 1-retailer newsvendor system under supply uncertainty. Okyay et al. (2014) analyze three different types of supply uncertainty: Random yield, random capacity and both. They
(1) How do the commitment order contracts influence the supplier's optimal input quantity decisions? (2) How do the distribution characteristics of the yield rate influence the SC members' optimal decisions under the commitment order contracts? (3) Is it possible to improve the performance of the SC by introducing the commitment order contracts? In this study, we model the above three commitment order contracts as Stackelberg games in which the retailer acts as the leader. The wholesale price of the regular order and the wholesale price of the commitment order are assumed to be exogenous. Through the model development, we highlight our key findings and derive the following insights. (1) Under the commitment order contracts, the retailer's commitment order quantity has a positive impact on the supplier's input quantity. However, our paper also shows that the supplier's input quantity may be independent on the retailer's commitment order quantity, or even decreases with the retailer's commitment order 399
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present the example in which random demand, capacity and yield are directly dependent. Xu et al. (2015) develop a SC with one buyer and two different suppliers. They consider a special case in which one supplier is the buyer's regular long distance supply source but still faces uncertain yield. Fang and Shou (2015) study the Cournot competition between two SCs subject to supply uncertainty. Baruah et al. (2016) propose an optimal order revision policy for a buyer facing typical demand and supply uncertainty. They demonstrate the value of soft orders as well as the supplier's inventory position information for the buyer. Jabbarzadeh et al. (2017) present a production-distribution planning model that is robust to common supply interruptions and demand variations. Du et al. (2018) consider a SC composed of a lossaverse supplier with yield randomness and a loss-averse retailer with demand uncertainty. They obtain the SC members' optimal decisions and examine how loss aversion and yield variance contribute to the SC performance. In this paper, we propose a SC model considering yield and demand uncertainty. The supplier's yield is uncertain and the actual output quantity is a random fraction of the input quantity. Such stochastic proportional yield models are widely used in many literature (Yano and Lee, 1995; Guo et al., 2018). Different contracts are proposed to coordinate the SC under yield uncertainty. Xu (2010) considers a SC in which the supplier's production yield is random and the manufacturer's demand is stochastic. Then, the option contract is proposed to improve the performance of the SC. Hsieh and Wu (2008) consider a special SC with one Original Equipment Manufacturer (OEM), one manufacturer, and one distributor under supply and demand uncertainty. They propose effective coordination mechanisms through information sharing between the members. Yin and Ma (2015) consider the retailer's requirement of a service level of the product supply from the manufacturer with random yield. The bonus contracts can achieve Pareto improvements. Hu and Feng (2017) model a SC with revenue sharing contracts and service requirement under supply and demand uncertainty. He and Zhao (2012) propose optimal contract-design schemes for a three-tier SC with both supply and demand uncertainty. In their contract-design schemes, wholesale price contracts and a returns policy are united to coordinate the SC. Giri and Bardhan (2017) also discuss the coordination of a three-echelon SC under uncertain demand and random yield in production. Peng et al. (2018) develop a SC under the carbon emission capand-trade scheme, the manufacturer faces yield uncertainty. A contract of revenue-sharing with subsidy on emission reduction is introduced to coordinate the SC. Similar works regarding SC coordination under yield uncertainty can also be found in Güler and Keski˙N (2013) and Giri et al. (2016).
amount of quantity of newly developed products. Both parties attain maximum profits under the concept of the synergy effect. Xu (2011) studies a stochastic inventory model with a multi-period where the buyer commits to purchase the product, with at least a certain percentage of the purchased capacity during the planning horizon. The buyer benefits from these commitments. Most of the above works focus on the commitments made by the buyer to the supplier. The supplier can also make commitments to the buyer (Ghijsen et al., 2010). Based on a tight partnership, the supplier and the buyer can make mutual commitments to improve the performance of the SC (Durango-Cohen and Yano, 2006; Chen et al., 2017). In addition to quantity commitment, price commitment is also valuable. Liu et al. (2014) find that if brand manufacturers provide a retail price markup commitment strategy, retailers are willing to follow this strategy. Commitment contracts can also be used in the SC under yield uncertainty. He and Zhang (2010) propose a minimum commitment contract, specifying a commitment quantity that the supplier must supply. If the actual output quantity is less than the commitment quantity, the supplier has two choices: Pay a penalty to the retailer or go to a secondary market for procurement. By contrast, our paper focuses on the commitment order contracts, under which the retailer first commits a minimum order quantity before the selling season, the supplier then determines the input quantity. After yield and demand uncertainty is observed, the retailer determines whether to place a second order. 2.3. The distinctiveness of this research The key contribution of this study lies in introducing three different types of commitment order contracts to improve the performance of the SC under yield and demand uncertainty. Yield rate is a random variable following uniform distribution defined in a general region. In the basic model, the SC operates under VMI, in which the supplier takes charge of inventory management. Commitment order contracts are introduced to improve the performance of the SC. Stackelberg game models are then constructed, and optimal decisions are obtained. Specifically, we discuss different boundary conditions needed to obtain members’ optimal decisions. Most related works can be found in Cho and Tang (2013) and Tang and Kouvelis (2014). Cho and Tang (2013) assume that the production capacity is a constant and the actual output is subject to random production yield. The authors introduce a linear demand curve and show how advance selling strategies impact the SC's competition structures. In their model, the manufacturer acts as the game leader by first determining the wholesale prices. The retailer must decide the pre-book quantity or regular-order quantity. Tang and Kouvelis (2014) use a modified pay-back-revenue-sharing contract to coordinate the SC under uncertain yield and demand. In their model, they assume that the random yield rate is distributed on [0,1]. Boundary constraints thus do not require discussion. Neither study introduces a commitment order contract to change the competition structure of the SC. Table 1 further helps position this paper and identify the contributions. In short, this research contributes to the literature in the following three ways.
2.2. Commitment contract Kumar et al. (1995) demonstrate that relationships with greater total interdependence exhibit higher trust, stronger commitment, and lower conflict than relationships with lower interdependence. Nyaga et al. (2010) think that credible commitments may develop when incomplete contracting exists because neither party can completely rely on contractual mechanisms to manage the relationship. Trust and commitment lead to improved satisfaction and performance of the SC. Zhao et al. (2007) develop an analytical model to quantify the cost savings of an early-order commitment in a two-level SC in which demand is correlated serially. Lian and Deshmukh (2009) study a SC with demand forecast updating with the buyer able to receive discounts for committing to purchase in advance. These contracts are used by automobile and contract manufacturers and are common in the fuel oil and natural gas delivery markets. Xie et al. (2010) introduce the early-order commitment (EOC), wherein the retailer commits to purchase a fixedorder quantity from a manufacturer. They find that the EOC-based wholesale price-discount scheme can reduce SC costs. Motivated by a case in the cosmetic industry, Li et al. (2016) study a special quantityflexibility contract arrangement in which the retailer commits an
(1) Most of the existing literature assumes that the yield rate is a random variable distributed in the region [0,1]. In our paper, the yield rate is a random variable following a uniform distribution defined in a general region. Constructing the model based on such an assumption enriches the extant literature on SC models considering yield uncertainty. (2) Different from most researches which focus on introducing complicated contracts into the Retailer-Managed Inventory (RMI) SC under yield uncertainty, we develop a VMI SC model and introduce the commitment order contracts. The commitment order contracts proposed in this paper only include one parameter and can be 400
International Journal of Production Economics 216 (2019) 398–412
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Table 1 Summary of some relevant literature. Literature
Stochastic proportional yield
Contracts
Key findings
Cai et al. (2017)
The yield rate is a random variable distributed in (0,1].
Option contract; subsidy contract
Tang and Kouvelis (2014) He and Zhang (2010)
The yield rate is a random variable distributed in [0,1]. The yield rate is a random variable with the mean value µ .
Pay-back-revenue-sharing contract
Yin and Ma (2015)
The yield rate is a random variable distributed in [0,1]. The yield rate is a random variable distributed in [0,1].
Bonus contracts
The option contact is effective in coordinating the SC, it can be easily applied in a VMI SC; the subsidy contract can coordinate the SC, but it is not feasible for most situations under a VMI SC. A modified pay-back-revenue-sharing contract can coordinate the SC under yield and demand uncertainty. Different parties of the SC prefer different level of commitment quantity, and the reduction of random yield risk is useful in improving the performance of the SC. The bonus contracts can achieve Pareto improvements.
Li et al. (2017) Chick et al. (2008)
Commitment contract (the supplier commits a minimum supply quantity)
Commitment-option contracts
Effects of demand uncertainty, production yield variability, instant procurement price fluctuation, and the price parameters of option contracts are discussed. A variant of the cost sharing contract can provide incentives to both SC members so that the SC achieves global optimization. Three strategies are compared from both SC members' perspectives, respectively.
The yield rate is a random variable with the mean value µ . The yield rate is a random variable distributed in [ , ¯], and the mean ¯ value is 1. The yield rate is a random variable distributed in [0,1]. The yield rate is a random variable distributed in [L , U ], 0 L < U 1. The yield rate is a random variable distributed in [0,1].
Cost sharing contract
Peng et al. (2018)
The yield rate is a random variable with the mean value 1.
Revenue-sharing with subsidy contract
Güler and Keski˙N (2013)
The yield rate is a random variable
This paper
The yield rate is a random variable distributed in [ , ], 0 .
Buy-back contract; revenue sharing contract; quantity discount contract; quantity flexibility contract Commitment order contracts (the retailer commits a minimum order quantity before the selling season)
Cho and Tang (2013)
Xu (2010) Hu and Feng (2017) He and Zhao (2012)
distributed in [ l,
h] ,
0<
l
<
h
< 1.
Advance selling strategy; regular selling strategy; dynamic selling strategy Option contract
Both SC members can be better off under the option contracts.
Revenue sharing contract
Conditions to coordinate the SC are obtained under the revenue sharing contract. In a three-level SC, a returns policy used by the manufacturer and the retailer, combined with the wholesale price contract used by the raw-material supplier and the manufacturer, can perfectly coordinate the SC. The proposed contract can coordinate the low-carbon SC with yield uncertainty perfectly and the carbon emission reduction level can achieve the level under the centralized case. The randomness in the yield does not change the coordination ability of the contracts but affects the values of the contract parameters.
Wholesale price contract combined with a returns policy
conveniently introduced into the VMI SC connected by a wholesale price contract. (3) Detailed boundary conditions are obtained for all possible optimal solutions of both members. Diversified sensitivity analyses are conducted in the paper to find out management insights of the commitment order contracts. Thus, the models can be easily accepted by the enterprises, and the results can support their operations management.
The SC members' optimal decisions are obtained given different conditions under yield and demand uncertainty. It is possible to introduce the commitment order contracts to improve the performance of the SC given specific conditions.
then is obtained by the retailer. The retailer places its final order Min {D , xK } . Anticipating the retailer's optimal decision, the supplier determines the optimal input quantity. The supplier's expected profit can be expressed as s
(K ) = w 0
3. Basic model
xK 0
Df (x ) g (D ) dDdx +
+ xK
xKf (x ) g (D) dDdx (1)
cK It is easy to find that
Consider a two-echelon SC with one supplier and one retailer. Let p denote the retail price and w0 denote the wholesale price. The supplier's unit production cost is c . We assume c < w0 < p . The supplier's yield is uncertain. If the supplier's input quantity is K , then its actual output quantity is xK . Here, x is a stochastic variable with a uniform distribution. Let f (x ) and F (x ) denote the density function and the cux mulative distribution function of x , respectively. Assume 0 . Here, and are both constants, F ( ) = 0 and F ( ) = 1. The mean + value of x is µ , i.e., µ = 2 . Demand D is also uncertain. Let g (D) and G (D) denote the density function and the cumulative distribution function of D . In the next, a basic game model is proposed to describe a two-echelon SC operating under VMI. And the decision process can be described as follows.
s
(K ) is concave in K . Let K denote the input
quantity determined by solving
µw0
c
w0
s (K )
K
xf (x ) G (xK ) dx = 0
= 0 . We then have (2)
Let K * denote the supplier's optimal input quantity, we then have the following theorem. Theorem 1. If µ w , then the supplier does not input any quantity, 0 c i.e., K * = 0 . If µ > w , then the supplier's optimal input quantity is 0 K* = K . With the support of demand information sharing from the retailer, the supplier obtains its optimal input quantity when no commitment order contract is introduced. According to Theorem 1, under yield and demand uncertainty, the supplier must relate the wholesale price and the mean value of yield rate together to judge whether to take part in c the operation of the SC or not. If and only if µ > w , the supplier is 0 willing to choose a positive input quantity. Thus, the supplier’s c
① Under yield and demand uncertainty, the supplier determines an optimal input quantity. Then, the products are produced. ② Both members find actual output quantity. Demand information
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expected profit is s (K *) , and the retailer’s expected profit is r (K *) . The total expected profit of the SC then is T (K *) = s (K *) + r (K *) . Such an operation mode equals a VMI mode. And it reflects the supplier’s supply support for the retailer. In the next, our work shows that commitment order contracts can be used to stimulate the input quantity of the supplier.
Lemma 1. Given µ , c and w0 , the supplier's expected profit changes with K as follows. (i) If µ
(ii) If µ
Lemma 1 can be interpreted as follows. Firstly, given a relatively low mean value of yield rate, the more quantity the supplier inputs, the more losses it bears. Then the supplier does not choose a positive input quantity whatever a large commitment order quantity is placed by the retailer. Secondly, given a relatively high mean value of yield rate, the supplier can maximize its expected profit with an optimal input quantity. In order to obtain explicit solutions to maximize the supplier’s expected profits, we ignore the special situation with = in the paper. We then have the following theorem.
4. The Retailer's commitment order contracts Assume that the retailer decides to offer commitment order contracts before the input quantity is determined by the supplier. According to the commitment order contracts, the retailer commits to purchase at least a quantity with the wholesale price w1. Then, the supplier determines an optimal input quantity and produces the products. After yield and demand uncertainty is observed, the retailer determines whether to place a second order with the wholesale price w0 by comparing the actual output quantity and the actual demand. In reality, two wholesale prices w1 and w0 are always determined through the negotiation between the members at the beginning of the trade. During the negotiation, characteristics of industry, market and product may all influence the relative size of w1 and w0 . In the next, we consider different situations according to the following three conditions: ① w1 = w0 ; ② w1 < w0 ; ③ w1 > w0 .
Theorem 2. If µ w , the supplier does not input any quantity, i.e., 0 K a * = 0 . If µ > wc and 1( ) wc , the supplier's optimal input quantity is c
0
, i.e., K xK
(K ) = w 0 + w 0 +
+
(xK
xK
(ii) When , i.e., K can be expressed as sa (K )
c w0
and
Here,
1(
)=
= w0 K xK
+ xK
(xK
(D
K
cK
K
(3)
, the supplier's expected profit
) f (x ) g (D ) dDdx +
xKf (x ) g (D) dDdx + K
sa (K ) = (µw0
K
, i.e., K
the supplier's optimal input quantity is K a * = K .
xf (x ) 1
( ))dx .
G x
0
c
c , ① w0 a*
in the region
[0,
1
. We then have
1) ,
the supplier's
c
+ 0
f (x ) g (D ) dDdx
cK
0
K a2 ( ) . ① Anticipating K a * = K , which is independent of , the retailer’s optimal commitment order quantity is 0. Then, the retailer’s expected profit can be expressed as
(4) (iii) When expressed as
xf (x ) G (xK a2 ( )) dx = 0 .
(i) When µ w , anticipating K a * = 0 , the retailer does not make a 0 commitment order. No member takes part in the operation of the SC. c (ii) When µ > w , the supplier’s optimal input quantity is K or
) f (x ) g (D ) dDdx )
+ 0
(
K a2 ( )
optimal input quantity K is independent of ; ② in the region [ 1, + ) , the supplier's optimal input quantity K a * increases with . Theorem 3 shows that if the retailer’s commitment order quantity is relatively small ( < 1), the supplier’s input quantity decision is not influenced by the retailer’s commitment order quantity. Then, the supplier makes the optimal decision according to the demand information shared by the retailer and the yield information, i.e., just like the VMI mode. If the retailer’s commitment order quantity is relatively large ( 1), then the more quantity the retailer commits, the more quantity the supplier inputs. This conclusion is very close to the reality, c since a high mean value of yield rate ( µ > w ) indicates a stable output, 0 and a large commitment order indicates a prosperous market. Anticipating the supplier’s optimal input quantities, the retailer determines its optimal commitment order quantities. The following situations should be considered.
K
+ w0
c , w0
Theorem 3. When µ >
) f (x ) g (D ) dDdx
) f (x ) g (D) dDdx
) >
w0
= w . Combining with Eq. (2), we conclude K = 0 the following theorem. 1 ( 1)
, the supplier's expected profit can be
(D
1(
c
It is easy to find that 1 ( ) decreases with . We have 1(0) = µ and c lim 1 ( ) = 0 . Thus, when µ > w , a unique 1 exists and satisfies
The above game can be analyzed by using backward induction. We first determine the supplier's optimal input quantities given the retailer's commitment order quantities. Because w1 = w0 , w1 is replaced by w0 in all equations in this section. By noting the actual quantity Min {xK , Max {D , }} ordered, we can express the supplier's expected profit as functions of K in different regions as follows.
sa
If µ >
+
K
0
K a * = K a2 ( ) , where K a2 ( ) satisfies µw0
4.1. w1 = w0
(i) When expressed as
c , the supplier's expected profit sa (K ) decreases with K . w0 c > w , the supplier's expected profit sa (K ) is concave in K . 0
, the supplier's expected profit can be
ra
(5)
c) K
xK
(0) = +
The supplier determines the optimal input quantity K that maximizes its profit function sa (K ) in the region K [0, + ) . Although the supplier's expected profit functions vary in different regions of K , the supplier's expected profit is differentiable and continuous in the whole region. The following lemma summarizes that how the supplier's expected profits change with K under different conditions. a*
(p
0
+ xK
w0 ) Df (x ) g (D ) dDdx (p
w0 ) xK f (x ) g (D ) dDdx
(6)
② Anticipating K a * = K a2 ( ) , the retailer’s expected profit can be expressed as
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International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
ra
+ xK a2 ( )
( )=
xK a2 ( )
+ +
K a2 ( )
(p
xK a2 ( ) 0
w0 ) Df (x ) g (D ) dDdx
w0 xK a2 ( )) f (x ) g (D ) dDdx
(pD
(7)
(p 1 ( K
a2
Here,
( 2
a1 i ))
c)
a1 i
is determined by
a1 i)
w0 G (
1
F
a1 i
K a2 (
a1 i)
0
Ta (K
a*
c ) K a*
) = (µp
xK a *
p
0
1(
2
a*
sb
, i.e., K xK
(K ) = w 1 + w 0 +
+
(xK
xK
f (x ) G (D ) dDdx .
, the supplier's expected profit can be
= w0
xK
+ + w1
K
+ xK
K K
+ 0
(xK
(D
(D
) f (x ) g (D) dDdx
cK
1(
(8)
w0
, the supplier's expected profit
K
sb
(K ) = (µw1
2
w1 . w0
2
()
2 (K )
=
1(
1(
( )>
w1 ; w0
=1
x2
2
) >
c , w0
K
2
( )>
w1 w0
>
2
()
2
1 K
sb (K )
and and
c w0
) 2
()
2
()
K
and
obtain
2
( ) g(x )dx .
K
K = K0
satisfies
K0
( )=1
sb (K )
(B)
w1 . w0
x 2K 3g (xK ) dx ,
3
0;
K = K0
and
G( )
is obtained from
2
K b20 ( )
F
,
K b20 ( )
c , if 1 w1
G( )
xK
1
f (x ) Dg (D) dDdx <
then the supplier's optimal input quantity is K
xK b20 ( )
( )>
w1 w0
>
2
( ),
f (x ) Dg (D ) dDdx > (w0 then
the
supplier's
sb (K )
K
K = K0
w0 G ( ) optimal
b*
w1 w0
= K ; if and
>0
w1 ) input
quantity
is
= ) ; otherwise, the supplier does not input any quantity, i.e., c c K b * = 0 . When µ > w , if 1( ) > w , then the supplier's optimal input b*
K b20 (
0
quantity is K b * = K ; otherwise, the supplier's optimal input quantity is b* = K b20 ( ) , where K b20 ( ) satisfies the following equation.
) f (x ) g (D ) dDdx ) + 0
>
1(
we
1
G( )
(C)
1
K
c , w0
)
w1 w0
G( )
And
c , w0
)
1
) f (x ) g (D ) dDdx
xKf (x ) g (D) dDdx +
, i.e.,
and
( )=1
and
) f (x ) g (D) dDdx
f (x ) g (D ) dDdx
cK
w0
(9) (iii) When K expressed as
c w0
)
c , w0
)
Theorem 5. When µ
(ii) When , i.e., K can be expressed as sb (K )
(B)
the first derivative of sb (K ) given in Eq. (9). c Lemma 2 shows that given w1 < w0 , when µ > w , an optimal input 1 quantity exists and maximizes the supplier’s expected profit. When c µ , it is very complicated to analyze the supplier’s optimal input w1 quantity decision. First, we can infer that the supplier’s expected profit is closely related to the retailer’s commitment order decision. Second, there exist two special cases. In Case ①, two conditions influence the supplier’s optimal input quantity decision. If one of the conditions is satisfied, there exists one local maximum in the region, and the supplier’s optimal input quantity can be obtained by comparing the boundary value and the local maximum. In Case ②, three conditions influence the supplier’s optimal input quantity decision. Any one of the conditions is satisfied, the supplier’s expected profit decreases with K . Then the supplier cannot benefit from any input quantity. According to Lemma 2, we have the following theorem.
c
Similar to Section 4.1, the game can be analyzed by using backward induction. We first determine the supplier's optimal input quantities given the retailer's commitment order quantities. Then, the supplier's expected profit can be expressed as functions of K in different regions as follows. K
1(
Here
4.2. w1 < w0
(i) When expressed as
c ; w0
> 0.
K = K0
2 (K 0 )
c
1(
(A)
denote the retailer’s optimal commitment order quantity. When µ w , 0 a* =0 . When µ > wc , a * {0, a } . Then, the total expected profit of the SC is
) >
is concave in K .
② If one of the following conditions is satisfied, then the supplier's expected profit decreases with K .
=0
K a2 ( 2)) = p . Given w1 = w0 , we let
1(
(A) K
Theorem 4. Anticipating K a * = K a2 ( ) , the retailer can find at least one optimal commitment order quantity in the set a { 1} Z , where = { a1i } , (i = 1,2, ..., n) , and a1i is determined by the following [ 1, 2 ) . equation in the region a1 i)
c , then the supplier's expected profit w1 c , then there are two scenarios: w1
sb (K )
The retailer determines the commitment order quantity a that maximizes its expected profit given in Eq. (7). And according to the [ 1, + ) . Then, the proof of Theorem 3, a must fall into the region following theorem summarizes the results.
K a2 (
(i) If µ >
(ii) If µ ① If one of the following conditions is satisfied, then the supplier's expected profit first decreases then increases, and finally decreases with K .
w0 ) f (x ) g (D ) dDdx
(pD
0
K a2 ( ) K a2 ( )
+
w0 ) xK a2 ( ) f (x ) g (D) dDdx
(p
, the supplier's expected profit can be
= w1
(10)
c) K
K b20 ( )
K b20 ( )
F
F (x ) dx
K b20 ( )
F (x ) dx
K b20 ( )
K b20 ( )
F
K b20 ( )
K b20 ( )
xf (x ) G (xK b20 ( )) dx
+c
Theorem 5 shows that, given w1 < w0 , even the mean value of yield rate c is relatively low, i.e., µ , the supplier is still willing to choose a positive w
The supplier determines the optimal input quantity K that maximizes its profit function sb (K ) in the region K [0, + ) . Similarly, the supplier's expected profit is differentiable and continuous in the whole region. The following lemma summarizes that how the supplier's expected profits change with K under different conditions. b*
1
input quantity if the retailer commits a proper quantity. When µ > w , if the 1 retailer’s commitment order quantity is relatively small, the supplier’s optimal input quantity equals K , just like the VMI mode; if the retailer’s commitment order quantity is large enough, the supplier may choose its optimal input quantity with the commitment order quantity considered. According to Theorem 5, we first let
Lemma 2. Given µ , c , w0 and w1, the supplier's expected profit changes with K as follows. 403
c
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al. 3(
)=1
derivative of given by
) is
3(
1 ( 1)
the region
c w0
=
[0,
only if µ >
xK
1
G( )
3( )
and
=
1(
and
1)
K b * increases with ; if 1 4 , then the supplier's optimal input [ 1, 4], and increases quantity K b * decreases with in the region [ 4, + ). with in the region Theorem 6 is analogous to Theorem 3, given a relatively high mean c value of yield rate ( µ > w ), if the retailer’s commitment order quantity 1 is relatively small ( 1), the supplier makes its input quantity decision according to the VMI mode. If the retailer’s commitment order quantity is large enough, then the supplier’s input quantity decision is influenced by the retailer’s commitment order quantity. Different from the situation with w1 = w0 , given w1 < w0 , the supplier adds a threshold commitment order quantity ( 4 ) to judge the relationship between the retailer’s commitment order quantity and the actual market demand. Then,
f (x ) Dg (D ) dDdx . Then, the first order xK
1
f (x ) Dg (D ) dDdx . Because
2
) decreases with , we conclude 1(
in the region
c w0
)
c . w0
Additionally, we always have
w1 , w0
then a unique
1(
1 c )> w 0
1(
is
in
[ 1, + ) if and when µ
c w0
)
c , w0
under which the supplier’s optimal input quantity is K = 0 . Re( ) [0, 1) membering that K = 1 , we conclude 3 > 0 in the region c c when w < µ w . The following two situations should be discussed: ① If
lim 3(
1
3(
0
) <
) >
1
exists and satisfies
3
holds in the region
w1 w0 w1 w0
) < holds in the region corollary. 3(
[0,
Because
2
( )=1
let
4(
4)
} {
0<
4
[0,
c w0
) >
2
()
a
w1 . w0
w1 w0
c w0
c , w1
<µ
1(
1
unique
exists
4
in the region
f (x ) Dg (D ) dDdx
and
c w0
)
w1 , and w0 w1 , then w0
We then have the following
}.
K b20 ( )
Given
② if lim 3 ( )
Then, we conclude
w1 w0
xK b20 ( )
) = w0
) <
< 1,
w1 w0
G ( 4) = and
3(
=
① If the retailer’s actual commitment order quantity is larger than Max { 1, 4 } , then the supplier ensures that the market demand is really large. Thus, the more quantity the retailer commits, the more quantity the supplier inputs. Max { 1, 4 } , the supplier is very cautious in ② If Min { 1, 4} determining its input quantity and two scenarios may be considered. In the first scenario, i.e., 1 > 4 , the supplier’s optimal input [ 4 , 1]. In the second quantity is independent of in the region scenario, i.e., 1 4 , the supplier’s optimal input quantity de[ 1, 4]. It is a little counter-intuitive creases with in the region that the supplier’s optimal input quantity decreases with the commitment order quantity. In fact, it is reasonable since the supplier adds a threshold commitment order quantity ( 4 ). In the region [ 1, 4], the supplier may think that the retailer only pursues a low wholesale price through the commitment order quantity, not really for a forecasting of an optimistic market demand. Thus, in the [ 1, 4], the more quantity the retailer commits, the more region pessimistic market anticipation the supplier supposes, and the less quantity the supplier inputs. If and only if the retailer’s commitment order quantity is larger than 4 , the supplier may really ensure an optimistic market demand.
there exists at least one positive quantity in the set
<µ Corollary 1. When commitment order 1(
1) .
3 ( 3)
c , w1
c w0
{
3) ;
[0,
b*
2
2
( )>
and w1 w0
satisfies
in the region
[ 4 , + ) . At the same time, (w 0
w0 G ( )
( )>
w1 w0
>
2
w1 )
1
F
K b20 ( )
.
( ), there must be
> 1, and falls into the region [ 1, 4 ) . Under specific settings of the parameters { , , c, w1, w0, p} , we can find at least one commitment = { i } [ 1, 4 ) , , where order quantity in the set (i = 1,2, ..., n) , and i satisfies 4 ( i ) > 0 . Thus, the supplier chooses a positive input quantity given the retailer’s commitment order quantity in the set . Because the supplier’s expected profit function is piece-wise continuous and differentiable, we summarize the supplier’s optimal input quantity decisions given any closed interval of commitment order quantity. We then have the following lemma. 4
Anticipating the supplier’s optimal input quantities, the retailer determines its optimal commitment order quantities. The following situations should be considered.
Lemma 3. Given the retailer's commitment order quantities, we obtain the supplier's optimal input quantities as follows. ① When
c w0
c , w1
<µ
(A) if
3 ( 1)
>
w1 , w0
then in the region
(i) When µ
3], w1 , w0
[0,
( )<
sb (K )
w1 , w0
K
K = K0
> 0 and
4(
) > 0 ; (D) otherwise, the sup-
plier's optimal input quantity is K = 0 . c ② When µ w , the supplier's optimal input quantity is K b * = 0 . ③ When µ >
0
c , w1
b*
(A) in the region
[0,
1],
[ 1, + ) , the sup-
Because
1 ( 1)
satisfies
4
3 ( 3)
satisfies 1
=
w1 w0
G ( 4) =
if and only if
w1 , w0
3 ( 1)
1
>
satisfies
w1 , w0
=
c , w0
it is easy to find
But it is hard to determine the relative sizes of
4
and
1
c , ① w1 b*
in the region
[0,
1],
and 3
given µ >
By analyzing the supplier’s optimal input quantity given µ > then have the following theorem. Theorem 6. When µ >
rb
the supplier's optimal
input quantity is K b * = K ; (B) in the region plier's optimal input quantity is b * = K b20 ( ) .
c , w1
the supplier’s optimal input quantity K b * is in the
set {0, K , K b20 ( )} . Let b1 *denote the retailer’s optimal commitment order quantity that maximizes its expected profit, then, ① Anticipating K b * = 0 , the retailer does not make a commitment order. No member takes part in the operation of the SC, i.e., b1 * = 0. ② Anticipating K b * = K , the retailer’s expected profit can be expressed as
the supplier's optimal input quantity is K b * = K ; (B) if 3 ( 1) [0, 1], the supplier's optimal input quantity is then in the region K b * = K ; (C) if there exists a region [ 1, 4], then the supplier's b* = K b20 ( ) optimal input quantity is given 2
c , w1
( )=
(pD
0
xK
+
+ xK
w1 ) f (x ) g (D ) dDdx +
((p
w0) D + (w0
((p
w1) xK + (w0
w1) ) f (x ) g (D) dDdx w1) ) f (x ) g (D) dDdx
(11)
It is easy to find that rb ( ) is concave in , and Eq. (11) is maximized by 4 . According to Lemma 3, b1 * must fall into the region [0, Min { 1, 3}] to induce the supplier to choose a positive input w w quantity. Remind that 3 satisfies 3 ( 3) = w1 if and only if 3 ( 1) > w1 , 0 0 and there is 3 4 if 3 exists. Hence, one and only one optimal b1 * can be found by the retailer in the set commitment order quantity { 1, 3 , 4} .
3
4. c . w1
we
the supplier's
optimal input quantity K is independent of ; ② in the region [ 1, + ) , if 1 > 4 , then the supplier's optimal input quantity
③ Anticipating pressed as
404
b*
= K b20 ( ) , the retailer’s expected profit can be ex-
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
rb
xK b20 ( )
K b20 ( )
( )=
0
K b20 ( )
+
K b20 ( )
+
K b20 ( )
+
K b20 ( )
w1 ) xK b20 ( ) f (x ) g (D ) dDdx
+ xK b20 ( )
induction. We first determine the supplier's optimal input quantities given the retailer's commitment order quantities. Then, the supplier's expected profit can be expressed as sc ( ) = sb ( ) in the regions
K
w1 ) f (x ) g (D ) dDdx ((p
w 0 ) D + (w 0
w1 ) ) f (x ) g (D ) dDdx
(i) If µ > (ii) If µ
(12) Combining with Eq. (A.3), the first order derivative of respect to is rb
( )
= (p 1 ( K b20 ( ))
c)
K b20 ( )
+ (w 0
w0 G ( )
w1 ) 1
F
rb
( ) with
)) = Here, 1 ( falls into the region K b20 ( )
G (xK b20 (
xf (x )(1 [ 1, 4], then w1
K b20 ( )
))) dx . Because b1 * w0 + w0 G ( ) 0 and
0 . Recalling
1(
K b20 ( 1)) =
c , w0
(p 1 ( K b20 (
b1 i ))
c)
(ii) When µ >
K b20 (
b1 i)
b1 i
c , w1 b2 *
+ (w0
w0 G (
b1 i)
then in the region
w1 ) 1
F
b1 i b2 K 0 ( b1i )
Theorem 9. When µ >
=0
µ
① Anticipating K b * = K , the retailer’s expected profit is given in Eq. (11). According to Lemma 3, b2 * must fall into the region [0, 1], one and only one optimal commitment order quantity can be found by the retailer in the set { 1, 4} . ② Anticipating b * = b20 ( ) , the retailer’s expected profit is given in Eq. (12). [ 1, + ) , According to Lemma 3, b2 * must fall into the region the following theorem then summarizes the results.
[0,
1],
the supplier's
(
c c , w1 w 0
, the commitment order contract is always efficient to
c
(i) When µ , anticipating K c * = 0 , the retailer does not make a w1 commitment order. No member takes part in the operation of the SC. c (ii) When µ > w , the supplier’s optimal input quantity is K or
b2 0(
1
f (x ) G (D ) dDdx .
K c2 ( ). ① Anticipating K c * = K , which is independent of , the retailer’s optimal commitment order quantity is 0. Then, the retailer’s expected profit is rc (0) = ra (0) . ② Anticipating K c * = K c2 ( ) , the retailer’s expected profit is rc ( ) = rb ( ) , which is given in Eq. (12). The retailer determines the optimal commitment order quantity c that maximizes its expected profit. And according to the proof of Theorem 9, c must fall [Max {0, 1}, + ) . Then, the following theorem into the region summarizes the results.
Similar to Section 4.2, the game can be analyzed by using backward
Theorem 10. Anticipating K c * = K c2 ( ) , the retailer can find at least {0, 1} one optimal commitment order quantity in the set c , where = { c1i } ,(i = 1,2, ..., n) , and c1i is determined by the following [Max {0, 1}, 5]. equation in the region
(p 1 ( K b20 (
Here,
µ
in the region
stimulate the supplier to take part in the operation of the SC, and the supplier’s input quantity increases with the retailer’s commitment order quantity. Anticipating the supplier’s optimal input quantities, the retailer determines its optimal commitment order quantities. The following situations should be considered.
= ) , the retailer can find at least one Theorem 7. Anticipating optimal commitment order quantity in the set b2 * { 1} , where = { b2i } ,(i = 1,2, ..., n) , and b2i is determined by the following [ 1, 5]. equation in the region
b*
c , ① w0 c*
optimal input quantity K is independent of ; ② in the region [ 1, + ) , the supplier's optimal input quantity K c * increases with c c . When w < µ w , the supplier's optimal input quantity K c * always 1 0 increases with . Theorem 9 is analogous to Theorem 3, but there are still some difc c ferences. Given a relatively low mean value of yield rate ( w < µ w ), 1 0 the supplier still inputs a positive quantity. Hence, in the region
the supplier’s optimal input quantity is in the set
b*
c , then the supplier's expected profit is concave in K . w1 c , then the supplier's expected profit decreases with K . w1
theorem.
{K , )} . Let denote the retailer’s optimal commitment order quantity that maximizes its expected profit. Then, K b20 (
) respectively.
c , then the supplier does not input any quantity, w1 c c and 1( ) > w , then the supplier's optimal input w1 0 c c quantity is K c * = K . If µ > w and 1( ) w , then the supplier's 0 1 c* c2 c2 ( ) = b20 ( ) . optimal input quantity is K = K ( ) ; here, c Recall that 1 exists if and only if µ > w , we then have the following 0
[ 1, 4], 1( K b20 ( 1)) > always holds. Hence, similar to the conclusion in Theorem 4, we summarize as follows: Anticipating b* = K b20 ( ) , the retailer can find at least one optimal commitment order quantity in the set b1 * { 1} , where = { b1i },(i = 1,2, ..., n) , and b1i is determined by the following equation in the region [ 1, 4]. c p
,+
Theorem 8. If µ i.e., K c * = 0 . If µ >
< 0 (from the proof of Theorem 6), it is easy to find
b2 1 ( K 0 ( ))
and K
,
Lemma 4 is analogous to Lemma 1, given a relatively high mean c value of yield rate ( µ > w ), the supplier can maximize its expected 1 profit with an optimal input quantity; given a relatively low mean value of yield rate, the supplier does not input any quantity. But different from the situation with w1 = w0 , given a higher w1, the supplier is more likely to participate in the operation of the SC even when the mean value of yield rate is relatively low. We then have the following theorem.
(13)
K b20 (
,K
Lemma 4. Given µ , c , and w1, the supplier's expected profit changes with K as follows.
w1 ) ) f (x ) g (D) dDdx
w0 ) xK b20 ( ) + (w0
((p
0,
The supplier determines the optimal input quantity K c * that maximizes its profit function sc (K ) in the region K [0, + ) . Then, the following lemma summarizes that how the supplier's expected profits change with K under different conditions.
w1 ) xK b20 ( ) f (x ) g (D) dDdx
(pD
xK b20 ( )
+
(p
(p
0
K b20 ( )
w1 xK b20 ( )) f (x ) g (D ) dDdx
xK b20 ( )
+
+
(pD
b2 i ))
5
c)
K b20 (
b2 i)
b2 i
+ (w 0
is determined by
1(
w0 G (
b2 i)
b2 0 ( 5))
w1 ) 1
F
b2 i b2 K 0 ( b2i )
=0
c
= p . Given w1 < w0 , we let
denote the retailer’s optimal commitment order quantity. When c c , b* = b1 *. When µ > w , b* = b2 *. Then, the total expected w 1
profit of the SC is
1
Tb (K
b*
) = (µp
c ) K b*
p
xK b *
0
4.3. w1 > w0
405
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
(p 1 ( K c2 (
c1 i ))
c)
K c2 ( c1 i
c1 i)
+ (w 0
w0 G (
c1 i)
w1 ) 1
F
c1 i
K c2 (
c1 i)
Table 3. Compared with the basic model, by introducing the commitment order contract with w1 = w0 , the SC members may obtain a Pareto improvement. Compared with the commitment order contract with w1 = w0 , the retailer can benefit from the commitment order contract with w1 < w0 . For the supplier, a lower wholesale price w1 of the commitment order may result in a lower expected profit. For example, given = 0.5 and = 0.8, the supplier's expected profit is 141.24 under the basic model, 175.56 under the commitment order contract with w1 = w0 , and 131.12 under the commitment order contract with w1 < w0 . However, under specific situations, a lower w1 can encourage the retailer to make a commitment, thus the supplier can benefit more. For example, given = 0.6 and = 1.0 , the supplier's expected profit is 495.90 under the basic model, 495.90 under the commitment order contract with w1 = w0 , and 501.19 under the commitment order contract with w1 < w0 . After introducing the commitment order contract with w1 < w0 , the total expected profit of the SC can also be improved. For example, given = 0.5 and = 0.9 , the total SC's expected profit is 1551.51 under the basic model, 1626.49 under the commitment order contract with w1 = w0 , and 1630.21 under the commitment order contract with w1 < w0 . We conclude that the SC may perform better under the commitment order contact with w1 < w0 . As to the commitment contract with w1 > w0 , it is a good way to stimulate the supplier to take part in the operation of the SC even the mean value of yield rate is very low. Take the chip industry as an example, because of a long R&D cycle and a low yield rate, few companies are willing to take part in the chip production in China. But the chip is an essential part of many electronic products. Then, the commitment order contract with a higher wholesale price w1 is an effective way to stimulate the supplier's input in the chip production. Our numerical analysis shows that given = 0.5 and = 0.7 , the supplier is still willing to participate in the operation of the SC. Thus, both the supplier and the retailer can obtain profits under a relatively low mean value of yield rate.
=0
Given w1 > w0 , we let c * denote the retailer’s optimal commitment c c , c *=0 . When µ > w , c * {0, c } . Then, order quantity. When µ w1 1 the total expected profit of the SC is c* xK c ) K c* p f (x ) G (D ) dDdx . Tc = (µp 0
5. Numerical analysis In this section, to gain further insights, we present a numerical study based on the following setting. We assume that demand D is normally distributed with mean = 50 and variance 2 = 52 , i.e., D~N (50, 52) . Let c = 30 , w0 = 50 , p = 80 and = 0.5, then the yield rate x is uniformly distributed in the region [0.5, ]. Specifically, in the situation with w1 < w0 , we let w1 = 49; in the situation with w1 > w0 , we let w1 = 51. Fig. 1 shows that given = 0.8, the supplier's expected profit is concave in K under the commitment order contracts when = 10 or = 60 . Then the results in Lemmas 1(ii), 2(i) and 4(i) are confirmed respectively. Special cases are shown in Fig. 1(c), given w1 < w0 and = 0.72 , the supplier's expected profit first decreases then increases, and finally decreases with K when = 10 or = 29, and the supplier's expected profit decreases with K when = 60 , as the conclusions illustrated in Lemma 2(ii). Furthermore, Fig. 2 shows the impact of the retailer's commitment order quantity on the supplier's input quantity decision, as the results illustrated in Theorem 6 and Theorem 9. Anticipating the supplier's decisions under different conditions, the retailer chooses the optimal commitment order quantities to maximize its expected profits, as shown in Table 2. In the next, we discuss the performance of three commitment order contracts by comparison. Table 2 shows the SC members’ optimal decisions given different and in four models (i.e., the basic model, and the commitment order contracts with w1 = w0 , w1 < w0 , w1 > w0 , respectively). The corresponding optimal expected profits are shown in
Fig. 1. The supplier's expected profits change with K under commitment order contracts.
406
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
Fig. 2. The supplier's optimal input quantities change with under commitment order contracts.
Table 2 The SC members’ optimal decisions given different The retailer's optimal commitment order quantity
= 0.5 Basic model
Commitment order contract with w1 = w 0 Commitment order contract with w1 < w 0 Commitment order contract with w1 > w 0
= = = = = = = = = = = = = = = =
0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0
– – – – – 55.42 54.88 53.79 – 56.14 55.74 54.83 54.56 54.70 – –
and
The supplier's optimal input quantity
= 0.6 – – – – 50.96 51.82 51.84 – 51.40 52.47 52.47 51.59 50.51 – – –
inventory management and must determines the optimal input quantity under yield and demand uncertainty. A real case involving Zhejiang Nali is used to illustrate the management problem in this paper. We find that the SC members are eager to introduce a simple contract to improve the performance of the SC facing yield and demand uncertainty. Specifically, they want to keep the regular wholesale price stable during the trading period. In this study, we introduce commitment order contracts to construct a cooperative relationship between the members. Under the commitment order contracts, the retailer promises to purchase at least a quantity with a special wholesale price. Three possible situations are investigated: ① The wholesale price of the commitment order equals the wholesale price of the regular order; ② the wholesale price of the commitment order is lower than the wholesale price of the regular order; ③ the wholesale price of the commitment order is higher than the wholesale price of the regular order. Under these contracts, Stackelberg game models between the supplier and the retailer are constructed, in which the retailer determines the commitment order quantity and the supplier determines the input quantity. By using backward induction, we obtain the supplier's optimal input quantities and the retailer's optimal commitment order quantities under different conditions. Sensitivity analysis for the retailer's commitment order quantity on the supplier's input quantity is also investigated. The result shows that, the retailer's commitment order quantity decision has a great influence on the supplier's input quantity decision. Specially, the supplier's input quantity may decrease with the retailer's commitment order quantity under specific conditions, thus a proper commitment order quantity should be chosen by the retailer to
in four models.
= 0.5 – 60.16 57.01 53.68 – 71.18 64.63 58.98 – 71.59 65.07 58.98 78.40 70.76 57.01 53.68
= 0.6 65.07 61.57 57.69 54.11 74.05 67.76 62.21 54.11 74.34 68.19 62.42 57.26 73.74 61.57 57.69 54.11
6. Discussion and conclusion In this paper, we consider a SC operating under yield and demand uncertainty. In the traditional mode, the supplier takes charge of Table 3 The SC members’ optimal expected profits given different
and
in four models.
The retailer's expected profit
Basic model
Commitment order contract with w1 = w 0
Commitment order contract with w1 < w0
Commitment order contract with w1 > w0
= = = = = = = = = = = = = = = =
0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0
= 0.5
= 0.6
– 1167.61 1184.88 1181.74 – 1254.66 1225.15 1203.75 – 1300.00 1268.23 1244.60 1245.06 1209.90 1184.88 1181.74
1263.60 1281.00 1278.74 1271.59 1336.35 1312.70 1288.95 1271.59 1383.90 1358.43 1322.50 1311.03 1289.19 1281.00 1278.74 1271.59
407
The supplier's expected profit
= 0.5
= 0.6
– 141.24 264.62 369.77 – 171.56 313.95 422.74 – 131.12 273.00 385.61 46.90 219.14 264.62 369.77
153.80 288.00 400.62 495.90 183.57 331.45 449.61 495.90 136.69 287.53 407.63 501.19 229.81 288.00 400.62 495.90
The SC's expected profit
= 0.5
= 0.6
– 1308.85 1449.50 1551.51 – 1430.22 153,910 1626.49 – 1431.12 1541.23 1630.21 1291.96 1429.02 1449.50 1551.51
1417.40 1569.00 1679.36 1767.49 1519.92 1644.15 1738.56 1767.49 1520.59 1645.96 1740.13 1812.22 1519.00 1569.00 1679.36 1767.49
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
stimulate the input quantity of the supplier. Our numerical analysis demonstrates that, with special sets of wholesale prices, the commitment order contracts can be used to improve the performance of the SC facing different random yield environments. Specially, when the supplier's yield rate is very low, the commitment order contract with w1 > w0 performs better, it can stimulate the supplier to participate in the operation of the SC. When the supplier's yield rate is relatively high, the commitment order contract with w1 < w0 performs better, it can encourage the retailer to make a commitment. When the yield rate is mid-high, the commitment order contract with w1 = w0 performs better. Hence, enterprises need to choose the appropriate commitment order contract according to the actual situation, thus they can effectively reduce the impacts from yield uncertainty and improve profits. The main contributions of the paper can be summarized as follows.
construct a tighter relationship between the retailer and the supplier based on commitment order contracts. The proposed contracts can be easily applied by the members through simple contract parameters. Based on this paper, given different sets of wholesale prices, the retailer and the supplier can rapidly determine their optimal commitment order decisions and input quantity decisions under yield and demand uncertainty. It is difficult to eliminate all possible stock shortages with yield and demand uncertainty. Most studies thus introduce replenishment tactics to improve the service level. Replenishment tactics can effectively improve the supply capability of the supplier. This paper does not consider replenishment tactics because of the following reasons: ① In most cases, replenishment tactics are difficult to implement during a selling season because of cost, time and other factors. ② Based on the model, traditional replenishment tactics actually can be absorbed, and corresponding optimal solutions can be conveniently obtained. ③ In future work, we want to introduce replenishment tactics that consider changes in cost, time and demand. There are several future directions for this research. Competition between multiple suppliers or retailers can be introduced into the model. Information among the SC members is asymmetric in most situations. When considering information asymmetry, the competition structures become more complex. Furthermore, a model with several trade periods would be very interesting.
(i) The yield rate is a random variable following uniform distribution defined in a general region. This definition very closely models practical problems related to yield uncertainty. (ii) The commitment order contracts proposed in this paper can be conveniently introduced into the VMI SC connected by a wholesale price contract. (iii) All optimal solutions are obtained for both members under different conditions. Hence, the models can be easily accepted by the enterprises to support their operations management. The retailer wants to improve the performance of procurement without incurring more inventory costs. The operation mode of VMI SC therefore is popular for releasing the retailer from the pressure of inventory management. However, it is difficult for the supplier to maintain a stable or sufficient quantity for the retailer under yield and demand uncertainty. In this paper, we propose an effective way to
Acknowledgments This research is supported by National Natural Science Foundation of China (71572184, 71874046, 71601169, 71702167, 71701184), Natural Science Foundation of Zhejiang Province (LY19G020011), Xinmiao Talent Project of Zhejiang Province (2019R403075).
Appendix Proof of Theorem 1. It is straightforward and details are omitted. □ Proof of Lemma 1. We consider the supplier's expected profits in different regions. (i) In the region K 1(
(
)=
1(
)
then
sa (K )
G x
sa (K )
the first order derivative of then µ
1(
decreases with K ; ② if
c and
sa (K )
K
K=
2
) . Because 1(
) >
, the first order derivative of
,
= µw0
K=
),
( ))dx ,
xf (x ) 1 c , w0
(ii) In the region K K
,+
c , w0
sa (K ) K2
then
sa (K )
sa (K )
< 0,
sa (K )
sa (K )
K
= w0 1 ( )
K=
K
= w0
c and
xf (x )(1
sa (K )
K
K
+
G (xK )) dx
c . Let
c < 0 , we have ① if
=
is concave in K .
given in Eq. (4) is
c , we have ① if µ
= w0 1 ( )
sa (K )
given in Eq. (3) is
c , w0
then
sa (K )
K sa (K )
= µw0
c
w0
K
xf (x ) G (xK ) dx . Because
decreases with K ; ② if µ >
c , w0
2
sa (K ) K2
< 0,
then we consider the
following two scenarios. (A) If
1(
c , w0
)
sa (K )
(iii) In the region K
is concave in K ; (B) if
1(
) >
c , w0
sa (K )
, according to Eq. (5), we have ① if µ
0,
increases with K . c , w0
then
sa (K )
decreases with K ; ② if µ >
c , w0
then
sa (K )
increases with K .
The supplier's expected profit is a piece-wise continuous and differentiable function of K . From the above discussions, we consider the following two situations. (i) If µ
(ii) If µ >
c , w0 c , w0
then
(A) When
1(
)
(B) When
1(
) >
sa (K )
decreases with K .
then we consider the following two scenarios. c , w0 c , w0
sa (K )
is concave in K and maximized in the region K
,
sa (K )
is concave in K and maximized in the region K
,+
Proof of Theorem 2. From the proof of Lemma 1, it is easy to find that. (i) If µ
c , w0
then the supplier's optimal input quantity is K a * = 0 . 408
.
). □
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
(ii) If µ >
and
c w0
µw0
c
1(
c , w0
)
by solving the first order derivative of
given in Eq. (4), a unique K a2 ( ) exists and satisfies
sa (K )
xf (x ) G (xK a2 ( )) dx = 0
w0
(A.1)
K a2 ( )
Then, K a * = K a2 ( ) . (iii) If µ > □
c w0
and
1(
c , w0
) >
by solving the first order derivative of
Proof of Theorem 3. When µ >
[0,
and
1)
1(
a unique
in the region
c w0
)
c , w0
independent of . In the region
exists and satisfies
1
given in Eq. (3), we obtain the supplier's optimal input quantity K a * = K .
sa (K )
1 ( 1)
c . w0
=
Because
1(
) decreases with , we have
[ 1, + ) . From Theorem 2, we conclude that in the region
[ 1, + ) , K
a*
K a2 ( )
= K ( ) . From Eq. (A.1), we find a2
=
1) ,
[0,
K
1(
) >
in the region
= K . Obviously, K is
a*
K a2 ( ) G ( ) 2G ( ) + (K a2 ( ))3
c w0
>0.□
x 2g (xK a2 ( )) dx K a2 ( )
Proof ra ( )
of
= (p 1 ( K
Here, Because 1(
K
Theorem
a2
a2
4.
( ))
c)
K a2 ( )) =
1(
K a2 ( )
> 0,
c . w0
( 1)) =
Combining
K a2 ( )
with
w0 G ( ) 1
F
Eq.
K a2 ( )) decreases with
Meanwhile, we have lim
1(
+
and
< 0 , we conclude
a
2;
=
first
order
in the region
K
a2
(B) when
must fall into the region
2),
[ 1,
ra ( )
lim
of
ra (
2
a2 1 ( K ( ))
K a2 ( )) is 1 ( 1)
=
c w0
exists and satisfies
a1 i ))
K a2 (
c)
a1 i
a1 i)
w0 G (
a1 i)
1
F
1(
K
a1 i
(
Eq.
(7)
is
x 2f (x ) g (xK a2 ( )) dx . c
K a2 ( 2)) = p . Obviously,
[ 1, + ) . Then, (A) when
2
c
0
is positive or negative, two
ra ( )
= 1
a1 i },
={
1.
>
[ 2 , + ) , p 1 ( K a2 ( ))
a
a2
in
K a2 ( )
=
< 0 , then = 1 ; (b) if there exists at least an extremum, let possible situations should be discussed: (a) If [ 1, 2 ) . solution set of determined by the following equation in the region
(p 1 ( K a2 (
given
)
and Eq. (A.1) together, we conclude
< 0 . Since we cannot determine whether
2
ra ( )
1(
[ 1, + ) . Considering
( )) = 0 . Given p > w0 , a unique a
derivative
.
K a2 ( )
The retailer's optimal commitment order quantity ra ( )
the
G (xK a2 ( ))) dx . The first order derivative of
xf (x )(1
1(
(A.1),
(i = 1,2, ..., n) denote the
=0
a1 i)
Considering (A) and (B) together, we conclude that at least one
a
can be found in the region
[ 1, + ) ; here
a
.□
{ 1}
Proof of Lemma 2. We consider the supplier's expected profits in different regions. (i) In the region K
,+
(ii) In the region K
,
sb (K )
K
= w0
K
F
w1 We 2
sb (K ) K2
2
K3
Here, 2
f
, the first order derivative of
()
F (x ) dx
K
sb (K )
K
( )1 K
let
2 (K )
( )=1
G( )
① When
2
( )>
K=
F
()
= µw0
G( )
>
2
sb (K )
c , w1
2
and
c
K w1 w0
x 2K 3g (xK ) dx
K
1 2
K
② When (A) If µ > (B) If µ
③ When
2
then
w1 , w0
1(
2
( ),
sb (K ) K2
) >
K=
()
w1 , w0
c.
= w0 1 ( )
c , w0
then
sb (K )
c w0
sb (K ) K2
is concave in K .
The
second
order
derivative
sb (K )
K
1(
)
x 2K 3g (xK ) dx ,
and
decreases
2 (K )
with
K.
is concave in K . There exists a unique K 0 satisfying 2 (K 0 ) =
0 and and
and
c w0
sb (K )
K
sb (K )
sb (K )
K
1(
K = K0
0,
0 and
c , w0
) 1(
)
sb (K )
K
sb (K )
is
We
obtain
2
sb (K )
c w0
and
w1 w0
and maximizing
( )=1 sb (K )
K
first increases then decreases with K ; (b) when sb (K )
K
K = K0
> 0,
decreases with K ; (c) when
sb (K ) 1(
G( )
and
. We consider the 1(
) >
c , w0
sb (K )
first decreases then increases, and finally
) >
c , w0
sb (K )
first decreases then increases
increases with K . We consider the following scenarios.
increases with K .
sb (K )
of
.
then we have the following conclusions. (a) When 2
) >
given in Eq. (9) is
then we have the following conclusions. (a) When
()
2 c , w1 c , w1
1(
( ) g(x )dx . Then, 3
decreases with K ; (b) when with K .
decreases with K ; ② if
sb (K )
(A.2) sb (K )
following scenarios. c (A) If µ > w , then we have the following conclusions. (a) When 1 increases with K . (B) If µ
then
c
K
=1
2
w1 w0
K
1
x2
c , w0
)
K
G( )
1
1(
xf (x ) G (xK ) dx
K
F (x ) dx
K
find
= w0
), from Eq. (8), we have ① if
1(
)
c , w0
sb (K )
decreases with K ; (b) when
decreases with K . We consider the following scenarios. 409
1(
) >
c , w0
sb (K )
is convex in K .
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
(A) If µ >
c , w1 c , w1
(B) If µ
then we have the following conclusions. (a) When then
1(
)
(iii) In the region K
and
c w0
sb (K )
c , w0
)
decreases with K .
c , w1
, from Eq. (10), we have ① if µ
0,
1(
then
is concave in K ; (b) when
sb (K )
decreases with K ; ② if µ >
sb (K )
c , w1
1(
) >
then
c , w0
sb (K )
sb (K )
increases with K .
increases with K .
From the above discussions, we consider the following three situations. (i) If µ >
c , w1
① When
1(
② When
1( c , w1
(ii) If µ ① When
1(
then we consider the following two scenarios.
) ) > )
K = K0
② When
1(
③ When 2
is concave in K and maximized in the region K
,
sb (K )
is concave in K and maximized in the region K
,+
c , w0
sb (K )
decreases with K in the regions K
> 0 hold, then c , w0
) >
1(
( )>
.
).
decreases with K in the region K
sb (K )
)
w1 w0
>
2
,+
,
0,
( ) and
sb (K )
K
,+
); in the region K
,
, if
2
( )>
w1 w0
0 ; (B)
K = K0
2
()
w1 ; w0
(C)
2
()
first decreases then increases with K in the region K
sb (K )
).
and one of the following conditions is satisfied,
c w0
and K
0,
>
2
( ) and
,
, and
first decreases then increases, and finally decreases with K .
sb (K )
is concave in K in the region K
sb (K )
(A)
sb (K )
then we consider the following three scenarios.
sb (K )
K
c , w0 c , w0
sb (K )
decreases with K .
w1 . w0
□
Proof of Theorem 5. From Lemma 2, it is easy to find that, (i) When µ
c , w1
① When i.e., II sb (KI )
1(
we consider the following three situations.
) >
= w0 (
(b) If 1 ② When
we discuss as follows. (A) The boundary value is
w1 w0
By comparing (a) If 1
c , w0
(1
and
sb (0)
1
G( )
sb (K
xKI
1
xK
f (x ) Dg (D) dDdx <
G( )
1
xK
f (x ) Dg (D) dDdx
c , w0
)
( )>
2
w1 w0
>
2
(K b20 ( )) = (w1
=(w1
K b20 ( )
w0 )
w0 + w0 G ( ))
1
F
K b20 ( )
K b20 ( )
K b20 ( )
= w1
F
F (x ) dx
By comparing (a) If w0 (b) If w0
2
sb (0)
xK b20 ( ) K b20 ( )
xK b20 ( ) K b20 ( )
③ When (A)
K b20 ( )
( )>
1( w1 w0
c w0
) >
2
and
F
K = K0
sb (0)
(K )
sb
and K b * = K .
(0) and K b * = 0 .
> 0 hold, we discuss as follows: (A) The boundary value is
c ) K b20 ( )
xK b20 ( )
K
) >
= 0 ; (B) the local
w0
xK b20 ( ) K b20 ( )
f (x ) G (D ) dDdx
= 0 given in Eq. (A.2), i.e.,
xf (x ) G (xK b20 ( )) dx
+c
K b20 ( )
(A.3)
)) , the supplier's optimal input quantity K w0 G ( )
w1 )
1
F
f (x ) Dg (D ) dDdx
w0 G ( )
w1 )
1
F
(w0
K b20 ( )
K b20 ( )
sb (K )
K = K0
0 ; (B)
2
()
w1 ; w0
(C)
2
()
w1 . w0
410
b*
can be obtained as follows.
, then , then
and one of the following conditions is satisfied, K b * = 0 . K
sb (0)
f (x ) Dg (D) dDdx
f (x ) Dg (D ) dDdx > (w0
( ) and
),
, i.e.,
K b20 ( )
b2 sb (K 0 (
sb
K b20 ( )
sb (K )
K b20 ( )
then
K
F (x ) dx
K b20 ( )
sb (K
F (x ) dx + (µw0
+ w0
K b20 ( )
then
sb (K )
,
Here, K b20 ( ) is determined by solving
w0
,+
f (x ) Dg (D ) dDdx ))
w1 , w0 w1 , w0
( ) and
maximum is obtained in the region K sb
= 0 ; (B) the local maximum is obtained in the region K
) , the supplier's optimal input quantity K b * can be obtained as follows.
G( )
1(
sb (0)
b2 sb (K 0 (
sb
)) >
(K b20 ( ))
sb (0)
sb
and K b * = K b20 ( ) .
(0) and K b * = 0 .
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
(ii) When µ > ① If
1(
) >
② If
1(
)
c , sb (K ) is concave in K . We consider the following two w1 c , the supplier's optimal input quantity K b * = K . w0 c , the supplier's optimal input quantity K b * = K b20 ( ) . □ w0
Proof of Corollary 1. When if
w1 . w0
lim 3 ( ) 1
{
1(
c w0
) >
Thus,
} {
c w0
<µ
) <
w1 w0
c , given 1( w1 c c <µ w , w0 1
when
3(
c , w0
) >
we have
there
exists
c , w1 K b20 ( )
[ 1, + ) , K b * = K b20 ( ) . From Eq. (A.3), we find = (w1
w0 + w0 G ( ))
2
+ w0 (K b20 ( ))
Given K b20 ( ) > , we conclude that (K b20 ( ))
[ 4 , + ) ; in the region
can obtain
3 K b20 ( )
3 K b20 ( )
=
(w1
4.
1
and
Hence, T ( 4 ) > 0 . Recalling
xf (x ) G (
4 ) dx
4],
[ 1,
K b20 ( 1)
1,
=
µ w1 . Because µ w
=µ
at
w1 w0
least
}
{ 0
one
<
3}
positive
if lim
3(
1
w1 ; w0
) >
commitment
{
3(
order
1],
w1 w0
) <
}
{ 0
quantity
in
< the
1}
set
K b * = K . Obviously, K is independent of ; ② in the region
w0 + w 0 G ( )) K b20 ( ) . M
K b20 ( )
x 2g (xK b20 ( )) dx > 0 and
because
K b20 ( ) 2
> 0 in the region
> K b20 ( )
>
4,
then w1
[ 1, + ) ; (B) if
4,
1
w0
w0 G ( ) > 0 .
then
K b20 ( )
0 in
G ( ) , we then
g (xK b20 ( )) dx = G ( K b20 ( ))
K b20 ( )
w0 + w0 G ( K b20 ( ))) 2 . Because K b20 ( ) > , we haveG ( K b20 ( 4)) > G ( 4 ) given
( ) 1
µ w1 is true given µ > w
>µ
1
x 2g (xK b20 ( )) dx
we then obtain G ( K b20 ( 1)) = G c w0
0
[0,
x 2g (xK b20 ( )) dx . Then, we discuss as follows: (A) If
w0 + w0 G ( K b20 ( ))) 2 . Let T ( ) = (w1
> (w 1
) <
① in the region
(K b20 ( ))3
the region
3(
}. □
Proof of Theorem 6. According to Lemma 3, when µ > Here,
{
situations.
0
( ). Meanwhile, we have we conclude G (x ) > G (
G x
c , w1
( )dx = µ
1
1
xf (x ) G x 4) .
c w0
1
Thus, we conclude
> G ( 4 ) . Hence, T ( 1) > 0 . [ 1, 4], at least one exists and satisfies < 0 . If Assumption I In the next, we first propose an assumption, i.e., Assumption I: In the region is true, then at least one exists and satisfies T ( ) < 0 . Because T ( 1) > 0 and T ( 4 ) > 0 , a region of (let denote this region) exists and satisfies
G(
K b20 ( 1))
T( )
T( )
< 0 . Since
K b20 ( )
= w0 g ( K b20 ( )) K b20 ( )
(a) If w0 g ( K b20 ( ))
< 0 , because w0 g ( K b20 ( )) > 0 , then
K b20 ( )
true, then
K b20 (
w0 + w0 G ( ))
K b20 ( )
in the region
<0
Proof of Theorem 7. According to the expression of
then
K b20 ( 5))
1(
rb ( )
[ 5, + ) , b2 *
=
1;
=
c . p
Thus,
< 0 and
b2 *
1(
K b20 (
=
5;
② If and
1
4,
1
4,
>
at least one
K b20 ( )
then
b2 1 ( K 0 ( ))
> 0 in the region
[ 1,
))
K b20 ( )
1
i.e., (p 1 ( K b20 (
4
<
[ 4 , + ) . Recalling
[ 1,
1(
Proof of Lemma 4. In the region K
,
b2 *
{ 1}
={ b2 i ))
to that of Lemma 1, and we omit the details. □
0 , i.e., K b20 ( ) decreases with
.
K b20 ( 1))
4],
[ 1,
=
and
1(
can be found in the region 2
sc (K ) K2
< 0 and
K b20 ( 5)) b2 *
))
c p
4,
>
c , w0
then w0
in the region
w0 G ( )
in the region
b2 i)
[ 1,
4];
=
c , p
w1 ) 1
K b20 ( )
F
5
[ 1, 5]. Then, (A) in the
denote the solution set of
w0 G (
w1 < 0 and
together with p > w0 , a unique
similar to that of Theorem 4: (a) If
0 in the region c w0
1
K b20 ( 1)) =
1(
K b20 (
i
b2 1 ( K 0 ( ))
, because w1 > w0 , we find
< 0 and
5], the proof is b2 i } ,(i = 1,2, ..., n ) K b2 ( b2 ) c ) 0 b2 i + (w 0
5 . Then, similar to the discussions in ①, at least one
Hence, we conclude that at least one
K b20 ( )
Thus
[ 5, + ) , and 1 (
can be found in the set { 1}
0 (from the proof of Theorem 6) and
0 in the region
[ 4 , + ) , i.e.,
b2 *
5],
4].
b2 1 ( K 0 ( ))
in the region
(B) in the region
[ 1,
On the other hand, in the region
given in Eq. (13), we discuss as follows: ① If
[ 1, + ) . Because c p
4 ]).
0.
(b) if there exists at least an extremum, let
the following equation in the region conclude that if
[ 1,
rb ( )
> 0 (from the proof of Theorem 6) in the region
exists and satisfies
(
0 . If Assumption I is true, then
)
Hence, Assumption I leads to the contradiction. Then [ 1, 4]. □
region
< 0.
< 0.
If Assumption I is true, then
K b20 ( )
K b20 ( )
.
w0 + w0 G ( K b20 ( ))) < 0 , then G ( K b20 ( )) < G ( 4 ) . Recall G ( K b20 ( 1)) > G ( 4 ) and G ( K b20 ( 4)) > G ( 4 ) . If G ( K b20 ( )) < G ( 4 ) is
(b) If 2 (w1
(w1
w0 + w0 G ( K b20 ( ))) , we consider the following two possible scenarios in the region
+ 2 (w 1
b2 i
K b20 ( b2i )
rb ( )
< 0,
determined by
= 0 . Hence, we
0 (from the proof of Theorem 6)
we conclude that
can be found in the set { 1}
5
only exists in the region .
[ 1, + ) . □ sc (K )
K
decreases with K . Then, the rest of the proof is similar
Proof of Theorem 8. It is similar to that of Theorem 2 and we omit the details. □ c Proof of Theorem 9. Given w1 > w0 , we have w >
holds. Then, we discuss as
region
[ 1, + ) ,
1 ( 1)
c c c c c c . When µ > w , a unique 1 exists and satisfies 1 ( 1) = w ; when w < µ w , 1( ) w always w1 0 0 1 0 0 c c c* [0, 1], 1 ( 1) w and K = K . Obviously, K is independent of . In the follows: (i) When µ > w , in the region 0 0 c c c < w and K c * = K c 20 ( ) = K b20 ( ) ; (ii) when w < µ w , K c * = K c 20 ( ) = K b20 ( ) . Here, K b20 ( ) increases with . □ 0 1 0 0
411
International Journal of Production Economics 216 (2019) 398–412
J. Cai, et al.
Proof of Theorem 10. It is similar to that of Theorem 4 and we omit the details. □
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