The Coordination and Preference of Supply Chain Contracts Based on Time-Sensitivity Promotional Mechanism

JMSE 2018, 3(3), 158–178 doi:10.3724/SP.J.1383.303009

http://www.jmse.org.cn/ http://engine.scichina.com/publisher/CSPM/journal/JMSE

Article

The Coordination and Preference of Supply Chain Contracts Based on Time-Sensitivity Promotional Mechanism Xin Fang 1,2,* and Fengjiao Yuan 2 1

Collaborative Innovation Center for Chongqing’s Modern Trade Logistics & Supply Chain, Chongqing Technology and Business University, Chongqing 400067, China

2

School of Business Planning, Chongqing Technology and Business University, Chongqing 400067, China

* Correspondence: [email protected] Received: 27 March 2018; Accepted: 6 December 2018; Published: 21 January 2019

Abstract: To encourage retailers to submit orders as soon as possible, manufacturers usually launch a time-sensitivity promotional mechanism that the earlier you order, the cheaper the wholesale price will be in advance of the selling season. This paper aims to investigate if the mechanism can improve supply chain performance. A dyadic decentralized supply chain system comprising a single manufacturer and a single retailer is viewed as a research framework. Initially, a benchmark model is proposed to provide a criterion-referenced for coordinating the supply chain in a non-standard distribution environment. Second, a time-sensitive wholesale price contract is constructed to confirm that the mechanism can coordinate the supply chain. However, the retailer accepts the entire forecast risk under the contract. An improved contract called a time-sensitive revenue-sharing contract is constructed based on the notion that the manufacturer shares partial forecast risk. The results show that participants can arbitrarily divide the optimal supply chain’s expected profit between the constructed price contracts; however, two differences exist between the contracts, that is, participants have contract preferences. Finally, a numerical analysis and a few management insights are given. Keywords: Time-sensitivity; Promotional; Supply chain; Contract; Distribution-free approach

1. Introduction In practice, it is common for manufacturers to launch a time-sensitivity promotional mechanism that the earlier a retailer orders, the cheaper the wholesale prices will be in advance of the selling season. Some examples include Trek Inc.’s bicycles selling models (Cachon, 2004), moon cake selling model (Jian et al., 2015, p64) and Caesar’s clothing shops selling models (Wang and Zhou, 2010, p1293). The mechanism is very alluring so that retailers submit their orders as soon as possible, and the manufacturer can save extra production costs that are 158

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usually called crashing cost (Liao and Shyu, 1991). However, the forecast risk of demand increases under the mechanism, and the retailer’s early orders result in either too much or too little inventory, thereby contributing to disposal costs, product stock-outs, or insufficient customer service levels (Louly and Dolgui, 2013). It is common knowledge that the longer the time interval between delivery and purchase, the larger is the variance in the forecasted demand (Blackburn, 1991; Iyer and Bergen, 1997). Due to ordering early, the perfect distribution information obtained by the retailer is often limited. It is difficult to specify the distribution function of the sources of uncertainty, particularly for uncertain demand. Sometimes the randomness of the demand pattern is so variable that market information is only an educated guess of the mean and the variance (Yu et al., 2013). When the retailer does not know the standard distribution function, estimation of mean and standard deviation of forthcoming demand is possible (Moon and Gallego, 1994). Therefore, it is a challenge to calculate the profit without having perfect information on demand. To overcome the challenge, a distribution-free approach is used to tackle the worst possible distribution of the demand for the mean and the variance. On the other hand, with a time-sensitivity promotion, the manufacturer must have the ability to control the production process. In other words, lead time, defined as the elapsed time between placing an order and receiving by the retailer, is controlled to manufacture by paying an additional crashing cost (Liao and Shyu, 1991). The extra crashing cost could be expenditure on equipment improvement, information technology, order expenditure, or special shipping and handling (Hsu and Lee, 2009). In recent years, many scholars and practitioners have begun to study or utilize it as a controllable tool to improve the performance of the supply chain. Deriving from the problem mentioned above, this paper aims to verify the following queries. (1) When is the optimal production time for an integrated supply chain? (2) Can the time-sensitivity promotional mechanism optimize the decentralized supply chain? (3) How are benefits and forecast risk shared between participants? A dyadic decentralized supply chain system comprising a single manufacturer and a single retailer is viewed as the research framework. The manufacturer launches a time-sensitivity promotional mechanism that evolves continuously with controllable lead time; the retailer only learns the mean and standard deviation value of the lead time demand instead of perfect information with standard distribution function. The manufacturer and retailer are in a Stackelberg game in which the manufacturer is a leader. After receiving an order from the retailer, the manufacturer completes the finished product and delivers it to the retailer. Before the sales season, the retailer has only one chance to order and must choose an order quantity to meet the realized demand in a decision epoch. Under the mechanism, the retailer is inherently exposed to the forecast risk of demand. If the realized demand drops below the order quantity, the retailer will not be able to sell all the goods and earn salvage per unit unsold at the end of the selling season; but any demand above the order quantity represents a foregone opportunity and the retailer incurs a cost per unit penalty for loss of goodwill. Furthermore, it is assumed that all the information is common knowledge for players, and the decision objective of each player is to maximize his/her own expected total profit. Initially, to solve the first query, a benchmark model is proposed with a non-standard distribution function from the perspective of an integrated supply chain. Some optimal decisions are calculated by analyzing the model. Under the optimal decisions, a balance is achieved between the forecasted risk and the production cost of the integrated supply chain. Second, a contract defined time-sensitivity wholesale price (abbreviated as TsWP) contract is constructed to verify that if the time-sensitivity promotional mechanism can optimize the

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decentralized supply chain, but we find that the retailer bears the forecast risk completely. Thus, an improved contract defined time-sensitivity revenue sharing (abbreviated as TsRS) contract is constructed based on the idea that the manufacturer sets a constant wholesale price for all time and shares the forecast risk with a proportion. Moreover, contracts are compared with the mean of the forecast risk and the preference of contracts is analyzed. Unlike the existing literature, this paper extends the research to a non-standard distribution environment to conquer the difficulty that retailers must know the standard distribution function of lead time demand, and explores how to determine optimal wholesale price in advance of the selling season for the supply chain. The remainder of the paper is organized as follows. Section 2 reviews the relevant literature on promotional mechanisms and distribution-free approach in operations and supply chain. Section 3 introduces some assumptions and notations employed in this paper. Sections 4 and 5 propose a benchmark model and construct two differential contracts to coordinate the decentralized supply chain based on the optimal decisions of the benchmark model, respectively. Section 6 provides a numerical case to illustrate the effectiveness of the model. Section 7 contains the concluding remarks.

2. Literature Review Two categories in the literature are related to this research. The first one is the promotional mechanism in operations and supply chain management. Promotional mechanisms have emerged as an important part of the marketing strategies, and many peculiarities are shown in economic activities (Lin, 2016). Blattberg and Neslin (1990) delineate three types of sales promotions: trade promotion from the manufacturer to the retailer, retailer promotion from the retailer to the consumers, and consumer promotion from the manufacturer to the consumers (manufacturer promotion). This paper focuses on manufacturer promotion. Lariviere and Porteus (2001) study a model with a “supplier selling to a newsvendor” using a single wholesale price contract in advance of the selling season. Cachon (2004) and Dong and Zhu (2007) study a model with push, pull, and advance-purchase discount contracts. With a push contract there is a single wholesale price and the retailer needs to place an order before the selling season. An advance-purchase discount has two wholesale prices: a discounted price for inventory purchased before the season and a regular price for replenishment during the selling season. Since then, (Jin et al., 2015) model a supply chain with one manufacturer and one retailer who have limited capital and face deterministic demand depending on the retail price and sales promotion. The result shows that only the combination (called a chain business model) of a consignment contract with the manufacturer ’s right of sales promotion or a wholesale price contract with the retailer’s right of sales promotion is better for both members. Tsao (2015) researches a decentralized supply chain in the presence of manufacturer promotional efforts and uncertain demand. The results demonstrate that the promotional cost-sharing policy motivates the manufacturer to increase promotional efforts and the retailer to order more products. Lin (2016) considers the price promotion with the reference price of a manufacturer to the retailer and finds that the reference price effects could mitigate “double marginalization” effects and improve channel efficiency. Tsao and Lu (2016) compare four trade promotions (off-invoice, scan-back, unsold-discount, and target rebate) in manufacturer-retailer supply chains. With an off-invoice policy, manufacturers offer discounts on the order quantity sold to retailers, and in a scan-back policy, manufacturers offer discounts on the actual quantity that retailers sell to end customers. Bai et al. (2017) study a model with revenue and promotional cost-sharing

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contract versus a two-part tariff contract. They assume that the time-varying demand in this system is affected by three endogenous variables that include promotional efforts provided by the retailer, product-selling price, and the sustainable level determined by the manufacturer. Fang (2018) studies the coordination of a wholesale price contract under the perspective of equality between enterprises, and demonstrates that the participants can share risk with a constructed contract. Yang et al. (2018) investigate the effect of product promotion which the manufacturer can promote via a direct channel (manufacturer’s promotion), and the retailer via a retail channel (retailer’s promotion) based on remanufacturing in dual-channel supply chains under cap-and-trade regulation. Pakhira et al. (2018) establish that if the supplier shares a part of the promotional cost then the channel profit as well as individual profits increase. The second one is the distribution-free approach in operations and supply chain management. To conquer the difficulty, a decision maker must know the standard distribution function of the lead-time demand, Scarf (1958) first suggestes a closed-form solution for a newsvendor problem with only the mean and standard deviation of the known demand. The ordering rule proposed by Scarf is practical and easy to use, but it is lengthy and quite difficult to understand. Therefore, Gallego and Moon (1993) simplify the proof of Scarf’s ordering rule for the newsvendor problem and extend the analysis to recourse constraints, fixed ordering cost, and random yields, etc. Moon and Choi (1995) study the classic newsboy model using a distribution-free newsboy problem with customer balking. Moon and Yun (1997) address a distribution-free job control problem to determine an optimal release time. Ouyang and Wu (1998) consider a mixed inventory model with both lead time and order quantity as the decision variables. They apply a minimax distribution-free approach to solve the problem. In a more recent paper, Alfares and Elmorra (2005) extend the results obtained by Gallego and Gallego and Moon (1993) via the incorporation of a shortage penalty cost. They present the optimal order quantity and a lower bound on the profit under the worst possible distribution of demand. Chu et al. (2005) extend the results of (Ouyang and Wu, 1997) considering a service level constraint and variable lead time. Lee and Hsu (2011) study the effect of advertising on the distribution-free newsvendor problem. They consider three cases: (1) demand has constant variance, (2) demand has constant coefficient of variation, and (3) demand has an increasing coefficient of variation. They also provid closed-form solutions or steps to solve the problem. Liao et al. (2011) extend the distribution-free newsvendor model in an environment of customer balking and fixed ordering costs. Moon et al. (2014) provid a continuous-review ( Q, r, L) inventory model with a fill rate service constraint and a negative exponential crashing cost function with a variable lead time. They extend the distribution-free continuous-review inventory model to minimize the total cost by using a negative exponential lead time crashing cost function, and derive a closed-form expression for the optimal order quantity, reorder point, and lead time. Kwon and Cheong (2014) study the optimal policies of retailers who operate their inventory with a single period model under a free shipping offer where a fixed shipping fee is exempted if an order quantity is greater than or equal to a given minimum quantity. They extend the base model to deal with a practically important aspect of inventory management when the exact distribution function of demand is not available. Braglia et al. (2017) study stochastic joint-replenishment problem with backorders-lost sales mixtures, and controllable major ordering costs and lead times with a distribution-free approach. To sum up, the existing literature on promotional mechanisms always assumes that wholesale price is given and fixed in advance and there is no explanation about when such wholesale prices should be given before the selling season. To the best of the authors’ knowledge, the time-sensitivity promotion problem has not

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been examined to date. Meanwhile, studying the coordination in a decentralized supply chain in this situation has not been previously considered. Therefore, in this paper, a time-sensitivity promotion problem is considered in a dyadic decentralized supply chain with a non-standard distribution function.

3. Assumptions and Notations 3.1. Assumption of Demand The demand forecasting evolution process exhibits how lead time affects supply-demand mismatch. If a manufacture can produce items at time 0 and deliver them to a retailer at time T only, then the time lag T can be called as normal lead time. Now, it is assumed that the manufacturer can produce items at any time between 0 and tmax . The retailer has forecasted the mean and standard deviation of lead time demand Xt at time t ∈ (0, tmax ], and then he/she places an order with the manufacturer at a decision epoch. Therefore, parameter tmax representing the maximum amount of lead time can be reduced and the decision epoch t represents the amount of compressed lead time. The longest lead-time alternative requires a production commitment at decision epoch 0. The process of these actions is presented in Figure 1.

t

Figure 1. Demand forecast-evolution process with lead time compression

Some assumptions about the forecast-evolution process can be assumed as follows by referring to scholars Blackburn (1991); Iyer and Bergen (1997); de Treville et al. (2014) and Jian et al. (2015). Assumption 1 σt0 = κ, t ∈ [0, tmax ] and κ < 0. Assumption 2 µt = µ, t ∈ [0, tmax ] and µ > 0. These assumptions are consistent with a certain economic activity. Due to ordering early or before the selling season, retailers will find it difficult to obtain the specific distribution function of market demand accurately; the market can only be described by “the quantity of the item is approximately at µ, with a margin of error is no more than σ”, namely, retailers estimate the mean and the standard deviation for the upcoming demand. Blackburn (1991) and Iyer and Bergen (1997) point that the less is the time between delivery and order placement, the variance of the forecasted demand will be lower. Therefore, we assume that the variance decreases with time (Assumption 1). Meanwhile, de Treville et al. (2014) and Jian et al. (2015) assume that the forecast updates are unbiased with time changing, such that for any subsequent times, the expected value of the forecast update is zero; hence, we assume that the mean value remains unchanged (Assumption 2). However, there is no denying that estimates of both mean value and variance can vary from time to time. This situation renders the solution of the model complex; therefore, it will be discussed in a future study.

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3.2. Assumption of Production Cost Curve The manufacturer generates crashing cost due to compressed lead time and increases its total production cost per-unit. In this paper, let parameter ct be the production cost at decision epoch t and ct = normal production cost + crashing cost. The normal production cost represents a general production cost without lead time reduction. The crashing cost is assumed as a general function in time t, which can be calculated by referring to Liao and Shyu (1991). Assumption 3 c0t > 0, t ∈ [0, tmax ]. In general, the second-order deviation of ct is usually assumed as a strictly convex function in time t. It is undeniable that this assumption is in accordance with the vast majority of the cost curve of firms (Priyan and Uthayakumar, 2015; Yang, 2010), but it is reasonably believed that for some firms production cost ct is a concave function in time t (Jian et al., 2015). In this paper, we consider two different types of production costs mentioned above. They are both increasing function in time t and the only difference between them is the convexity and concavity of function with respect to time t. Without loss of generality, the production cost function with a strict convexity in time t is noted as Type One. Conversely, the production cost function with a concavity in time t is noted as Type Two. Fortunately, our investigation shows that in Proposition 1 of Section 4, there are some different optimal decisions between the two types of production cost.

3.3. Notations The notations used in this paper are summarized in Table 1. Table 1. Notations used throughout this paper

p

Retail price

g

Goodwill cost

v

Salvage value

T

Normal lead time Maximum amount of compressed lead time, tmax ≤ T

tmax t

Decision epoch, t ≤ tmax

qt

Order quantity at decision epoch t

ct

Production cost at decision epoch t

µt

Mean of demand at decision epoch t

σt

Standard deviation of demand at decision epoch t

δ

Ratio of integrated supply chain revenue the retailer keeps, δ ∈ (0, 1)

wt

Wholesale price with TsWP contact

φt

Ratio of retailer shares the sales revenue with TsRS contact, φt ∈ (0, 1)

E( x )

Mathematical expectation

( x )+

max( x, 0)

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4. Mathematical formulation 4.1. Benchmark model In this section, a benchmark model is investigated from the perspective of an integrated supply chain. Furthermore, optimal decisions that are viewed as a criterion for the game of participants are calculated from the model. The integrated supply chain’s expected profit E ∏C (qt , t) at decision epoch t for a produce of qt units is given by E ∏C (qt , t) = ( p + g − ct ) qt − ( p + g − v) E (qt − DT )+ − gµ (1) Where, the term E (qt − DT )+ denotes the expected salvage of unsold items. In the existing literature, equation (1) can be analyzed under the assumption that lead time demand follows a standard distribution function, such as normal distribution function and uniform distribution function. However, it is very difficult to know the perfect distribution information in practical situations, especially in the early ordering phase. Hence, demand that is defined as a standard distribution function is relaxed in this paper and considered as a distribution-free approach to solve the mentioned-above problem. In addition, our model has great practical significance. For instance, in the absence of past statistical data, which is irrelevant to new products or products with a short lifecycle, it is difficult to specify the distribution function of the sources of uncertainty, particularly for uncertain demand. In addition, sometimes, the randomness of demand is so variable that market information is obtained from an educated guess of the mean and the variance (Yu et al., 2013). The model only assumes that the density function of demand belongs to the class Ω of least favorable distribution function Ft with mean µ and standard deviation σt . As the distribution form of demand Xt is non-standard, the exact value of E (qt − DT )+ cannot be calculated. According to the description of the distribution-free approach proposed by Gallego and Gallego and Moon (1993), the following inequality for any Ft ∈ Ω is obtained. q  1 E ( q t − DT )+ ≤ σt2 + (qt − µ)2 + (qt − µ) (2) 2 Thus, equation (1) can be transformed as follows by using in inequality (2). 1 E ∏C (qt , t ) ≥ ( p + g − ct ) qt − ( p + g − v ) 2

q

σt2

2



+ (qt − µ) + (qt − µ) − gµ = EπC (qt , t)

(3)

Let EπC (qt , t) be the lower bound of E ∏C (qt , t). It guarantees that the integrated supply chain gains revenue for complete demand distribution including worst possible demand distribution with the same mean and standard deviation (Gallego and Moon, 1993). Thus, this paper focuses on analyzing all the questions mentioned above in the Introduction based on this lower bound function. It is clear that EπC (qt , t) is concave in qt as 3/2  1 <0 ∂2 EπC (qt , t)/∂q2t = − σt2 ( p + g − v)/ σt2 + (qt − µ)2 2

(4)

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Hence, there exists a uniquely optimal production q∗t for the integrated supply chain. By setting ∂EπC (qt , t)/∂qt equals to zero, q∗t can be calculated as follows: q∗t = µ + σt



1 − Rt 2

s

1 ; R t (1 − R t )

0 ≤ Rt < 1

(5)

Where, Rt = (v − ct )/(v − ( p + g)). Parameter Rt represents relative risk, which is a rate between production risk and the risk of loss of potential revenue per unit. Meanwhile, its value increases in ct for dRt /dct > 0 and increases in time t for dRt /dt > 0 with the assumption 3. Hence, there exists a unique decision epoch to that satisfies Rto = 0.5, and further infers that equation q∗t ≥ µ holds for all t ≤ t0 . Iyer and Bergen (1997) state that if q∗t < µ with a controllable lead time, then the supply chain has low service levels, and it does not expect to observe such low service levels in practice. Hence, without loss of generality, it is assumed that inequality Rt ≤ 0.5 holds for all t ∈ [0, tmax ] in this paper. Furthermore, taking the first order partial derivative of q∗t in t and in Rt . It is obtained that dq∗t /dt < 0 holds for both assumption 1 and assumption 3 and that dq∗t /dRt < 0 holds. Therefore, the optimal production quantity q∗t decreases in t and in Rt , respectively. Substituting equation (5) into equation (3), the equation can be revised as follows. EπC (t) = ( p − ct ) µ − σt (v − ( p + g))

q

R t (1 − R t )

(6)

Next, the optimal decision epoch from equation (6) will be analyzed. Proposition 1. The optimal decision epoch t∗ depends on both the type of production cost curve and demand forecasting process. Specifically, for all t ∈ [0, tmax ], (1) If either c00t ≤ 0 or both c00t > 0 and Υt ≥ q∗t c00t holds, then the optimal decision epoch t∗ has the following three situations: If limt→0+ Ψt ≥ 0, then t∗ = tmax . ∗ If limt→tmax − Ψ t ≤ 0, then t = 0. ∗ If limt→0+ Ψt · limt→tmax − Ψ t < 0, then t ∈ {0, tmax }. (2) If both c00t > 0 and Υt < q∗t c00t hold, then the optimal decision epoch t∗ has the following two situations. ∗ If limt→tmax − Ψ t ≥ 0, then t = tmax . ∗ If limt→tmax − Ψ t < 0, then t ∈ { t ∈ R | ∂Eπ C ( t ) /∂t = 0 }.    2 1/2 σt (c0t ) 1 0 1 −R Where, Υt = √ 1 p + g − v − κc ; ( ) t t 2 4 R (1− R ) R t (1− R t )

t

t

Ψt = κ (v − p − g)

q

(1 − Rt ) Rt − q∗t c0t .

Proof. See Appendix A. According to proposition 1, the optimal production quantity of integrated supply chain q∗t can be calculated by equation (5); further, the optimal expected profit of integrated supply chain EπC (t∗ ) can be deduced by equation (6). Obviously, the expected profitability of an integrated supply chain can be significantly increased in the set of (q∗t∗ , t∗ ). Meanwhile, since lead time viewed as a controllable tool is reduced to the optimal position

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and the accuracy of demand forecasting is further improved, the forecast risk of the integrated supply chain can be decreased. Coordination between supply chain participants is one of the main issues in managing a decentralized supply chain (Cachon, 2003). Let us review the main thinking of the time-sensitivity promotion that the earlier you order, the cheaper the unit purchase price will be. In this promotional process, the manufacturer releases a mechanism with continuous time to allure the retailer to order early, aiming to optimize his/her own profit. However, there exists only one set of optimal decisions from proposition 1. Therefore, the optimal decision between the integrated supply chain and the participants are conflicting with each other. In this case, coordination mechanisms can be helpful. Top-level coordination can reduce the forecast risk throughout the supply chain and increase the supply chain profitability. A contract coordinates the supply chain if the profitability of a decentralized supply chain under the mentioned mechanism is equal to that of an integrated supply chain (GüLer and Keskin, 2013).

4.2. Constructing TsWP contract To solve the confliction of optimal decision, a TsWP contract is constructed based on the idea that the earlier you order, the lower the wholesale price. That is, the manufacturer sets a wholesale price that is a function of continuous time. Assume that the manufacturer charges the retailer wt at time t per unit purchased. Then, the lower bound estimate on the retailer’s expected profit function Eπr (qt , t; wt ) and the manufacturer’s Eπs (qt , t; wt ) can be represented as follows by using inequality (2): 1 Eπr (qt , t; wt ) = ( p + g − wt ) qt − ( p + g − v) 2

q

σt2

2



+ (qt − µ) + (qt − µ) − gµ

Eπs (qt , t; wt ) = (wt − ct ) qt

(7) (8)

Let wt be the wholesale price with continuous time for any δ, δ ∈ (0, 1). wt = δct + (1 − δ) [( p + g) ϑt + v (1 − ϑt )]  where, ϑt = µ −

σt 2

q

Rt 1− R t



(9)

/qt .

Proposition 2. The TsWP contract with parameter wt can coordinate the decentralized supply chain and the optimal profit can be arbitrarily divided for participants by δ. Proof. Substituting wt into equation (7), it can be reduced to as follows. Eπr (qt , t; wt ) = δ [ EπC (t∗ ) + gµ] − gµ

(10)

The retailer’s optimal decision epoch for any given qt equals the integrated supply chain’s optimal decision epoch. Thus, the set pair (q∗t∗ , t∗ ) is optimal if q∗t∗ is optimal for the givent∗ , which clearly holds for δ > 0. The upper bound on δ must ensure that the manufacturer earns a nonnegative profit. Proved. It can be seen from equation (9) that, on the one hand, the wholesale price wt trends toward the production cost ct when the parameter δ closes to one, which means that the manufacturer’s wholesale price is equal to

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his/her production cost and the retailer obtains the total profit of an integrated supply chain. On the other hand, parameter wt trends to ( p + g) ϑt + v (1 − ϑt ) when parameter δ closes to zero, which means that the manufacturer sets a higher wholesale price and the retailer gains lower profit. Although the constructed contract can coordinate the decentralized supply chain, there is a drawback that the retailer bears the whole forecast risk (discussed later in Section 5). Hence, the retailer will refuse to join the supply chain and the cooperation will be interrupted when the fluctuation in demand becomes extremely and unsteady.

4.3. An improved contract In this section, a contract named time-sensitivity revenue sharing (TsRS) is constructed to conquer the drawback of the TsWP contract mentioned above. Distinguished from the TsWP contract, the TsRS contract is constructed based on the idea that the manufacturer fixes the wholesale price for all time and the retailer shares partial forecast risk that is caused by purchasing later. Under the TsRS contact, the manufacturer charges a constant w0 per-unit purchased for all time plus the retailer gives the manufacturer a percentage of his/her revenue. Let φt be the fraction of supply chain revenue the retailer keeps. Then the lower bound estimate on the expected profit function of the retailer Eπr (qt , t; w0 , φt ) and manufacturer Eπs (qt , t; w0 , φt ) can be represented as follows by using inequality (2), respectively. Eπr (qt , t; w0 , φt ) = (φt p + g − w0 ) qt −

1 [φt ( p − v) + g] 2

q

 σt2 + (qt − µ)2 + (qt − µ) − gµ

1 Eπs (qt , t; w0 , φt ) = [(1 − φt ) p + w0 − ct ] qt − (1 − φt ) ( p − v) 2

q

σt2

2

+ (qt − µ) + (qt − µ)

(11)

 (12)

Let parameter φt be the fraction of supply chain revenue the retailer keeps with fixed wholesale price w0 for any δ, δ ∈ (0, 1). ( p − v ) v + δ ( c t − v ) − w0 (1 − δ) gϑt − (13) φt = 1 − ( p − v ) ϑt + v ( p − v ) ϑt + v Proposition 3. The TsRS contract with parameters {w0 , φt } can coordinate the decentralized supply chain and the optimal profit can be arbitrarily divided for participants by δ. Proof. Substitute φt into equation (11), then it can be reduced to as follows. Eπr (qt , t; w0 , φt ) = δ [ EπC (t∗ ) + gµ] − gµ

(14)

The retailer’s optimal decision epoch for any given qt equals the integrated supply chain’s optimal decision epoch. Thus, the set (q∗t∗ , t∗ ) is optimal if q∗t∗ is optimal for givent∗ , which clearly holds for δ > 0. The upper bound on δ ensures that the manufacturer earns a nonnegative profit. Proved.

It can be seen that parameter φt trends to 1 − [( p − v) v − w0 + ct − v]/[( p − v) ϑt + v] when parameter δ closes to one for any given constant wholesale price w0 , which means that the retailer obtains the total profit of integrated supply chain and the manufacturer gains zero profit according to equation (14). Furthermore, parameter φt trends to 1 − [( p − v) v − w0 + gϑt ]/[( p − v) ϑt + v] when parameter δ closes to zero for any given

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constant wholesale price w0 , indicating that the manufacturer obtains the total profit of integrated supply chain and the retailer gains zero profit according to equation (14).

5. Contract preferences It is seen that both constructed contracts can coordinate the dyadic decentralized supply chain according to proposition 2 and proposition 3 for an optimal decision point t∗ . Meanwhile, the contracts are mutually equivalent because the retailer and the manufacturer can divide the optimal integrated supply chain’s expected profit by δ and 1 − δ, respectively. Namely, the following equations hold for any given constructed contract. Eπr∗ = δ [ EπC (t∗ ) + gµ] − gµ; Eπs∗ = (1 − δ) [ EπC (t∗ ) + gµ] . However, there are two differences between a TsWP contract and a TsRS contract. Initially, a difference of procurement price is charged. Under the TsWP contract, in equation wt > ct must hold for all time, and the procurement price w0 may be less than ct for all time under the TsRS contract. The second difference is in sharing forecasted risk. With a TsWP contract the retailer’s transfer payment is wt qt . This is a realized profit for the manufacturer before the selling season and there is no variation in the manufacturer’s profit, no matter what the realization of demand, but the retailer’s profit varies with the realization of demand. Hence, the retailer bears the whole forecast risk and the manufacturer probably prefers this contract to coordinate the decentralized supply chain, especially when the demand volatility is higher. With a TsRS contract the retailer’s transfer payment is (w0 qt + φt · retailer 0 s revenue), where values of wt , qt and φt are marked at point t∗ . This means that the manufacturer’s profit comprises two parts. The first one is w0 qt , and it is a profit realized in advance of the selling season, no matter what the realization of demand is. However, the second part relates to the retailer’s revenue, and the retailer’s revenue varies with the realization of demand. Hence, the manufacturer bears some forecast risk and could prefer a TsRS contract to motivate the retailer to take part in the supply chain, and he/she will obtain a higher expected profit.

6. Numerical analysis In this section, a numerical analysis is presented to calibrate the effectiveness of the proposed model mentioned above. The numerical analysis includes two aspects: (1) The optimal decisions analysis. This part mainly contains the optimal decision epoch, optimal production quantity, and coordination and designing of the contract parameters. (2) The performance analysis between non-standard distribution and standard distribution function is given to verify the effectiveness of the proposed model. The structure of demand, which is used by Chen and Chuang (2000), is described as follows: the mean and variance value of demand Xt are µ and σt , respectively. Where µ is a positive value, σt = σ0 + (σT − σ0 ) t/tmax ,t ∈ [0, tmax ], parameters σ0 and σT represent standard variance at time t = 0 and t = tmax , respectively. Two different types of production cost are considered to illustrate the impact of optimizing decision making. Type One, ct = c0 + 0.5αt2 and Type Two, ct = c0 + αt, where parameter c0 represents the production cost at time t = 0, and parameter α represents the marginal cost per-unit.

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To simplify calculation, we use the term relative lead time (Song et al., 2013) to refer to a given lead time as a proportion of this maximum lead time value, and to simplify calculation, without loss of generality, it is assumed that the term is normalized to tmax = 1, scaling volatility and drift accordingly. In addition, this paper considers four decimals to approximate accuracy of the work because the calculated results may be irrational numbers. The other parametric values are shown in Table 2. Table 2. Parametric values

Parameter

Value

Parameter

Value

Parameter

Value

Parameter

Value

p g

10 3

v c0

1 4

σ0 σT

30 10

µ

200

6.1. Analysis of type one 6.1.1. Optimal decision Since c00t = α > 0 holds in type one, that is, the production cost is a strictly convex function in time t, the optimal decisions of the integrated supply chain depend on the signs of both Υt − q∗t c00t and Ψt according to proposition 1 mentioned above in Section 3. Furthermore, the constraint condition Rt ≤ 0.5 restricts that the inequality p + g + v − 2ct ≥ 0 must hold. It is calculated that Υt − q∗t c00t < 0 for any α > 0 for all time t and that limt→tmax − Ψ t > 0 when α > 0.5204, respectively. Thus, the expected profit function is a monotone increasing in t when α ≤ 0.5204. For any α > 0.5204, the optimal decision epoch is calculated by solving the first-order optimal condition of the expected profit function defined in proposition 1. Table 3 lists the optimal decisions with some given α as an example. To demonstrate the impact of the marginal cost per unit time on the integrated supply chain’s optimal decision effectively, we depict the optimal expected profits changing with time t when α ∈ {0.2, 0.8, 1.4, 2.0}, shown in Figure 2. The abscissa that corresponds to the asterisk (*) represents the optimal decision epoch. The results can objectively illustrate our proposition 1 mentioned above in Section 3. Table 3. Optimal decisions with some given α in type one

α

t∗

ct∗

q t∗

EπC∗ × 103

Rt∗

α

t∗

ct∗

qt∗

EπC∗ × 103

Rt∗

0.2 0.4 0.6 0.8 1.0

1.0 1.0 0.86 0.63 0.50

4.10 4.20 4.22 4.16 4.12

205.52 205.28 206.71 209.34 210.93

1.13 1.12 1.09 1.08 1.07

0.35 0.36 0.37 0.36 0.35

1.2 1.4 1.6 1.8 2.0

0.41 0.35 0.31 0.27 0.24

4.10 4.09 4.08 4.07 4.06

212.00 212.74 213.32 213.76 214.12

1.07 1.06 1.06 1.06 1.06

0.35 0.35 0.34 0.34 0.34

6.1.2. Supply chain coordination In this section, the parameters of both constructed contracts are designed in type one. For a given traditional newsvendor solving, the participants achieve Pareto improving with a coordination contract. That is, both manufacturer and retailer are at least well off, and one party is better off after adopting the contract. Without loss of generality, it is assumed that the wholesale price w0 equals to seven at decision epoch zero. Hence, in the

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1140 α = 0.2

α = 0.8

0.1

0.2

α = 1.4

α = 2.0

1120 1100

Profits →

1080 1060 1040 1020 1000 980 960 940

0

0.3

0.4

0.5 t→

0.6

0.7

0.8

0.9

1

Figure 2. EπC (q∗t , t) changes with t under some given α in type one

classical newsvendor environment, the retailer’s order quantity is 200, and the manufacturer’s and retailer’s profits are 420 and 600, respectively. However, we can see from Table 3 that the integrated supply chain’s profits are better than the traditional newsvendor’s with different α. The participants have a space of Pareto improving according to either proposition 2 or proposition 3. Table 4 shows the variation range of each contracted contract that participants can achieve Pareto improving. The arrowhead in the table indicates the direction of the contract parameter that keeps increasing the retailer’s profit. Figure 3 depicts the corresponding parameters variation range, in which the contract can coordinate the decentralized supply chain when the contract parameter lies in the gray area. Table 4. The coordination parameters with w0 = 7 in type one

α

δ

wt

φt

α

δ

wt

φt

0.2 0.4 0.6 0.8 1.0

0.59→0.65 0.60→0.65 0.60→0.64 0.61→0.64 0.61→0.64

7.54→7.02 7.55→7.12 7.45→7.12 7.30→7.03 7.20→6.97

0.4799→0.4784 0.4738→0.4721 0.4682→0.4666 0.4650→0.4637 0.4631→0.4619

1.2 1.4 1.6 1.8 2.0

0.61→0.64 0.61→0.64 0.61→0.645 0.62→0.64 0.62→0.64

7.15→6.93 7.11→6.90 7.07→6.89 7.05→6.87 7.03→6.86

0.4618→0.4607 0.4608→0.4598 0.4601→0.4591 0.4595→0.4586 0.4591→0.4582

6.2. Analysis of type two 6.2.1. Optimal decision Since c00t = 0 holds in type two, that is, the production cost is a concave function in time t, the optimal decisions of integrated supply chain are only decided by Ψt according to proposition 1 mentioned above in Section 3. Furthermore, the constraint condition Rt ≤ 0.5 restricts that the inequality p + g + v − 2ct ≥ 0 must

δ

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0.66 0.65 0.64 0.63 0.62 0.61 0.6 0.59 0.2

171

Coordination area Lower bound Upper Bound

0.4

0.6

0.8

1

1.2

α→

1.4

1.6

1.8

2

7.6 7.5 7.4 7.3 7.2 7.1 7 6.9 6.8 0.2 0.4

0.48

Coordination area Upper bound Lower Bound

Coordination area Upper bound Lower Bound

0.475 0.47 t

wt

(a)

0.465 0.46

0.6 0.8

1 α

1.2

1.4

1.6 1.8

2

0.455 0.2 0.4

0.6 0.8

(b)

1 α

1.2

1.4

1.6 1.8

2

(c)

Figure 3. The area of coordination of contract parameters with some given α in type one

Table 5. Optimal decisions with some given α in type two

α

t∗

ct∗

qt∗

EπC∗ × 103

Rt∗

α

t∗

ct∗

q t∗

EπC∗ × 103

Rt∗

0.1 0.2 0.3 0.4 0.5

1 1 1 1 1

4.05 4.10 4.15 4.20 4.25

205.65 205.52 205.40 205.28 205.16

1.13 1.11 1.09 1.07 1.05

0.34 0.35 0.36 0.36 0.37

0.6 0.7 0.8 0.9 1.0

0 0 0 0 0

4.00 4.00 4.00 4.00 4.00

217.32 217.32 217.32 217.32 217.32

1.04 1.04 1.04 1.04 1.04

0.33 0.33 0.33 0.33 0.33

hold. It is calculated that limt→0+ Ψt ≥ 0 when the equality α ≤ 0.4782 holds and limt→T − Ψt ≤ 0 when the equality α ≥ 0.5350 holds. Namely, the optimal decision epoch equals to time tmax when α ≤ 0.4782, and equals to time when α ≥ 0.5350. Moreover, the optimal expected profit is a convex function in time t when α ∈ (0.4782, 0.5350). Table 5 lists the optimal decisions with some given α as an example. To effectively demonstrate the impact of the marginal cost per unit time on the optimal decision, we depict the optimal expected profit changing with time t when α ∈ {0.45, 0.48, 0.51, 0.54}, shown in Figure 4. The abscissa that corresponds to the asterisk (*) represents the optimal decision epoch. Thus, it is seen that (1) EπC (t) is a monotone increasing function in t when α = 0.45(< 4782); thus, the optimal amount of lead time compression equals to tmax ; (2) EπC (t) is a convex function in t when α = 0.48 ∈ (0.4782, 0.5350) and the inequality EπC (tmax ) > EπC (0) holds; therefore, the optimal amount of lead time compression equals to tmax ; (3) EπC (t) is a convex function in t when α = 0.51 ∈ (0.4782, 0.5350) and the inequality EπC (tmax ) < EπC (0) holds; hence, the optimal amount of lead time compression equals to 0; and (4) EπC (t) is a monotone decreasing function in t when α = 0.54 (> 0.5350); therefore, the optimal amount of lead time compression equals to 0.

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We can find that a critical point αCP exists, which can also be called a cost-different frontier that the optimal decision epoch is positioned in time zero when the inequality α ≥ αCP holds and lies in time tmax when the inequality α ≤ αCP holds. Given the situation where α = αCP , the optimal decision epoch that equals to either at time 0 or at time tmax yields the equivalent expected profit. The value of αCP can be calculated by solving the equation EπC (0) = EπC (tmax ). Parameter αCP changing with tmax is shown in Table 6. Correspondingly, the two-dimensional plane, which consists of per-unit time and marginal cost, is divided into two areas by critical points. The areas are shown in Figure 5. The area A is composed of the set of all the elements of α > αCP , and the area B is composed of the set of the elements of α < αCP . Hence, it is easy to judge that the optimal decision epoch equals to time zero when α belongs to area A and equals to time tmax when α belongs to area B. Table 6. Cost-indifferent frontiers with the reduction of tmax in type two

tmax

αCP

tmax

αCP

tmax

αCP

tmax

αCP

tmax

αCP

0.1 0.2

0.4811 0.4841

0.3 0.4

0.4870 0.4898

0.5 0.6

0.4927 0.4955

0.7 0.8

0.4983 0.5011

0.9 1.0

0.5038 0.5066

1056 α = 0.45

1054

α = 0.48

α = 0.51

α = 0.54

1052

Profits →

1050 1048 1046 1044 1042 1040 1038 1036 0

0.1

0.2

0.3

0.4

0.5 t→

0.6

0.7

0.8

0.9

1

Figure 4. EπC (q∗t , t) changes with t under different α in type two

6.2.2. Supply chain coordination The parameters of the both constructed contracts are designed in type two in this part. The coordination analysis of this part is similar to Section 6.1.2. For a given traditional newsvendor solving, the participants achieve Pareto improving with a coordination contract. That is, both manufacturer and retailer are at least as well off, and one party is better off after adopting the contract. Without loss of generality, it is assumed that the wholesale price w0 equals to seven at decision epoch zero. Hence, in the classical newsvendor environment, the order quantity of the retailer is 200, and the manufacturer and retailer’s profits are 420 and 600, respectively. However, we can see from Table 3 that the integrated supply chain profits are better than the traditional

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0.51 αCP 0.505

α→

0.5

A

0.495

0.49 B 0.485

0.48 0.1

0.2

0.3

0.4

0.5 0.6 t max →

0.7

0.8

0.9

1

Figure 5. Areas of changing the monotonicity of EπC (q∗t , t) in type two

newsvendor’s with different α. The participants have a profit space to achieve Pareto improving according to either proposition 2 or proposition 3. Table 7 shows the variation range for each constructed contract, in which the participants can achieve Pareto improving. The arrowhead in the table indicates the direction of the contract parameter that keeps increasing the retailer’s profit. The negative sign in the table indicates the revenue transfer to the retailer. Figure 6 depicts the corresponding parameter variation range, in which the contract can coordinate the decentralized supply chain when the contract parameter lies in the gray area. Table 7. The coordination parameters with w0 = 7 in type two

α

δ

wt

φt

α

δ

wt

φt

0.1 0.2 0.3 0.4 0.5

0.59→0.65 0.60→0.65 0.60→0.64 0.61→0.64 0.62→0.64

7.49→6.97 7.44→7.02 7.39→7.07 7.35→7.12 7.30→7.17

0.4828→0.4816 0.4797→0.4785 0.4765→0.4754 0.4733→0.4724 0.4700→0.4694

0.6 0.7 0.8 0.9 1.0

0.62→0.64 0.62→0.64 0.62→0.64 0.62→0.64 0.62→ 0.64

6.87→6.76 6.87→6.76 6.87→6.76 6.87→6.76 6.87→6.76

0.4550→0.4544 0.4550→0.4544 0.4550→0.4544 0.4550→0.4544 0.4550→0.4544

6.3. Efficiency analysis Thus, the optimal decisions from the integrated supply chain and the coordination of the decentralized supply chain are analyzed with a non-standard distribution function. Hence, the efficiency of this distribution function is very important to the decision maker. To evaluate the efficiency, we assume that demand Xt has a probability density function Ft ( x ) with mean µ and standard deviation σt and that Ft ( x ) denotes the probability
density function for uniform distribution U ( at , bt ) and for normal distributionN µ, σt2 , respectively, where √ √ at = µ − 3σt , b = µ + 3σt .

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0.66 Coordination area Lower bound Upper Bound

0.65 0.64 δ

0.63 0.62 0.61 0.6 0.59

0.1

0.2

0.3

0.4

0.5

0.6

α

0.7

0.8

0.9

1

wt

(a)

0.485 Coordination area Upper bound Lower Bound

Coordination area Upper bound Lower Bound

0.48 0.475 0.47 t

7.5 7.4 7.3 7.2 7.1 7 6.9 6.8 6.7 0.1 0.2

0.465 0.46 0.455 0.3

0.4

0.5 0.6 α

0.7

0.8

0.9

0.45 0.1 0.2

1

(b)

0.3

0.4

0.5 0.6 α

0.7

0.8

0.9

1

(c)

Figure 6. The area of coordination of contract parameters with some given α in type two Table 8. Performance analysis of non-standard distribution with Uniform and Normal

Type One Mean Max. Min. Mean square of Residual

Type Two

Uniform

Normal

Uniform

Normal

0.9767 0.9889 0.9696 4.4465e-005

0.9769 0.9901 0.9694 5.0319e-005

0.9735 0.9890 0.9640 1.3106e-004

0.9735 0.9901 0.9637 1.4457e-004

Gallego and Moon (1993) suggested a framework to calibrate the effectiveness of the non-standard distribution function of the newsvendor problem. A performance measure, the expected value of additional information (EVAI), is introduced, which is determined by considering the difference in the optimal revenue when demand distribution is perfectly  known and the optimal revenue generated by the distribution free ∗ ∗ ˆ approach, that is, EVAI = EπC qˆtˆ∗ , t − EπC (q∗t∗ , t∗ ). A similar performance measure that is often used   is Ratio = EπC (q∗t∗ , t∗ )/EπC qˆ∗tˆ∗ , tˆ∗ . It is the relative measure of revenue that a firm yields operating at non-standard control parameters (q∗t∗ , t∗ ) to the corresponding revenue at the optimal control parameters   qˆ∗tˆ∗ , tˆ∗ when demand distribution is perfectly known. Gallego and Moon (1993) first introduced this performance measure and it has been used in many related studies to calibrate the performance of the distribution-free approach, which is also used in this paper. In this paper, 100 samples within α ∈ (0, 2) in type one and α ∈ (0, 1) in type two are generated following the uniform distribution and normal distribution approach proposed by Gallego and Moon (1993), respectively.

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Table 8 summarizes the results of the numerical experiment. By knowing the demand distribution, a supply chain can achieve an additional mean revenue gain, which is about 2.33 and 2.31% in type one and about 2.65 and 2.65% in type two, if unknown demand distribution may have followed uniform, normal distribution, respectively, relative to the corresponding optimal revenues when demand distribution is perfectly known. The minimum value gained in the supply chain can yield the knowledge of demand behavior, which follows uniform and normal distribution. These gains are 3.04 and 3.06 % in type one and 3.60 and 3.63 % in type two. In summary, the efficiency of the non-standard distribution can be sensitive to the unknown demand distribution. Moreover, the numerical experimentations have provided support that the investigation of non-standard distribution is robust in this context, and has a significant implication for real life business decisions in revenue management.

7. Conclusions In practice, to allure retailers to submit their orders as soon as possible, manufacturers usually launch a time-sensitivity promotional mechanism before the selling season, in which the earlier you order, the cheaper the wholesale price will be. However, the mechanism brings a decisional confliction between the forecast risk and the production cost. Hence, the purpose of this paper is to investigate how to solve the confliction to achieve a win-win outcome for the participants. The conclusions of our investigation are as follows. First, a benchmark model that determinates the optimal decision epoch and optimal production quantity is proposed to provide a criterion-referenced for coordinating a dyadic decentralized supply chain. In the model, two different types of production cost curves proposed by Jian et al. (2015) are discussed under a non-standard distribution environment. The investigation shows that only a set of optimal decisions exists, which can balance the forecast risk and production cost. Second, a TsWP contract is constructed to confirm the promotional mechanism released by the manufacturer that coordinates the decentralized supply chain and achieves a win-win outcome for participants. The TsWP contract can resolve the decision confliction mentioned above. However, we find that the retailer bears the entire forecast risk under the TsWP contract, and cooperation between the participants is interrupted when the demand fluctuation becomes enormous and unstable. Particularly, the retailer will refuse to join the supply chain when the predicted demand is highly volatile and the TsWP contract is adopted. Third, to overcome the drawback of the TsWP contract, an improved contract, defined as TsRS contract, is constructed based on the notion that the retailer shares partial forecast risk caused by purchasing later. This research shows that the participants can arbitrarily divide the optimal profit of an integrated supply chain under any constructed contracts. Finally, the preference of the two constructed contracts is discussed. The contracts are mutually equivalent from the perspective of expected profit, but have two differences. One is the means of sharing forecast risk and the other is the different ways of charging procurement price. Therefore, the manufacturer most probably selects the TsWP contract to coordinate the decentralized supply chain, especially when the volatility of demand is smaller. For a higher forecast risk, the manufacturer could select the TsRS contract to stimulate the retailer to take part in the supply chain and obtain a higher expected profit. However, the retailer prefers to choose the TsRS contract regardless of the high or low forecast risk.

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Our findings provide a helpful guideline and decision-making tool for firms who are game leaders. They can learn from the finding when and how incorporation of the non-standard distribution environment ensues. For example, as a game leader, the manufacturer should first find the optimal decision of the integrated supply chain, and then “shoot the retailer” by charging an infinitely high wholesale price or revenue sharing ratio, unless the retailer orders at the desire decision epoch, then the retailer will do that with either of the constructed contract. There are some expandable assumptions to enrich the paper further, such as taking other forms of demand variance and/or inventory cost into account, which form part of our future research. Acknowledgments: This work is supported by Research Project of Humanities and Social Science of Ministry of Education (Grant No. 18YJC630030). The Key Projects of The National Social Science Foundation (Grant No. 17AGL007). Project of Chongqing Science and Technology Bureau (Grant No. cstc2018jsyj-jsyjX0014), Chongqing Engineering Research Center for Processing, Storage and Transportation of Characterized Agro-Products (Grant No. KFJJ2016026). Conflicts of Interest: The authors declare no conflict of interest.

Appendix A Proof for Proposition 1. Proof. It is deduced that Υt > 0 according to 0 ≤ Rt < 0.5, assumption 1 and assumption 2, immediately. The sign of Ψt depends on algebraic operation. Furthermore, the first and second-order partial derivative of EπC (t) on t is dEπC (t)/dt = Ψt and d2 EπC (t)/dt2 = Υt − q∗t c00t , respectively. A1. We can deduce that d2 EπC (t)/dt2 ≥ 0and if either c00t ≤ 0 or both c00t > 0 and Υt − q∗t c00t ≥ 0 holds and further deduce that the function dEπC (t)/dt is a non-decreasing function on time t. Therefore, the optimal decision epoch t∗ has the following three situations. A1.1. If lim Ψt ≥ 0 holds, then EπC (t) is a non-decreasing function on time t, so the optimal decision t →0+

epoch t∗ satisfies that t∗ = tmax . A1.2. If lim Ψt ≤ 0 holds, then EπC (t) is a non-increasing function on time t, so the optimal decision − t→tmax

t∗

epoch satisfies that t∗ = 0. A1.3. If lim Ψt · lim Ψt < 0 holds, then EπC (t) is a convex function on time t, so the optimal decision t →0+

− t→tmax

epoch t∗ satisfies that t∗ ∈ {0, tmax }. A2. We can deduce that d2 EπC (t)/dt2 < 0 if both c00t > 0 and Υt − q∗t c00t < 0 hold, and further deduce that the function dEπC (t)/dt is a non-increasing function on time t. Therefore, the optimal decision epoch t∗ has the following two situations. A2.1. If lim Ψt ≥ 0 holds, then EπC (t) is a non-decreasing function on time t, so the optimal decision − t→tmax

epoch t∗ satisfies that t∗ = tmax . A2.2. If lim Ψt < 0 holds, then EπC (t) is a concave function on time t, so the optimal decision epoch t∗ − t→tmax

satisfies that dEπC (t∗ )/dt = 0. Proved.

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