The effect of decision rights allocation on a supply chain of perishable products under a revenue-sharing contract

Int. J. Production Economics xxx (xxxx) xxx

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International Journal of Production Economics journal homepage: http://www.elsevier.com/locate/ijpe

The effect of decision rights allocation on a supply chain of perishable products under a revenue-sharing contract Tal Avinadav Department of Management, Bar-Ilan University, Ramat-Gan, 5290002, Israel

A R T I C L E I N F O

A B S T R A C T

Keywords: Perishable products EOQ Supply chain Revenue sharing contract Decision rights allocation

Using the framework of an economic order quantity (EOQ) model, we study marketing and operational decisions in a two-echelon supply chain in which a retailer and a manufacturer use a revenue-sharing contract to sell a perishable product. The demand function is sensitive to price, sales effort and the age of the product on shelf. We take into account the issue of decision rights allocation with respect to both sales-effort investment and replenishment policy. In particular, we investigate and compare the performance of the individual parties and of the total supply chain across six scenarios: three in which the investment in sales effort is made by the retailer, and three in which the investment in sales effort is made by the manufacturer, where in each group the cycle length is set either by the retailer, by the manufacturer or in a cooperative manner. We show that when one party determines the cycle length, there are certain conditions under which a two-part tariff contract can be used by the other party in order to influence the cycle length decision, and thus to increase its profits. In addition, we show that there are cases in which both parties benefit when the manufacturer is responsible for investing in sales effort, such that it is in the retailer’s interest to give up her decision right in this regard.

1. Introduction A perishable product can be defined as any product that either de­ teriorates or becomes obsolete in the course of time. This definition encompasses a large variety of products, including vegetables and fruits, bread, fresh meat and fish, dairy products, beverages, industrial food, ammunition, batteries, printer ink, medical equipment, blooddonations, drugs, and more. In many cases, product freshness is an important consideration in consumers’ purchase decisions with regard to perishable products. In particular, Sarker et al. (1997) claim that consumers tend to feel less confident purchasing perishable products whose expiration dates are approaching. Consumers’ preference for fresher products can have serious impli­ cations for retailers, given that a perishable product cannot be sold once its expiration date passes, and the retailer must absorb the loss (see, e.g., Donselaar and Broekmeulen, 2012). Such loss can be substantial: For example, according to the United States Department of Agriculture (USDA), food waste in 2010 was estimated at 30–40 percent of the food supply, which corresponds to approximately 133 billion pounds of food and 161 billion dollars (https://www.usda.gov/oce/foodwaste/faqs. htm). Global trends indicate a further increase of waste in the future, which can be explained by the increasing number of supermarkets, as

well as the growth of household incomes, which increases the demand for perishable products (Parfitt et al., 2010). Therefore, it is essential to consider the negative effect of products’ age on demand when making operational and marketing decisions within a supply chain of perishable products. Accordingly, herein, we use the framework of an economic order quantity (EOQ) model to study inventory and marketing decisions of a manufacturer and a retailer who produce and sell a perishable product. We assume that the demand function is sensitive to price, to the age of the product on the shelf, and to the level of effort invested (by either party) in the marketing of the product (‘sales effort’). The important innovation of this study is in using a game theoretic approach to extend the retailer’s EOQ optimization problem to consider interactions within a two-echelon supply chain. Specifically, we investigate a scenario in which the retailer and the manufacturer interact via a revenue-sharing contract in which the two parties decide how to split the total future revenue from selling a product. In recent years, this type of contract has become popular in numerous industries (Avinadav et al., 2015a,b; Avinadav et al., 2017a,b; Avinadav et al., 2019; Chernonog, 2020), most prominently in online commerce platforms (e.g., web stores at Amazon. com, Alibaba.com, and eBay.com) and in distribution platforms for virtual products (e.g., Google Play and Apple iTunes). Revenue-sharing

E-mail address: [email protected]. https://doi.org/10.1016/j.ijpe.2019.107587 Received 10 October 2018; Received in revised form 13 August 2019; Accepted 13 December 2019 Available online 18 December 2019 0925-5273/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Tal Avinadav, Int. J. Production Economics, https://doi.org/10.1016/j.ijpe.2019.107587

T. Avinadav

International Journal of Production Economics xxx (xxxx) xxx

contracts are also applied in retailing with consignment contracting; for example, the Sara Lee Corporation adopted a consignment marketing channel for the sale of L’eggs pantyhose, and many supermarkets use consignment contracts when selling direct-store-delivery products such as dairy products, baked goods, soft drinks, and so on (Chen et al., 2011). Our model assumes that, in establishing the terms of the contract, the manufacturer and the retailer must determine the price of the product and their respective revenue shares, the duration of the inventory cycle, and the level of sales effort to be invested. We consider the possibility that responsibility for the latter two decisions might be allocated in different ways across the two parties; this approach constitutes another key contribution of our study. In particular, we evaluate and compare the performance of the individual parties and of the total supply chain across six scenarios: three in which the sales effort is the responsibility of the retailer (retailer-investor model) and three in which the sales effort is the responsibility of the manufacturer (manufacturer-investor model), where in each scenario the cycle length is set either by the retailer, by the manufacturer or in a cooperative manner. Our investigation of the cooperative decision with regard to the cycle length is motivated by the concept of Collaborative Planning, Forecasting and Replenishment (CPFR), a commonly-used operational approach wherein manufac­ turers, distributors, and retailers work together to develop and imple­ ment inventory management processes (see an example of an application of this concept within a supply chain of a perishable product in Fang Du et al., 2009). In particular, we show that under the above framework:

demand), and thus, the ordering policy should take it into account. Examples of studies that incorporate this consideration include: (i) the works of Haijema and Minner (2016, 2019), who studied stock-age-dependent order policies and compared them with stock-level-dependent order policies for perishable items with a deter­ ministic lifetime and a deterministic lead-time; (ii) the works of Kouki et al. (2016), and Kouki et al. (2018), who studied ordering policies for continuous review systems of perishables with random lifetimes, posi­ tive lead-times, backordered demands or lost sales. The issue of age-dependent demand in an EOQ framework, which is at the core of the current research, has been discussed in several studies in the inventory literature. Avinadav and Arponen (2009) analyzed the properties of the profit function for a case in which the demand rate is polynomial in the remaining time-to-expiry. Avinadav et al. (2013) extended the latter model to include a linear price effect on the demand rate, and analyzed the sensitivity of the optimal solution to the model parameters. Maihami and Nakhai Kamalabadi (2012) investigated a model where the demand rate is linear in price and exponential in the product’s age with partial backlogging. Valliathal and Uthayakumar (2011) used a similar model with general price and age effects on de­ mand. Avinadav et al. (2014) extended the study of Avinadav et al. (2013) by considering general price and age effects with multiplicative and additive demand forms. Avinadav, Chernonog, Lahav et al. (2017)a, b considered dynamic decisions of a retailer who seeks to determine the selling price and promotion expenditures associated with a perishable product, as well as to set the order quantity and the inter-replenishment time. Dobson et al. (2017) took into account consumers’ assessment of a product’s quality over the course of its lifetime, and assumed that the demand rate is a linearly decreasing function of the age of the product. Feng et al. (2017) proposed an inventory model that stipulates the de­ mand explicitly in a multivariate function of price, freshness and dis­ played stocks. All these studies addressed the case of a single decision-maker. The innovation of our study in this domain is in tak­ ing into account the interaction between two parties in a supply chain, using a game theoretic approach.

1. By using a cooperative decision about the cycle length with a sidepayment, Pareto improvement can be achieved (as compared with the case in which a single party determines the cycle length). 2. When one party determines the cycle length, a two-part tariff con­ tract (revenue share plus fixed ordering cost or fixed subsidy per order) can be used by the other party in order to influence the cycle length decision, and thus to increase its profits. 3. For a given cycle length, the retailer takes a larger revenue share for herself when she is responsible for investing in sales effort than when she is not. To compensate, the manufacturer sets a higher selling price under retailer-investor models than he does under manufacturer-investor models. 4. There are cases in which both parties benefit when the investment in sales effort is borne solely by the manufacturer, so it is in the re­ tailer’s interest to give up her decision right.

2.2. Decision rights allocation Decision rights allocation is a known mechanism for improving ef­ ficiency in supply chains (Feng and Zhang, 2014). An example of a de­ cision rights allocation arrangement is a vendor-managed inventory (VMI) partnership, in which a retailer might determine her order quantity but delegate other replenishment decisions—e.g., the epochs at which inventory is replenished—to the supplier (Bernstein et al., 2006). Another example is given by Ru and Wang (2010), who addressed the question: “Who should control inventory in the supply chain?” The au­ thors concluded that it is beneficial both to the supplier and to the retailer when the inventory decision is delegated to the supplier rather than to the retailer in the channel. Studies considering decision rights allocation include the work of Dong and Zhu (2007), who identified optimal decision rights allocation policies in a scenario in which a retailer is offered two opportunities to order inventory from a manufacturer (one in advance of the selling season and one after demand is observed). Other studies have consid­ ered cases in which delegation of power to make decisions entails cost allocation (Corbett, 2001). For example, Jin et al. (2015) considered two popular contract types between a manufacturer and a retailer (a wholesale price contract and a consignment contract with revenue sharing) and investigated the implications of assigning the responsibility (and costs) for sales promotion to each party. They showed that there are only two cases in which both parties benefit: the combination 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. Chernonog and Avinadav (2019) considered three wholesale price contracts, in which responsibility for investing in advertising of a perishable product is borne, respectively, by the manufacturer, the

2. Literature review Previous research related to this paper can be broadly grouped into three categories: product perishability, decision rights allocation, and revenue-sharing contracts. We discuss each of these categories in what follows. 2.1. Product perishability The topic of product perishability has been studied extensively in the operations management and production economics literature (see sur­ veys of Nahmias, 1982; Weatherford and Bodily, 1992; Gallego and van Ryzin, 1994; Goyal and Giri, 2001; and recently of Karaesmen et al., 2011; Bakker et al., 2012). Perishable products garner academic interest because of their specific attributes and their increasing market share (see recent studies in Chen et al., 2014; Herbon et al., 2014; Herbon, 2017; and Chua et al., 2017). Though in some cases consumers are indifferent to the age of the product, as long as it has not yet expired (see, for example, Chung and Erhun, 2013; Li et al., 2017), there are many cases in which the product decays and its quality deteriorates, during lead time or while the product is on the shelf (see, for example, de Keizer et al., 2017). In the latter cases, demand might be affected by the products’ freshness (First-In-First-Out demand versus Last-In-First-Out 2

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retailer, or both. To our knowledge, the latter paper was the first to combine decision rights allocation with product perishability. The cur­ rent study also combines these issues, but instead of using a wholesale price contract, we investigate a revenue-sharing contract. Moreover, in the above studies, the decision rights refer to a single decision, whereas in this study we analyze decision rights with respect to two decisions (the investment in sales effort and setting the cycle length).

through a bargaining process. In line with many works in the domain of perishables (You, 2005; Tsao and Sheen, 2008; Avinadav and Arponen, 2009; Minner and Transchel, 2010; Avinadav et al., 2013, 2014; Kouki et al., 2015; Avi­ nadav et al., 2017a,b), we assume that backlogging of unmet demand is not allowed. This assumption is appropriate for the context of perishable products because perishable items are usually purchased to satisfy im­ mediate needs of the consumer, and the likelihood that consumers are willing to wait for replenishment is low. This means that shortages either lead to loss of sales, or, in cases in which a retailer stocks multiple comparable brands (e.g., milk of competitive dairy producers), the customer will simply purchase a similar product by a different brand, such that shortage of a single brand followed by backlogging is less likely to occur (see, e.g., Krishna, 1992, p. 268). Further, for the purpose of this study, we assume a deterministic demand function. Although in the real world demand can almost never be predicted with certainty, deterministic models are acknowledged as being useful in examining certain aspects of real-world problems (according to Hadley and Whitin, 1963, p. 29, “the results obtained from these models yield, qualitatively, the proper sort of behavior—even when the deterministic demand assumption is removed”). Given that demand is deterministic, an optimal cycle length has to be smaller than the shelf life of the product, such that products do not perish within the cycle (see Corollary 1 in Avinadav et al., 2013). According to Proposition 1 of Avinadav et al. (2014), the assumptions above imply that the optimal time for replen­ ishment is exactly when inventory is exhausted to avoid leftovers and shortages. Consequently, under the optimal ordering policy, the elapsed time measured from the most recent replenishment indicates the in­ ventory age; all units in stock have exactly the same age; and the re­ tailer’s order quantity for given values of T, p and s is equal to the total RT quantity demanded during the entire cycle, 0 λðp; s; tÞdt. Four cost types are considered in the supply chain: a constant holding cost rate per unit, h; a unit production cost, c; an investment in sales effort per cycle, sT, and the parties’ fixed order costs per cycle, Ki, i 2 fR; Mg. The holding cost is paid by the retailer, whereas the unit production cost is paid by the manufacturer. In what follows, we investigate two scenarios that differ in the identity of the party who invests in sales effort. To provide a unified model formulation for the two scenarios, we introduce a binary variable B, which equals 1 if the retailer invests in sales effort, and 0 if the manufacturer invests in sales effort. Conse­ quently, the retailer’s and manufacturer’s profits per cycle under the optimal ordering policy are, respectively, Z T ððð1 ηÞp htÞλðp; s; tÞÞdt (1) Π R ¼ KR BsT þ

2.3. Revenue-sharing contracts In recent years, revenue-sharing contracts have been gaining popu­ larity in practice and, consequently, in research. One appealing char­ acteristic of this type of contract is that it offers the ability to coordinate the supply chain, a property studied by Wang et al. (2004), Cachon and Lariviere (2005), Li and Hua (2008), and Zhang et al. (2010). Additional studies in this vein include the work of Hu et al. (2017), who investi­ gated supply chain decisions and performance in a one-retailer and two-manufacturer supply chain incorporating both revenue-sharing contracts and product substitution. Hu and Feng (2017) considered a revenue-sharing contract in a supply chain comprising a single supplier and a single buyer, where uncertainty occurs both on the demand side and on the supply side, and found the parties’ optimal policies and conditions for coordination. Cai et al. (2017) analyzed a VMI supply chain with customers who are sensitive to service-level. They estab­ lished a dynamic game relationship under a revenue-sharing contract and proposed three flexible subsidy contracts to obtain coordination. Becker-Peth and Thonemann (2016) analyzed behavioral aspects of revenue-sharing contracts. They extended the classical normative deci­ sion model by incorporating reference-dependent valuation into the decision model and showed how this valuation affects inventory de­ cisions. The current study expands the literature on revenue-sharing contracts by combining it with the two streams discussed above, namely, product perishability and decision rights allocation, within an EOQ framework. 3. Model formulation and preliminary results 3.1. Model formulation Consider a two-echelon supply chain consisting of a single manu­ facturer with ample production capacity (“he”, denoted by subscript M) and a single retailer (“she”, denoted by subscript R), who are producing and selling a perishable product. The parties interact by using a revenuesharing contract in which the retailer is the leader and the manufacturer is the follower. Specifically, under this contract, the retailer offers the manufacturer a revenue share η, and then the manufacturer sets the product’s selling price p. The sales-effort investment rate, s, which ac­ celerates demand, can be determined either by the manufacturer or by the retailer. The demand rate, denoted by λðp; s; tÞ, is affected by three factors: the selling price, the sales-effort investment rate, and the age of the product. Whereas the first two factors are directly controlled by the parties, the last factor is only partially controlled, since the parties can limit the maximal age of sold units by dictating a cycle length, T. In­ ventory is replenished according to an EOQ ordering regime, according to which an order of fixed size is made at every fixed time interval (i.e., after a time duration of T). In line with prior EOQ models of perishable items both under deterministic demand (Avinadav and Arponen, 2009; Valliathal and Uthayakumar 2011; Maihami and Nakhai Kamalabadi 2012; Avinadav et al., 2013, 2014; Avinadav et al., 2017a,b; Chernonog, 2020) and under stochastic demand (Kouki et al., 2015), we assume that the age of an item at delivery time is normalized to zero (i.e., newly-arrived stock is fresh). In what follows, we investigate three scenarios that differ in the way that T is determined: (i) the retailer is exclusively responsible for determining the cycle length; (ii) the manufacturer is exclusively responsible for determining the cycle length; and (iii) the two parties determine the cycle length jointly

0

Z ΠM ¼

KM

ð1

BÞsT þ ðηp

cÞ

T

λðp; s; tÞdt: 0

(2)

We further define the long-run average profit rates of the parties:

πR � Π R =T and πM � Π M =T, which are the objectives to be maximized by

each party. According to Pekgün et al. (2017), in order to increase profitability, a firm should take a global perspective and coordinate its marketing and operational decisions rather than make each type of decision indepen­ dently. In reality, however, different divisions of large companies often fail to communicate on important business decisions (Chatterjee et al., 2002; Hausman et al., 2002). In fact, it is not uncommon for actual conflict to arise between the marketing and operations functions of a firm because of misalignment of goals (Sardana et al., 2016). Thus, it is likely that, in practice, a firm’s decisions regarding the cycle length—a logistical aspect determined by the operations function—are made separately from decisions regarding pricing and sales effort, which are the jurisdiction of the firm’s marketing function. Accordingly, we as­ sume that, in cases in which a given firm is (fully or partially) respon­ sible for determining the cycle length, it makes this decision prior to

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making its other marketing decisions. As will be elaborated below (see sections 4.1 and 5.1), this chosen sequence of decisions enables us to capture a common practice of periodic inventory control (see, e.g., Porteus, 1990; Herbon, 2014; Avinadav and Henig, 2015; Minner and Transchel, 2010, 2017), in which the cycle length is dictated exoge­ nously. This is the case, for example, in supermarkets that receive multiple products in a unified delivery, such that for certain products in the set the delivery period is effectively arbitrary rather than determined through an optimization process. We leave the opposite order of de­ cisions, as well as simultaneous decisions, for future research. Table 1 depicts the sequence of decisions in the six investigated models, which differ in: (i) the identity of the investor in sales effort; and (ii) the identity of the decision maker with regard to the cycle length. Since all these models follow sequential games, equilibrium is obtained by backward induction. Note that under each of the six models, the cycle length is determined in the first step of the game (either by the retailer and by the manufacturer or as a cooperative decision), and thus is investigated last. Moreover, Models 1–3 (retailer-investor, denoted by the superscript RI), as well as Models 4–6 (manufacturer-investor, denoted by the superscript MI), share the same solution methodology with respect to finding conditional equilibrium for a given T, ðsE ðTÞ ; ηE ðTÞ ; pE ðTÞÞ, where E2{RI,MI} denotes the equilibrium for each in­ vestor’s identity. We introduce the three objective functions according to which the equilibrium cycle length is extracted. According to Model 1 or Model 4, the retailer sets T to maximize her long-run average profit rate at con­ ditional equilibrium, � π ER ðTÞ � Π ER ðTÞ T: (3)

decisions (price and sales-effort), and g(t) is the effect of the product’s age on the demand. It is clear that f (p,s) is strictly decreasing in p and non-decreasing in s. The perishability of the product implies that de­ mand tends to decline as product units age (due to perceived value reduction from the consumers’ perspective), so g(t) is a non-increasing function of t.1 RT For simplicity of presentation, let GðTÞ � 0 gðtÞdt be the age effect on the demand over the entire cycle, which is an increasing function of RT T; let τðTÞ � 0 tgðtÞdt =GðTÞ be the average time a unit spends on the shelf before being sold; and let gðTÞ � GðTÞ=T be the average age effect on the demand rate over the entire cycle.2 3.2. Preliminary results Lemma 1. function. Proof.

τðT1 Þ ¼

τðT1 Þ R T2

¼

0

R T1 0

T2 T1 T1 gðtÞdt

�

R T1 RT RT gðtÞdt þ T12 τðT1 ÞgðtÞdt tgðtÞdt þ T12 tgðtÞdt 0 � R T2 GðT2 Þ gðtÞdt 0

tgðtÞdt ¼ τðT2 Þ GðT2 Þ

(ii) Since g(t) is a non-increasing function, then RT GðT1 Þ þ T12 gðtÞdt T1 gðT1 Þ þ ðT2 T1 ÞgðT1 Þ gðT2 Þ ¼ � T2 T2 T1 gðT1 Þ þ ðT2 T1 ÞgðT1 Þ � ¼ gðT1 Þ: T2

Finally, according to Model 3 or Model 6, the manufacturer and the retailer make a cooperative decision about T to maximize �μ �1 μ J E ðTÞ � π ER ðTÞ ⋅ πEM ðTÞ ; (5)

By Lemma 1, the average amount of time a unit spends on the shelf before being sold is greater when the cycle is longer, as expected. By Lemma 1(ii), the average sales rate from the beginning of the cycle decreases (or at least not increases) when the cycle length increases. The following preliminary results are used in the following sections to compare the equilibria across Models 1–3 and across Models 4–6. Let XðTÞ and YðTÞ be functions of T, where TX � argmaxfXðTÞg, TY �

where μ 2 ½0; 1� is the bargaining power of the retailer according to the asymmetric Nash bargaining approach. We adopt the multiplicative demand form, which is commonly used in the literature (e.g., Aust and Buscher, 2012; Huang et al., 2013; Jin et al., 2015; Chernonog et al., 2015; Avinadav et al., 2015a,b, Cherno­ nog and Avinadav, 2019),

T

argmaxfYðTÞg and TB � argmaxfðXðTÞÞμ ðYðTÞÞ1 μ g with μ 2 ½0; 1�.

(6)

T

T

Lemma 2. If XðTÞ and YðTÞ are quasi-concave functions then minðTX ;TY Þ � TB � maxðTX ; TY Þ:

where f (p,s) is the demand component affected by the marketing

Proof. For T < minðTX ; TY Þ both XðTÞ and YðTÞ are increasing func­ tions, and for T > maxðTX ; TY Þ both XðTÞ and YðTÞ are decreasing func­

Table 1 Sequence of decisions according to the various models. Sales effort

Let T1 < T2 .

RT RT R TgðtÞdt (i) Since τðTÞ � 0 GðTÞ ¼ T, then T12 τðT1 ÞgðtÞdt � R T2 T1 tgðtÞ dt, implying

According to Model 2 or Model 5, the manufacturer sets T to maxi­ mize his long-run average profit rate at conditional equilibrium, � π EM ðTÞ � Π EM ðTÞ T: (4)

λðp; s; tÞ ¼ f ðp; sÞ⋅gðtÞ;

(i) τðTÞ is an increasing function; (ii) gðTÞ is a non-increasing

tions. The claim is proved, since ðXðTÞÞμ ðYðTÞÞ1 μ is monotone increasing in XðTÞ and YðTÞ. By Lemma 2, Models 1–3 and Models 4–6, each within its group, can be ranked according to cycle length, as stated in the following Theorem.

Cycle length Retailer sets

Manufacturer sets

Cooperative decision

Retailer Investor (RI), B¼1

Model 1 1. Retailer sets T, η and s 2. Manufacturer sets p

Manufacturer Investor (MI), B¼0

Model 4 1. Retailer sets T and η 2. Manufacturer sets p and s

Model 2 1. Manufacturer sets T 2. Retailer sets η and s 3. Manufacturer sets p Model 5 1. Manufacturer sets T 2. Retailer sets η 3. Manufacturer sets p and s

Model 3 1. Parties bargain for T 2. Retailer sets η and s 3. Manufacturer sets p Model 6 1. Parties bargain for T 2. Retailer sets η 3. Manufacturer sets p and s

Theorem 1. If πER ðTÞ and π EM ðTÞ , E2{RI,MI}, are quasi-concave func­ tions, then the equilibrium cycle length under a cooperative decision is in 1 Our model, which is primarily developed for perishable products with age dependent demand, captures also the case of non-perishable products by setting gðtÞ ¼ 1; 8t. Therefore, the results in this paper are also valid for nonperishable products. 2 For non-perishable products (i.e.gðtÞ ¼ 1; 8t) , according to (6), demand is independent of the time elapsed from the beginning of the cycle and equals λðp; s; tÞ ¼ f ðp; sÞ. In such a case, gðtÞ ¼ 1, GðTÞ ¼ T and τðTÞ ¼ T=2; 8t.

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T. Avinadav

between those obtained when either the retailer or the manufacturer makes the decision alone.

4. Retailer-investor models For the retailer-investor models, the solution methodology proceeds according to the following two stages: First, we assume that T is given and find conditional equilibrium values for the remaining decision variables. As discussed in Section 3.1, these values are identical across Models 1–3. Then, we find the equilibrium value of T separately for each model; these values are dependent on the identity of the party respon­ sible for determining T (manufacturer, retailer, or both). Because it is too complex to obtain these values analytically, we use numerical ex­ amples to characterize the supply chain at equilibrium.

Proof. Let Tj be the cycle length under Model j 2 f1; …; 6g at equi­ librium. By Lemma 2, (i) minfT 1 ; T 2 g � T 3 � maxfT 1 ; T2 g; (ii) minfT4 ; T 5 g � T 6 � maxfT 4 ; T5 g. n o Let Ti � argmax ðXðTÞÞμi ðYðTÞÞ1 μi ; i 2 f1; 2g T

Lemma 3. Proof.

If 0 � μ1 � μ2 � 1 then XðT1 Þ � XðT2 Þ and YðT1 Þ � YðT2 Þ .

By the definitions of T1 and T2 , μ1

1 μ1

μ1

ðXðT1 ÞÞ ðYðT1 ÞÞ

4.1. Conditional equilibrium for a given T (7)

1 μ1

� ðXðT2 ÞÞ ðYðT2 ÞÞ

The following optimization problem is defined to obtain the condi­ tional equilibrium values of η, s and p for a given value of T: 9 8 ZT = < max Π R ¼ KR sT þ ððð1 ηÞpðη; s; TÞ htÞλðpðη; s; TÞ; s; tÞÞdt ; ηðTÞ;sðTÞ : 80 9 T Z < = S:t: pðη; s; TÞ ¼ argmax Π M ¼ KM þ ðηp cÞ λðp; s; tÞdt : ; p

and ðXðT2 ÞÞμ2 ðYðT2 ÞÞ1

μ2

� ðXðT1 ÞÞμ2 ðYðT1 ÞÞ1

μ2

(8)

:

By multiplying the above two inequalities and using algebraic ma­ nipulations, we get: ðXðT2 ÞÞμ2

μ1

ðYðT1 ÞÞμ2

μ1

� ðXðT1 ÞÞμ2

μ1

ðYðT2 ÞÞμ2

μ1

; which implies

XðT2 Þ XðT1 Þ � : (9) YðT2 Þ YðT1 Þ �μ1 � �μ1 � XðT1 Þ XðT2 Þ YðT2 Þ � ; so, combined with (9), we get From (7), YðT YðT2 Þ YðT1 Þ 1Þ YðT2 Þ YðT1 Þ

� 1.

�

From (8), get

XðT2 Þ XðT1 Þ

YðT1 Þ XðT1 Þ

�1

�

μ2

�

YðT2 Þ XðT2 Þ

�1

μ2

XðT2 Þ XðT1 Þ;

(11) For simplicity of presentation and analysis, let ΦðTÞ � Proposition 1. c ΦðTÞ

so, combined with (9), we

� 1.

Straightforward from Lemma 3.

j

Let πi be the profit of party i 2 fR; Mg under Model j 2 f1; …; 6g at equilibrium. Corollary 1. (i) π 1R � π3R � π2R and π2M � π3M � π1M ; (ii) π4R � π6R � π5R and π 5M � π 6M � π4M .

T ¼

�α 1 !1 1 β � �� 1 α 1 and pRI ðTÞ ¼ 1ΦðTÞ αg T ΦðTÞ α 1.

aβ

Corollary 2. (i) ηRI ðTÞ and sRI ðTÞ are decreasing functions, whereas pRI ðTÞ is an increasing function; (ii) the manufacturer’s revenue from selling a unit depends only on the product cost and the price elasticity and equals c=ð1 α 1 Þ.

Proof. (i) Straightforward from Theorem 2, since π1R ¼ π3R jμ¼1 and π2R ¼ π3R jμ¼0 ; π2M ¼ π3M jμ¼0 and π1M ¼ π 3M jμ¼1 ; (ii) similarly to (i). Corollary 1 compares the profits of a given party across different levels of his or her involvement in setting the cycle length, while keeping the identity of the investor in sales effort constant. Yet it is also inter­ esting to compare the profits of each party across scenarios in which the parties’ involvement in setting the cycle length is kept constant, whereas the identity of the investor varies. However, the latter comparison re­ quires additional assumptions about the demand and further analysis. In line with many earlier studies in supply chain management liter­ ature (e.g., Guiomar et al., 2006; Chen et al., 2011; Wang et al., 2012; Wang et al., 2013), we model fðp; sÞ as an iso-elastic function of both p and s in a Cobb-Douglas form: f ðp; sÞ ¼ ap s :

,s

�

At conditional equilibrium for a given value of T, ηRI ðTÞ ¼

a maximum. Substituting pðη; s; TÞ in the objective function in (11) and solving the maximization problem results in a unique point that satisfies the necessary condition. Since this point, denoted by ðηRI ðTÞ;sRI ðTÞÞ, also satisfies the sufficient condition, it is the conditional equilibrium for a given T. Then, the equilibrium selling price is obtained by pRI ðTÞ ¼ pðηRI ; sRI ; TÞ. By Proposition 1, whereas the price elasticity (α) affects the equi­ librium values of each of the decision variables p, η and s, the sales-effort elasticity (β) affects only s.

Theorem 2. For a given party that invests in sales effort, the more it is involved in setting the cycle length (i.e., has higher bargaining power), the higher its profit is.

α β

RI

αcþðα 1ÞhτðTÞ . α 1

Proof. First, we calculate the manufacturer’s best response by solving the first constraint in (11). The necessary condition yields a unique so­ lution pðη;s;TÞ ¼ ηð1 cα 1 Þ, which also satisfies the sufficient condition for

By Lemma 3, Models 1–3 and Models 4–6, each within its group, can be ranked according to the gain created for each supply chain member, as stated in the following Theorem.

Proof.

0

0<η<1

Proof. (i) The claims are proved since, by Lemma 1(i), ΦðTÞ is an increasing function and, by Lemma 1(ii), gðTÞ is a non-increasing func­ tion. (ii) The manufacturer’s revenue from selling a unit is ηRI ðTÞpRI ðTÞ ¼ c=ð1 α 1 Þ. 4.2. Setting the cycle length T The next step is to find the equilibrium value of T for Models 1–3, i.e., the value of T that maximizes either (3), (4) or (5), respectively. RI Proposition 2. If πRI R ðTÞ and π M ðTÞ are quasi-concave functions, then the equilibrium values of the decision variables under Model 3 are between those obtained under Models 1 and 2.

(10)

where a (>0) is the base demand, α (>1) is price elasticity, and β (2(0,1)) is sales-effort elasticity.In what follows we proceed to analyze the three retailer-investor models (1–3) and the three manufacturerinvestor models (4–6).

Proof. By Theorem 1, the equilibrium value of T under Model 3 is between the equilibrium values of T obtained under Model 1 and Model 2. The claim is proved, since, by Corollary 2, ηRI ðTÞ, pRI ðTÞ and sRI ðTÞ are monotone in T. 5

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International Journal of Production Economics xxx (xxxx) xxx

Since the objectives in (3)–(5) are complicated for analytical inves­ tigation in T, we use numerical examples to provide a conjecture on the behavior of each function. To do so, we use parameter values that are close to those in Avinadav et al. (2014): a ¼ 1000, α ¼ 2, β ¼ 0:5, h ¼ 0:1, c ¼ 1; KM ¼ 300, KR ¼ 500 and μ ¼ 0:5; these values are referred to hereafter as the parameters’ “base values”. In addition, we assume a polynomial effect of product age on demand gðtÞ ¼ 1 ðt=EÞn with E ¼ 10 and n 2 f0:5; 1; 1:5; 2g, which captures convex (0 < n < 1), linear (n ¼ 1) and concave (n > 1) declines of demand over time. We note that, in fact, a lower value of n means higher sensitivity to age, which means a stronger perishability effect, and E reflects the product’s expiration date. RI RI Fig. 1 depicts πRI R ðTÞ, π M ðTÞ and J ðTÞ for the various values of n. It is observed that when the perishability effect is stronger (i.e., n is smaller), at equilibrium, the cycle length and the value of the objective functions shrink. Moreover, all three objectives are quasi-concave functions, and thus Theorem 1 and Proposition 2 hold, as can be seen in Figs. 2 and 3. We continue with a sensitivity analysis of the equilibrium and of the corresponding profits for Models 1–3. Note that, with the exception of KM and KR (the manufacturer’s and the retailer’s fixed order costs, respectively), each parameter has a monotone effect on these values. Fig. 2 shows the decisions and profits of the supply chain members at equilibrium as a function of KM , and Fig. 3 shows these values as a function of KR . In each figure, the other parameters are kept at their base values. From Fig. 2 we conclude that when the retailer sets the cycle length (Model 1), a higher fixed order cost results in a longer cycle length, a higher revenue share for herself and a lower sales effort investment rate. This pattern of decisions can be interpreted as a protective strategy against losses. We further observe an increase in the selling price; this serves as a means for the manufacturer to compensate himself for the reduction in his revenue share. It is clear that the retailer’s profit de­ creases in KR . Interestingly, up to a certain value of KR (specifically, the value that produces a cycle length that is optimal for the manufacturer; the point where the red curve crosses the horizontal blue line in Fig. 2 (a)), the manufacturer’s profit increases in KR . Beyond this value, however, the manufacturer’s profit decreases in KR . It seems that it is in the manufacturer’s best interests for the retailer to pay certain fixed ordering cost (in our example, KR ¼ 220 with πRI M � 3304), which he can achieve either by charging her a fixed price (if she currently pays less than 220) or by giving her a subsidy (if she currently pays more than 220). Note that when the manufacturer has to charge a fixed fee in order to increase KR he benefits twice: directly from the fee and indirectly from adjusting the cycle length. On the other hand, of­ fering a subsidy to the retailer is not always beneficial to the

(a)

manufacturer, since the loss from the subsidy may offset the profits gained from influencing the cycle length. For example, as depicted in Fig. 2(f), shifting from the base value KR ¼ 500 with π RI M � 3255 (when the retailer sets the cycle length) to KR ¼ 220 with πRI M � 3304 by giving a subsidy of 500 220 ¼ 280 results in a profit increase of only 3304 3255 ¼ 49, which is not beneficial to the manufacturer. We thus conclude that, under certain conditions, a two-part tariff contract may be beneficial for the manufacturer. For Model 2, in which the manufacturer sets the cycle length, the plots of T; s; p; η and πM (Fig. 2(a)–(d) and (f)) are independent of KR , as expected from Proposition 1 and (4). On the other hand, the retailer’s profit decreases in her fixed ordering cost. For Model 3, we observe that the plots in Fig. 2(a)–(g) under Model 3 are in between those obtained for Models 1 and 2, in accordance with Theorem 1, Corollary 1 and Proposition 2. Note that the cooperative decision with regard to the cycle length (Model 3) produces higher total profit of the entire supply chain, although each party gains less than it would have if it were the sole decision maker. This means that by using a cooperative decision with a side-payment, Pareto improvement can be achieved (see, e.g., Avinadav et al., 2017a,b). Fig. 3 shows similar trends to those presented in Fig. 2, except that the roles of Model 1 and Model 2 are reversed. This result makes intu­ itive sense, since the fixed ordering costs of the two parties have similar effects on equilibrium cycle length, and thus on other measures of the supply chain. By comparing Fig. 2(f) with Fig. 3(e), we conclude that the retailer can also use a two-part tariff contract in order to influence the cycle length decision, and thus to increase her profits. Table 2 illustrates the effects of additional parameters on the two parties’ equilibrium strategies and profits. Specifically, the table shows the (monotone) direction of change for each variable when the value of a focal parameter increases in a domain that includes the parameter’s base value � 50%, and the other parameters are kept at their base values. We observe that the cycle length decreases when the base demand, the saleseffort elasticity or the holding cost increases. This is because the fixed ordering cost becomes relatively low compared with the increasing holding cost per cycle. This result is in line with the classical EOQ model, in which increasing demand or increasing holding costs leads to reduction in the cycle length. In contrast, as price elasticity increases, demand decreases, and thus, the cycle length increases. When either customer sensitivity to age decreases (i.e., n increases) or the product’s expiration date increases, the perishability effect diminishes, and the cycle length asymptotically converges to that without age effect. We further observe that the cycle length increases in the production cost. This observation can be explained as follows: when the production cost

(b)

(c)

RI RI Fig. 1. (a) πRI R ðTÞ; (b) π M ðTÞ; (c) J ðTÞ for a ¼ 1000, α ¼ 2, β ¼ 0:5, h ¼ 0:1, c ¼ 1; KM ¼ 300, KR ¼ 500, μ ¼ 0:5 and E ¼ 10 for various values of n.

6

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International Journal of Production Economics xxx (xxxx) xxx

(a)

(b)

(c)

(e)

(f)

(d)

(g)

Fig. 2. At equilibrium: (a) cycle length; (b) investment rate in sales effort; (c) product’s selling price; (d) retailer’s revenue share; (e) retailer’s profit; (f) manu­ facturer’s profit; and (g) the total supply chain’s profit as functions of the retailer’s fixed order costs.

increases, the manufacturer has to increase the selling price, thereby reducing demand; thus, according to the EOQ model, the cycle length increases.

5.1. Conditional equilibrium for a given T The following optimization problem is defined to obtain the condi­ tional equilibrium values of η, s and p for a given value of T: 8 9 ZT < = max Π R ¼ KR þ ððð1 ηÞpðη;TÞ htÞλðpðη;TÞ;sðη;TÞ;tÞÞdt ; ηðTÞ : 0 8 9 T Z < = S:t: ðsðη;TÞ;pðη;TÞÞ ¼ argmax Π M ¼ KM sT þ ðηp cÞ λðp; s;tÞdt : ; s;p

5. Manufacturer-investor models The solution methodology for the manufacturer-investor models is similar to that of the retailer-investor models and proceeds according to the following two stages: First, we assume that T is given and find conditional equilibrium values for the remaining decision variables; these values are identical across Models 4–6. Then we find equilibrium for each model separately by finding T.

0

0<η<1

(12) For simplicity of presentation and analysis, let Proposition 3. 7

ΘðTÞ � αcþðααþβ1Þh1τðTÞ.

At conditional equilibrium for a given value of T, ηMI ðTÞ ¼

T. Avinadav

International Journal of Production Economics xxx (xxxx) xxx

(a)

(b)

(c)

(e)

(f)

(d)

(g)

Fig. 3. At equilibrium: (a) cycle length; (b) investment rate in sales effort; (c) product’s selling price; (d) retailer’s revenue share; (e) retailer’s profit; (f) manu­ facturer’s profit; and (g) the total supply chain’s profit as functions of the manufacturer’s fixed order costs. Table 2 Changes in values of decision variables and profits at equilibrium in Models 1–3, when each parameter value increases in a domain that includes its base value � 50%, and the other parameters are kept at their base values. Parameter

T

η

p

s

πR

πM

a

↓ ↑ ↓ ↓ ↑ ↑ ↑

↓ ↓ ↓ ↑ ↑ ↑ ↑

↓ ↓ ↓ ↑ ↑ ↑ ↑

↑ ↓ ↑ ↓ ↓ ↑ ↑

↑ ↓ ↑ ↓ ↓ ↑ ↑

↑ ↓ ↑ ↓ ↓ ↑ ↑

α β h c n E

base demand price elasticity sales-effort elasticity holding cost rate per unit unit production cost age effect shape parameter product’s expiration date

c ΘðTÞ

� , sMI T ¼

�

aβc

α 1g

�α �1 1 β � �� α 1 T 1ΘðTÞ and pMI ðTÞ ¼ 1ΘðTÞ α 1.

Proof. First, we calculate the manufacturer’s best response by solving the first constraint in (12). The necessary condition yields a unique so­ 1 0 � �1 � � C B aβc � � ηð1 α 1 Þ α 1 β lution, the couple ðsðη;TÞ;pðη;TÞÞ ¼ B ;ηð1 cα 1 ÞC c @ α 1g T A, which also satisfies the sufficient condition for a maximum. Substituting sðη; TÞ and pðη; TÞ in the objective function in (12) and solving the maximization problem results in a unique point that satisfies the neces­ sary condition. Since this point, denoted by ηMI ðTÞ, also satisfies the sufficient condition, it is the conditional equilibrium for a given T. Then, 8

T. Avinadav

International Journal of Production Economics xxx (xxxx) xxx MI Proposition 4. If πMI R ðTÞ and π M ðTÞ are quasi-concave functions, then the equilibrium values of the decision variables under Model 6 are between those obtained under Models 4 and 5.

the equilibrium selling price and sales effort investment are obtained by sRI ¼ sðηMI ; TÞ and pMI ðTÞ ¼ pðηMI ;TÞ. By Proposition 3, the price and sales-effort elasticities affect all the decisions. Note that this property differs from the case of the retailerinvestor models, in which sales-effort elasticity has no effect on the selling price and revenue shares (see Proposition 1).

Proof. By Theorem 1, the equilibrium value of T under Model 6 is between the equilibrium values of T under Model 4 and Model 5. The claim is proved, since, by Corollary 3, ηMI ðTÞ, pMI ðTÞ and sMI ðTÞ are monotone in T. As in the case of the retailer-investor models, we use numerical ex­ amples to provide a conjecture on the behavior of each of the objective functions (3)–(5). To do so, we use the base parameter values given in MI MI section 4.2. Fig. 4 depicts πMI R ðTÞ, π M ðTÞ and J ðTÞ for the various values of n. It is observed that all three objectives are quasi-concave functions, and that when the perishability effect is stronger (i.e., n is smaller), at equilibrium, the cycle length and the values of the objective functions shrink. This result is similar to that obtained when the retailer is the investor (see Fig. 1). As in the case of the retailer-investor models, with the exception of KM and KR , each parameter has a monotone effect on the values of the profits and the decision variables at equilibrium. Fig. 5 shows the decisions and profits of the supply chain members at equilibrium as a function of KM , and Fig. 6 shows these values as a function of KR . In each figure, the other parameters are kept at their base values.

Corollary 3. (i) ηMI ðTÞ and sMI ðTÞ are decreasing functions, whereas pMI ðTÞ is an increasing function; (ii) the manufacturer’s revenue from selling a unit depends only on the product cost and the price elasticity and equals c= ð1 α 1 Þ . Proof. The proofs are similar to those of Corollary 2. Comparing Corollary 2 with Corollary 3 reveals that, regardless of the investor’s identity, the cycle length has a negative effect on the re­ tailer’s revenue share and on the level of sales-effort investment, whereas it has a positive effect on the selling price of the product. Comparing Proposition 1 with Proposition 3, for a given cycle length, we state: Corollary 4. (i) ηMI ðTÞ > ηRI ðTÞ; (ii) pMI ðTÞ < pRI ðTÞ; (iii) � ηMI ðTÞpMI ðTÞ ¼ ηRI ðTÞpRI ðTÞ ¼ c= 1 α 1 ; (iv) sMI ðTÞ < sRI ðTÞ if and only if τðTÞ > hc

�� �α 1 þ αβ1

�

�� 1 þ α11 .

It is important to note that when the manufacturer is the investor, the party/parties who set the cycle length enjoy higher profits than they do when the retailer is the investor. In fact, comparing the manufacturer’s and the retailer’s profits in Fig. 2(e)–(f) with those in Fig. 5(e)-(f), and comparing those in Fig. 3(e)–(f) with those in Fig. 6(e)-(f), we observe that the party who does not set the cycle length also enjoys higher profits than he or she does when the retailer is the investor. Thus, in our example, a Pareto improvement can be achieved by giving the manu­ facturer the right to invest regardless of the identity of the party who determines the length of the cycle (retailer, manufacturer or both). It is clear that if each party gains more profit when the manufacturer is the investor than when the retailer is the investor, then the total profit of the supply chain is also larger under a manufacturer-investor, as can be seen by comparing Figs. 2(g), 3(g) and 5(g) and 6(g). The trends with regard to cycle length, sales-effort investment, selling price, and revenue shares of the parties do not differ between the manufacturer-investor and the retailer-investor models (see Figs. 2, 3, 5 and 6(a)-(d)). Accordingly, Table 2, which shows how varying each parameter value directionally affects the two parties’ equilibrium stra­ tegies and profits under Models 1–3, is valid also for Models 4–6. To avoid repetition, we refer the reader to the corresponding analysis at the end of Section 4.

Proof. (i) and (ii) are obtained in a straightforward manner from Propositions 1 and 3, since ΦðTÞ > ΘðTÞ. (iii) Straightforward from (ii) of Corollaries 1 and 3; (iv) From Propositions 1 and 3 we get �α 1 1 0� B @

αþβ 1 α 1

hτðTÞ α α 1þ c

sMI ðTÞ sRI ðTÞ

¼

1 β

C A , and the claim is proved by solving

sMI ðTÞ sRI ðTÞ

< 1.

By Corollary 4, when the retailer is the investor, she gives a smaller share of the revenue to the manufacturer, and he, in turn, increases the selling price as compensation. However, the manufacturer’s revenue from selling a unit ðηpÞ is independent on the identity of the investor in sales effort. The corollary further suggests that the investment in sales effort can be either larger or smaller when the retailer is the investor compared with the case of a manufacturer investor. Specifically, beyond a certain cycle length, the sales efforts investment is larger under the retailer investor model compared with that under the manufacturer investor model. 5.2. Setting the cycle length T The next step is to find the equilibrium value of T for Models 4–6, i.e., the value of T that maximizes either (3), (4) or (5), respectively.

(a)

(b)

(c)

MI MI Fig. 4. (a) πMI R ðTÞ; (b) π M ðTÞ; (c) J ðTÞ for a ¼ 1000, α ¼ 2, β ¼ 0:5, h ¼ 0:1, c ¼ 1; KM ¼ 300, KR ¼ 500, μ ¼ 0:5 and E ¼ 10 for various values of n.

9

T. Avinadav

International Journal of Production Economics xxx (xxxx) xxx

(a)

(b)

(c)

(e)

(f)

(d)

(g)

Fig. 5. At equilibrium: (a) cycle length; (b) investment rate in sales effort; (c) product’s selling price; (d) retailer’s revenue share; (e) retailer’s profit; (f) manu­ facturer’s profit; and (g) the total supply chain’s profit as functions of the retailer’s fixed order costs.

6. Managerial implications and future research

declining age effect. We found that regardless of the identity of the investor in sales effort: (i) when one party determines the cycle length, a two-part tariff contract can be used by the other party in order to influence the cycle length decision, and thus to increase its profits; (ii) the cycle length has a negative effect on the manufacturer’s revenue share and on the sales effort investment, whereas it has a positive effect and on the selling price of the product; (iii) a cooperative decision with regard to the cycle length produces higher total profit of the entire supply chain compared with a decision made by one party, although each party gains somewhat less profit than when making the decision alone. On the other hand, (i) the effect of the sales-effort elasticity on conditional equilibrium de­ pends on the identity of the investor: when the manufacturer is the investor, this elasticity affects the selling price and revenue shares, whereas when the retailer is the investor it does not; (ii) when the

In this work we analyzed a two-echelon supply chain operating under a revenue-sharing contract for the sale of a perishable product. We investigated six different models, which differed according to the identity of the investor in sales effort (either the manufacturer or the retailer) and the identity of the party who determines the cycle length (manufacturer, retailer, or cooperative decision). First, we provided general results related to the parties’ decisions and profits in the different scenarios considered. Assuming a multiplicative form of de­ mand made up of a marketing effect (price and sales effort) and a product age effect, we derived conclusions regarding the average amount of time a unit spends on shelf and regarding the average sales rate; these conclusions are not influenced by the effects of marketingrelated decision variables on demand and are valid for a general 10

T. Avinadav

International Journal of Production Economics xxx (xxxx) xxx

(a)

(b)

(c)

(e)

(f)

(d)

(g)

Fig. 6. At equilibrium: (a) cycle length; (b) investment rate in sales effort; (c) product’s selling price; (d) retailer’s revenue share; (e) retailer’s profit; (f) manu­ facturer’s profit; and (g) the total supply chain’s profit as functions of the manufacturer’s fixed order costs.

retailer is the investor, she gives a smaller share of the revenue to the manufacturer, and he, in turn, increases the selling price as compensa­ tion (compared with the case of a manufacturer-investor); (iii) both the manufacturer and the retailer enjoy higher profits when the manufac­ turer is the investor in sales effort (compared with the case of a retailerinvestor). We propose three directions for future research. The first is to find equilibrium for different orders of decisions, for example, when the parties first make marketing-related decisions and then the operational ones, or make both types of decisions simultaneously. The second is to investigate a scenario under which the marketing and operational di­ visions have different objectives, such as maximum revenue vs. mini­ mum cost, respectively. Finally, a good avenue for further research

would be to consider heteronomous customer behavioral (FIFO and LIFO demands) under stochastic EOQ model. References Aust, G., Buscher, U., 2012. Vertical cooperative advertising and pricing decisions in a manufacturer–retailer supply chain: a game-theoretic approach. Eur. J. Oper. Res. 223 (2), 473–482. Avinadav, T., Arponen, T., 2009. An EOQ model for items with a fixed shelf-life and a declining demand rate based on time-to-expiry technical note. Asia Pac. J. Oper. Res., 26(06), 759-767. Avinadav, T., Chernonog, T., Lahav, Y., Spiegel, U., 2017a. Dynamic pricing and promotion expenditures in an EOQ model of perishable products. Ann. Oper. Res., 248(1–2), 75-91..

11

T. Avinadav

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Avinadav, T., Chernonog, T., Perlman, Y., 2015a. The effect of risk sensitivity on a supply chain of mobile applications under a consignment contract with revenue sharing and quality investment. Int. J. Prod. Econ., 168, 31-40.. Avinadav, T., Chernonog, T., Perlman, Y., 2015b. Consignment contract for mobile apps between a single retailer and competitive developers with different risk attitudes. Eur. J. Oper. Res., 246(3), 949-957.. Avinadav, T., Chernonog, T., Perlman, Y., 2017b. Mergers and acquisitions between riskaverse parties. Eur. J. Oper. Res., 259(3), 926-934.. Avinadav, T., Henig, M. I. (2015) Exact accounting of inventory costs in stochastic periodic-review models. Int. J. Prod. Econ., 169, 89-98.. Avinadav, T., Chernonog, T., Ben-Zvi, T., 2019. The effect of information superiority on a supply chain of virtual products. Int. J. Prod. Econ. 216, 384–397. Avinadav, T., Herbon, A., Spiegel, U., 2013. Optimal inventory policy for a perishable item with demand function sensitive to price and time. Int. J. Prod. Econ. 144 (2), 497–506. Avinadav, T., Herbon, A., Spiegel, U., 2014. Optimal ordering and pricing policy for demand functions that are separable into price and inventory age. Int. J. Prod. Econ. 155, 406–417. Bakker, M., Riezebos, J., Teunter, R.H., 2012. Review of inventory systems with deterioration since 2001. Eur. J. Oper. Res. 221 (2), 275–284. Becker-Peth, M., Thonemann, U. W., 2016. Reference points in revenue sharing contracts—how to design optimal supply chain contracts. Eur. J. Oper. Res., 249(3), 1033-1049.. Bernstein, F., Chen, F., Federgruen, A., 2006. Coordinating supply chains with simple pricing schemes: the role of vendor-managed inventories. Manag. Sci., 52(10), 14831492.. Cachon, G. P., Lariviere, M. A., 2005. Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manag. Sci., 51(1), 30-44.. Cai, J., Hu, X., Tadikamalla, P. R., Shang, J., 2017. Flexible contract design for VMI supply chain with service-sensitive demand: revenue-sharing and supplier subsidy. Eur. J. Oper. Res., 261(1), 143-153.. Chatterjee, S., Slotnick, S. A., Sobel, M. J., 2002. Delivery guarantees and the interdependence of marketing and operations. Prod. Oper. Manag., 11(3), 393-410. (). Chen, J.M., Cheng, H.L., Chien, M.C., 2011. On channel coordination through revenuesharing contracts with price and shelf-space dependent demand. Appl. Math. Model. 35, 4886–4901. Chen, X., Pang, Z., Pan, L., 2014. Coordinating inventory control and pricing strategies for perishable products. Oper. Res., 62(2), 284-300.. Chernonog, T., 2020. Inventory and marketing policy in a supply chain of a perishable product. Int. J. Prod. Econ. 219, 259–274. Chernonog, T., 2020. Strategic information sharing in online retailing under a consignment contract with revenue sharing. Ann. Oper. Res. Submitted for publication. Chernonog, T., Avinadav, T., 2019. Pricing and advertising in a supply chain of perishable products under asymmetric information. Int. J. Prod. Econ. 209, 249–264. Chernonog, T., Avinadav, T., Ben-Zvi, T., 2015. Pricing and sales-effort investment under bi-criteria in a supply chain of virtual products involving risk. Eur. J. Oper. Res., 246 (2), 471-475.. Chua, G.A., Mokhlesi, R., Sainathan, A., 2017. Optimal discounting and replenishment policies for perishable products. Int. J. Prod. Econ. 186, 8–20. Chung, Y.T., Erhun, F., 2013. Designing supply contracts for perishable goods with two periods of shelf life. IIE Trans., 45(1), 53-67.. Corbett, C. J., 2001. Stochastic inventory systems in a supply chain with asymmetric information: cycle stocks, safety stocks, and consignment stock. Oper. Res., 49(4), 487-500.. Dobson, G., Pinker, E.J., Yildiz, O., 2017. An EOQ model for perishable goods with agedependent demand rate. Eur. J. Oper. Res. 257 (1), 84–88. Dong, L., Zhu, K., 2007. Two-wholesale-price contracts: push, pull, and advancepurchase discount contracts. Manuf. Serv. Oper. Manag., 9(3), 291-311.. Donselaar, K. H. van, Broekmeulen, R.A.C.M., 2012. Approximations for the relative outdating of perishable products by combining stochastic modeling, simulation and regression modeling. Int. J. Prod. Econ. 140, 660–669. Fang Du, X., Leung, S. C., Long Zhang, J., Lai, K. K., 2009. Procurement of agricultural products using the CPFR approach. Supply Chain Manag.: Int. J., 14(4), 253-258.. Feng, L., Chan, Y.L., C� ardenas-Barr� on, L.E., 2017. Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date. Int. J. Prod. Econ. 185, 11–20. Feng, T., Zhang, F., 2014. The impact of modular assembly on supply chain efficiency. Prod. Oper. Manag., 23(11), 1985-2001.. Gallego, G., van Ryzin, G., 1994. Optimal dynamic pricing of inventory with stochastic demand over finite horizons. Manag. Sci. 40, 999–1020. Goyal, S.K., Giri, B.C., 2001. Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res. 134 (1), 1–16. Guiomar, M.H., Sihem, T., Georges, Z., 2006. The impact of manufacturers’ wholesale prices on a retailer’s shelf-space and pricing decisions. Decis. Sci. J. 37 (1), 71–90. Hadley, G., Whitin, T.M., 1963. Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, NJ. Haijema, R., Minner, S., 2016. Stock-level dependent ordering of perishables: a comparison of hybrid base-stock and constant order policies. Int. J. Prod. Econ. 181, 215–225. Haijema, R., Minner, S., 2019. Improved ordering of perishables: the value of stock-age information. Int. J. Prod. Econ. 209, 316–324. Hausman, W. H., Montgomery, D. B., Roth, A. V., 2002. Why should marketing and manufacturing work together?: some exploratory empirical results. J. Oper. Manag., 20(3), 241-257..

Herbon, A., 2014. Dynamic pricing vs. acquiring information on consumers’ heterogeneous sensitivity to product freshness. Int. J. Prod. Res., 52(3), 918-933.. Herbon, A., 2017. Should retailers hold a perishable product having different ages? The case of a homogeneous market and multiplicative demand model. Int. J. Prod. Econ. 193, 479–490. Herbon, A., Levner, E., Cheng, T.C.E., 2014. Perishable inventory management with dynamic pricing using time–temperature indicators linked to automatic detecting devices. Int. J. Prod. Econ. 147, 605–613. Hu, B., Feng, Y., 2017. Optimization and coordination of supply chain with revenue sharing contracts and service requirement under supply and demand uncertainty. Int. J. Prod. Econ., 183, 185-193.. Hu, B., Xu, D., Meng, C., 2017. Inconsistency of a retailer’s optimal policies and channel performance under revenue sharing contracts. Int. J. Prod. Econ., 183, 53-65.. Huang, J., Leng, M., Parlar, M., 2013. Demand functions in decision modeling: a comprehensive survey and research directions. Decis. Sci. J. 44 (3), 557–609. Jin, Y., Wang, S., Hu, Q., 2015. Contract type and decision right of sales promotion in supply chain management with a capital constrained retailer. Eur. J. Oper. Res., 240 (2), 415-424.. Karaesmen, I., Scheller-Wolf, A., Deniz, B., 2011. Managing perishable and aging inventories: review and future research directions. In: Kempf, K., Keskinocak, P., Uzsoy, R. (Eds.), Planning Production and Inventories in the Extended Enterprise: A State of the Art Handbook. Springer, New York, pp. 393–436. de Keizer, M., Akkerman, R., Grunow, M., Bloemhof, J.M., Haijema, R., van der Vorst, J. G., 2017. Logistics network design for perishable products with heterogeneous quality decay. Eur. J. Oper. Res. 262 (2), 535–549. Kouki, C., Jemaï, Z., Minner, S., 2015. A lost sales (r, Q) inventory control model for perishables with fixed lifetime and lead time. Int. J. Prod. Econ. 168, 143–157. Kouki, C., Babai, M.Z., Jemai, Z., Minner, S., 2016. A coordinated multi-item inventory system for perishables with random lifetime. Int. J. Prod. Econ. 181, 226–237. Kouki, C., Babai, M. Z., Minner, S. 2018. On the benefit of dual-sourcing in managing perishable inventory. Int. J. Prod. Econ., 2014, 1-17. Krishna, A., 1992. The normative impact of consumer price expectations for multiple brands on consumer purchase behavior. Mark. Sci. 11 (3), 266–286. Li, R., Chan, Y.L., Chang, C.T., C� ardenas-Barr� on, L.E., 2017. Pricing and lot-sizing policies for perishable products with advance-cash-credit payments by a discounted cash-flow analysis. Int. J. Prod. Econ. 193, 578–589. Li, S., Hua, Z., 2008. A note on channel performance under consignment contract with revenue sharing. Eur. J. Oper. Res., 184(2), 793-796.. Maihami, R., Nakhai Kamalabadi, I., 2012. Joint pricing and inventory control for noninstantaneous deteriorating items with partial backlogging and time and price dependent demand. Int. J. Prod. Econ. 136 (1), 116–122. Minner, S., Transchel, S., 2010. Periodic review inventory-control for perishable products under service-level constraints. Spectrum 32 (4), 979–996. Minner, S., Transchel, S., 2017. Order variability in perishable product supply chains. Eur. J. Oper. Res., 260(1), 93-107.. Nahmias, S., 1982. Perishable inventory theory: a review. Oper. Res. 30 (4), 680–708. Parfitt, J., Barthel, M., Macnaughton, S., 2010. Food waste within food supply chains: quantification and potential for change to 2050. Philos. Trans. R. Soc. B 365, 3065–3081. Pekgün, P., Griffin, P. M., Keskinocak, P., 2017. Centralized versus decentralized competition for price and lead-time sensitive demand. Decis. Sci. J., 48(6), 11981227.(). Porteus, E. L., 1990. Stochastic inventory theory. In Heyman, D., Sobel, M. (Eds.), Handbook in Operations Research and Management Science Vl 2 Stochastic Models. (North-Holland). Ru, J., Wang, Y., 2010. Consignment contracting: who should control inventory in the supply chain? Eur. J. Oper. Res. 201 (3), 760–769. Sardana, D., Terziovski, M., Gupta, N., 2016. The impact of strategic alignment and responsiveness to market on manufacturing firm’s performance. Int. J. Prod. Econ., 177, 131-138.. Sarker, B.R., Mukherjee, S., Balan, C.V., 1997. An order-level lot size inventory model with inventory-level dependent demand and deterioration. Int. J. Prod. Econ. 48, 227–236. Tsao, Y.C., Sheen, G.J., 2008. Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments. Comput. Oper. Res. 35 (11), 3562–3580. Valliathal, M., Uthayakumar, R., 2011. Optimal pricing and replenishment policies of an EOQ model for non-instantaneous deteriorating items with shortages. Int. J. Adv. Manuf. Technol. 54 (1–4), 361–371. Wang, Y. Y., Wang, J. C., Shou, B., 2013. Pricing and effort investment for a newsvendortype product. Eur. J. Oper. Res., 229(2), 422-432.. Wang, Y., Jiang, L., Shen, Z. J., 2004. Channel performance under consignment contract with revenue sharing. Manag. Sci., 50(1), 34-47.. Wang, Y.Y., Lau, H.S., Wang, J.C., 2012. Defending and improving the ‘‘slotting fee’’: how it can benefit all the stakeholders dealing with a newsvendor product with price and effort-dependent demand. J. Oper. Res. Soc. 63 (12), 1731–1751. Weatherford, L.R., Bodily, S.E., 1992. A taxonomy and research overview of perishableasset revenue management: yield management, overbooking, and pricing. Oper. Res. 40 (5), 831–844. You, P.S., 2005. Inventory policy for products with price and time-dependent demands. J. Oper. Res. Soc. 56 (7), 870–873. Zhang, D., De Matta, R., Lowe, T.J., 2010. Channel coordination in a consignment contract. Eur. J. Oper. Res. 207 (2), 897–905.

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