Optimal decision for a fuzzy supply chain with shrinkage under VaR criterion

Abstract

This paper investigates the problems of making optimal decisions on pricing and shelf-space for a fuzzy supply chain with one perishable product, with the help of Radio Frequency Identification Device (RFID) technology to reduce shrinkage. Based on the criterion of Value-at-Risk (VaR) and its minimization, we introduce one centralized decision model and three decentralized decision models to obtain the respective optimal decisions of both the manufacturer and the retailer, by analyzing the fuzzy uncertainties and the relationship among the demand, retail price and shelf-space. The corresponding optimal strategies of the two participants are obtained along with one example.

Keywords

Fuzzy variable supply chain Radio Frequency Identification Devices (RFID)Value-at-Risk (VaR)game theory

1 Introduction

In the traditionally optimal decisions of supply chains, it is often assumed that the physical sale quantity is equal to the demand, and the demand is identical with the order quantity (or inventory). However, due to the theft, the metamorphism, the errors made by staffs plus other reasons in the real supply chains, it is inevitable that the physical sale quantity is less than the demand (or the order quantity). A realistic example is a supply chain with perishable products, such as short-life products and seasonal products, and this discrepancy can be widespread. Agrawal and Sharda [1] showed from nearly 370,000 inventory records gathered from multiple stores owned by leading retailers, the discrepancy between physical sale quantity and inventory is as much as 65%.

Radio Frequency Identification Devices (RFIDs) is an emerging technology that is increasingly being used to track products and to manage demand and/or inventory. Due to its capability, RFID has found application in retail stores. Wal-Mart is a powerful propellant using over 50-billion RFIDs per year from 2005 to 2007. Accordingly, the premier sell-side research and brokerage firm Bernstein had analyzed and estimated that Walmart saved more than 83.5-billion dollars per year. Ilias [2] had provided a hierarchical model to evaluate the impact of RFID on supply chain, and showed that RFID can increase inventory by 33.8% and increase stock by 45.6%. Cui et al. [3] had proposed an RFID-based investment evaluation model for the joint replenishment and delivery problem (JRD), and illustrated that the saved cost rate accounted for 7.82%, which due to the ability of RFID to share information in real time, and to detect and trace multi-item simultaneously.

Robertoa and Franco [4] had applied a tensor approach in general modelling and designed an abstract algebraic framework for efficient analysis of RFID. Fan et al. [5] had presented a split-path schema-based data storage model to store and process more efficiently massive RFID data produced in supply chain management systems. Fan et al. [6] had studied the decisions of Internet of Things (IoT) with misplacement and shrinkage, and showed that when the fixed cost and tag cost of RFID are shared between the manufacturer and the retailer, the later is more sensitive to the sharing proportion. Zhang et al. [7] had analyzed the service level of a supply chain with lead-time and misplaced inventory, and showed that whether RFID can improve the service level depends on the tag cost of RFID in the centralized system, but only when the supplier bears the cost of RFID.

Some other papers had studied the supply chain with RFID from the perspective of minimizing the risk or loss. Under the criterion of Conditional Value-at-Risk (CVaR), Zhu et al. [8] had explored an optimal ordering policy subject to inventory shrinkage, and showed that RFID implementation is practical for high price products. Xu and Zhao [9] investigated the effect of RFID on inventory shrinkage in the retail supply chain with the CVaR criterion, and showed that for the centralized system, the sales-available rate, the recovery rate and the tag cost are the main driving factors in evaluating the benefit of RFID. Zhu et al. [10] used the criterion of CVaR to describe the retailer’s risk-averse indictor and the criterion of VaR to describe the supplier’s risk-neutral attitude; this has led to the establishment of a risk-sharing contract to coordinate the dual-channel supply chain.

All the above researches consider only the RFID tag cost, but not the influence of RFID on the demand which is assumed to be a random variable or to be a function of the retail price only. In fact, a smart shelf that is equipped to detect the RFID tags affixed directly to individual items [11, 12] can also help the customers to better understand the product. For example, RFID can be used for automatically detecting goods, detecting and eliminating inaccuracies, investigating where these inaccuracies arise. Therefore, the shelf-space with RFID detection has become an important influence factor on the demand. Szmerekovskya and Zhang [11] had studied the effect of RFID on the coordination problem of supply chain when shelf-space is limited. Szmerekovsky et al. [12] had investigated whether it is possible for the manufacturer and the retailer to derive economic benefits from the item-level of an RFID system. They had studied a manufacture-leader game model (when the manufacturer has more bargaining power) and a retailer-leader game model (when the retailer has more bargaining power), and the results showed that in the latter case the retailer may be forced to adopt RFID even if it does not maximize the system revenue.

There are also many uncertain factors embedded in the supply chain including manufacturing cost, shelf-space cost, customer demand and so on, which will definitely affect the effectiveness of the supply chain management. In case of little or no relative historical data, or under the circumstances of some newborn supply chains, the probability density and distribution of the above uncertain parameters might not be available. Lau and Lau [13] had suggested that it may be more appropriate to describe those uncertain parameters based on expert knowledge and forecast. Fuzzy theory, originally introduced by Zadeh [14] and perfected by Liu and Liu [15], can provide an effective way to deal with optimization problems with uncertainty of subjective vagueness, where the uncertainty should be described as fuzzy variables. In fact, fuzzy theory has gradually become a powerful mathematical tool to deal with supply chain problems under uncertainty; for instances, pricing decision [16], inventory [17], coordination [18] and suitability evaluation [19].

Up to now, as far as we know, no work has been found to study the supply chain decision of pricing and shelf-space with RFID and the criterion of VaR under fuzzy environments. So, in this paper, under the VaR criterion, we will study the optimal decision of a supply chain with RFID and uncertainties associated with the custom demand, manufacturer cost and shelf-space cost, which are characterized as fuzzy variables [15]. We will investigate how the manufacturer and the retailer make their own optimal decisions about whole-sale prices, retail prices and shelf-space under fuzzy uncertain environments. We shall also focus on the market power (the power/ability of a firm to profitably raise the market price of a good or service over marginal cost) balance between the manufacturer and the retailer, and the effects of the market power on the equilibrium prices, shelf-space and loss. Four scenarios are discussed, including one centralized case and three decentralized cases: that is, Manufacturer leader; Retailer leader; and simultaneous decision (i.e., each part in the system has an equal market power). Motivated by [16], where the optimal criterion was to maximize the α-optimistic value of the revenues, in this paper four optimal programming models will be built to minimize the VaR with an threshold value of both the manufacturer and the retailer, and the analytical equilibrium solutions are obtained.

The rest of this paper is organized as follows: Section 2 presents some necessary preliminaries, descriptions and notations. The optimal programming models of one centralized decision and three decentralized decisions are established in Section 3. One numerical example is presented to illustrate the effectiveness of the proposed model in Section 4. Finally, conclusions and some economic insights are provided in Section 5.

2 Modeling framework

2.1 Preliminaries

Let ξ be a fuzzy variable defined on a credibility space $(Θ, P (Θ), Cr)$ , where Θ is a nonempty set, $P (Θ)$ is the power set of Θ, and Cr is a credibility measure defined based on the possibility measure Pos [15] $Cr {ξ \geq x} = \frac{1}{2} (Pos {ξ \geq x} + 1 - Pos {ξ < x}) .$ (1)

For any given α in (0, 1], the α-pessimistic value $ξ_{α}^{L}$ and the α-optimistic value $ξ_{α}^{U}$ of ξ are defined as [20] $ξ_{α}^{L} = inf {λ | Cr {ξ \leq λ} \geq α},$ (2) $ξ_{α}^{U} = sup {λ | Cr {ξ \geq λ} \geq α} .$ (3)

Fuzzy variables ξ₁, ξ₂, …, ξ_n are said to be independent if and only if $Cr {⋂_{i = 1}^{n} {ξ_{i} \in B_{i}}} = min_{1 \leq i \leq n} Cr {ξ_{i} \in B_{i}} .$ (4)

Bai and Liu [21] defined a Credibility Value at Risk (CVaR) as $Va R_{α} (ξ) = inf {λ | Cr {ξ \leq λ} \geq α,}, α \in (0, 1],$ where ξ is a regular fuzzy variable different from the lower VaR and upper VaR defined by possibility measure [22].

In fact, from an economic point of view, Value-at-Risk (VaR) is a measure of the risk or loss. It is the opposite number of the lower α-quantile of the revenue function. VaR is positive if the lower α-quantile of the revenue is negative, otherwise, it is negative if the lower α-quantile of the revenue is positive. For this reason, we define a new definition of Value-at-Risk as follows.

Definition 1. Let ξ be a fuzzy variable defined on the credibility space $(Θ, P (Θ), Cr)$ , and π (ξ) be a revenue function of ξ. Then the Value-at-Risk (VaR) of π with a threshold value of α, denoted as VaR_α (π), is defined as the opposite number of the lower α-quantile of π (ξ), i.e., $Va R_{α} (π) = - inf {λ | Cr {π (ξ) \leq λ} \geq α},$ (5) which gives the lowest value λ such that the loss will not exceed the given threshold value -VaR_α (π) with a threshold value of α [8]. Here, [·] _α means the α-quantile of [·].

Example 1. If a fuzzy revenue π (ξ) has a VaR with the threshold value of α, then Cr {π (ξ) ∈ (- ∞, - VaR]} = α. In other word, the value of π (ξ) lies in interval [- VaR, + ∞) with the credibility 1 - α as illustrated in Fig. 1.

Fig.1

Example of VaR.

Remark 1. VaR_α (π) is the opposite number of the lower α-quantile of π. VaR_α (π) ≥0 means that Cr {π ≤ 0} ≥ α, otherwise Cr {π ≤ 0} < α.

2.2 Problem description

Consider a symmetric-information supply chain with one manufacturer and one retailer. The manufacturer makes one perishable product with a manufacture cost c_m and a RFID tag cost c_t, and then wholesales the product to the retailer at the wholesale price w, who in turn sells the product with a retail price p to the end customers on a smart shelf (in order to enhance customs’ effectiveness in shopping as discussed before). The retailer would install the smart shelf with space s and a shelf-space cost $g (s) = c_{s} s^{2},$ (6) to stimulate demand and to reduce shrinkage, where c_s is the shelf-space cost coefficient.

A linear demand function is adopted, which decreases with the retail price p and increases with the shelf-space s, i.e., $D (p, s) = a - b_{1} p + b_{2} s,$ (7) where a represents the primary market demand, b₁ is the retail price elastic coefficient and b₂ is the shelf-space elastic coefficient.

The fuzzy parameters a, b₁, b₂, c_m, c_s and c_t are assumed to be nonnegative and independent of each other. Furthermore, it is assumed that there is no shortages and nor leftovers. On the other hand, there is a demand shrinkage in the supply chain due to inefficiency and other reasons. The rate between the actual demand and the potential demand is denoted by γ, and the shrinkage rate is 1 - γ, namely, (1 - γ) D (p, s) are unavailable to the end customers due to shrinkage. However, there is a recovery rate, denoted as β. That is, when RFID is used, β (1 - γ) D (p, s) can also be purchased by end customers, and other (1 - β) (1 - γ) D (p, s) may remain as non-availability. The shrinkage of demand, and the recovery of shrinkage are summarized in Fig. 2.

Fig.2

Demand with versus without RFID.

Let δ = γ + β (1 - γ) . Then, based on the above assumptions, the respective revenue functions of the manufacturer and retailer in the case of adopting RFID can be expressed as $\begin{matrix} π_{m} (w, p, s) & = & (w - c_{m} - c_{t}) D (p, s) \\ = & (w - c_{m} - c_{t}) (a - b_{1} p + b_{2} s), \end{matrix}$ (8) $\begin{matrix} π_{r} (p, s) & = & p δ D (p, s) - wD (p, s) - g (s) \\ = & (δ p - w) (a - b_{1} p + b_{2} s) - c_{s} s^{2}, \end{matrix}$ (9) and the revenue function of the whole supply chain in the centralized decision case can be described as $\begin{matrix} π_{c} (w, p, s) \\ = π_{m} (w, p, s) + π_{r} (p, s) \\ = (δ p - c_{m} - c_{t}) (a - b_{1} p + b_{2} s) - c_{s} s^{2} . \end{matrix}$

We assume that both the manufacturer and the retailer would like making optimal decisions with the purpose of achieving the minimum of VaR with a given threshold value α of their own revenue, respectively. Details of the criteria of minimizing VaR, one can see reference [23].

The following notations are adopted throughout this paper.

w is the manufacturer’s wholesale price decision variable;

p is the retailer’s retail price decision variable;

s is the retailer’s shelf-space decision variable.

c_m is the manufacture cost per product;

c_t is the tag cost per product of RFID;

c_s is the cost per product for adopting RFID.

a represents the market base;

b₁ is the price elastic coefficient;

b₂ is the shelf-space elastic coefficient.

γ is the sales-available rate without RFID;

β is the recovery rate of shrinkage with RFID;

δ is the sale-available rate with RFID.

π_c, π_m and π_r are the profits of the whole system, the manufacturer and the retailer, respectively.

For the sake of simplify, we denote ${c_{m}}_{α}^{U} + {c_{t}}_{α}^{U}$ as $C_{α}^{U}$ (i.e. $C_{α}^{U} = {c_{m}}_{α}^{U} + {c_{t}}_{α}^{U}$ ), which will be used in what follows.

3 Analytical results

Depending on the relative scale of the manufacturer and the retailer in the supply chain we can have the following 4 scenarios, including one central system and three discrete systems. The main difference between a central and discrete system is whether the manufacture and the retailer are one unit or not. Specifically, in the central system, the manufacturer and the retailer are performed as one unit, and the wholesale price becomes an inner variable irrelevant to the objective function. In the discrete systems, the manufacturer and the retailer are individuals, and both of them will make their own optimal decisions independently. One point about the three discrete scenarios we would like to make is the relative scales of the manufacture and the retailer. 1) If the manufacturer has relative bigger scale, then the manufacturer will have the power/ability to make decision first. (Consequently, the game between the manufacturer and the retailer is the Manufacturer Leader Stackelberg one); If the retailer has relative bigger scale, then the retailer will have the power to make decision first (Consequently, the game is the Retailer Leader Stackelberg one); 3) If each part in the system has an equal scale or so, the they will make decision simultaneously, and the game is the Nash game.

3.1 Scenario C-central system

In the central decision supply chain, the decision-maker will only need to choose a proper retail price p and a shelf-space variable s to minimize the VaR of the whole system with a threshold value of α (α ∈ (0, 1)). This can be formulated as follows. $min_{(p, s)} Va R_{α} (π_{c}) = - inf {λ | Cr {π_{c} \leq λ} \geq α},$ (10) where VaR_α (π_c) is the VaR of the revenue π_c for the whole system with a threshold value of α.

Proposition 1. Let $p_{c}^{*}$ , $s_{c}^{*}$ and ${VaR}_{α}^{c *}$ be the optimal retail price, the optimal shelf-space decision variable and the minimum VaR of the C scenario with RFID, respectively. Then $p_{c}^{*} = \frac{2 δ a_{α}^{L} {c_{s}}_{α}^{U} + [2 {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}] C_{α}^{U}}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}},$ (11) $s_{c}^{*} = \frac{δ {b_{2}}_{α}^{L} [δ a_{α}^{L} - {b_{1}}_{α}^{U} C_{α}^{U}]}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}},$ (12) and ${VaR}_{α}^{c *} = - \frac{{c_{s}}_{α}^{U} {[δ a_{α}^{L} - {b_{1}}_{α}^{U} C_{α}^{U}]}^{2}}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}},$ (13) provided that $4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} > δ^{2} ({b_{2}}_{α}^{L})^{2} .$

Proof. For each fixed feasible solution (p, s), ${VaR}_{α}^{*} (π_{c})$ should be the minimum value of VaR_α (π_c) (i.e., the whole system’s VaR with at least threshold value α) [16]. According to the definition of pessimistic value (2) and the definitions of VaR in equation (5), we can obtain that VaR_α (π_c) equals to the opposite number of the α-pessimistic value of π_c. Therefore, model (10) is equivalent to the following model $min_{(p, s)} {VaR}_{α} (π_{c}) = - [π_{c} (p, s)]_{α}^{L} .$ (14)

Denote VaR_α (π_c) as ${VaR}_{α}^{c}$ . According to the assumption that the non-negative fuzzy variables a, b₁, b₂, c_m, c_s and c_t are independent with each other and the Theorem 2 of [24] and the Proposition 1 of [25], we have ${VaR}_{α}^{c} = {c_{s}}_{α}^{U} s^{2} - (δ p - C_{α}^{U}) (a_{α}^{L} - {b_{1}}_{α}^{U} p + {b_{2}}_{α}^{L} s) .$ (15)

The first-order and second-order partial derivatives of ${VaR}_{α}^{c}$ with respect to (p, s) are obtained as $\frac{\partial {VaR}_{α}^{c}}{\partial p} = 2 δ {b_{1}}_{α}^{U} p - δ {b_{2}}_{α}^{L} s - δ a_{α}^{L} - {b_{1}}_{α}^{U} C_{α}^{U},$ (16) $\frac{\partial {VaR}_{α}^{c}}{\partial s} = - δ {b_{2}}_{α}^{L} p + 2 {c_{s}}_{α}^{U} s + {b_{2}}_{α}^{L} C_{α}^{U},$ (17) $\frac{\partial^{2} {VaR}_{α}^{c}}{\partial p^{2}} = 2 δ {b_{1}}_{α}^{U}, \frac{\partial^{2} [π_{c} (p, s)]_{α}^{L}}{\partial s^{2}} = - 2 {c_{s}}_{α}^{U},$ and $\frac{\partial^{2} {VaR}_{α}^{c}}{\partial p \partial s} = \frac{\partial^{2} [π_{c} (p, s)]_{α}^{L}}{\partial s \partial p} = - δ {b_{2}}_{α}^{L} .$

Since the Hessian matrix is positive-defined, ${VaR}_{α}^{c}$ is convex under (p, s). Let (16) and (17) equal to zero for optimum solution, one then obtain (11) and (12) as the optimal strategy of the central decision with RFID. Using (11) and (12), the optimum value of the integrated system can be obtained as (13).

Corollary 1. Let $p_{c}^{* *}$ be the optimal retail price of the C scenario without RFID corresponded to $p_{c}^{*}$ in (11), and the optimal VaR in absence of RFID corresponded to ${VaR}_{α}^{c *}$ in (13); then $p_{c}^{* *} = \frac{γ a_{α}^{L} + {b_{1}}_{α}^{U} {c_{m}}_{α}^{U}}{2 γ {b_{1}}_{α}^{U}},$ and ${VaR}_{α}^{c * *} = - \frac{{(γ a_{α}^{L} - {b_{1}}_{α}^{U} {c_{m}}_{α}^{U})}^{2}}{4 γ {b_{1}}_{α}^{U}} .$

3.2 Scenario M-manufacturer leader system

Under this scenario, the manufacturer has larger scale and bigger market/bargaining power. So, the manufacturer will first set the wholesale price w which the retailer observes. Then, the retailer makes a retail price p and chooses a shelf-space decision variable s according to its own observation of w. Finally, the retailer and the manufacturer bear their VaRs. Consequently, the Manufacturer-leader (M) optimal model can be established as

$\begin{matrix} min_{w} Va R_{α} (π_{m} (w, p^{*} (w), s^{*} (w))) \\ = - inf {λ | Cr {π_{m} (w, p^{*} (w), s^{*} (w)) \leq λ} \geq α} \\ s . t . (p^{*}, s^{*}) solves \\ min_{(p, s)} Va R_{α} (π_{r}) = - inf {λ | Cr {π_{r} \leq λ} \geq α}, \end{matrix}$ (18) where VaR_α (π_m) and VaR_α (π_r) are the α-VaR of the manufacturer and the retailer, respectively.

Model (18) describes an optimal decision problem of a two-stage supply chain, and the game structure between the manufacturer and the retailer is a Manufacturer-leader Stackelberg one. However, the resolving process of model (18) is an inverted one, which means that we should get the optimal decision of the retailer firstly, and then obtain the optimal decision of the manufacturer.

Proposition 2. In model (18), the optimal response decision of the retailer to the wholesale w, denoted by (p^* (w), s^* (w)) can be obtained as $p^{*} (w) = \frac{2 δ a_{α}^{L} {c_{s}}_{α}^{U} + [2 {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}] w}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}},$ (19) and $s^{*} (w) = \frac{δ {b_{2}}_{α}^{L} [δ a_{α}^{L} - {b_{1}}_{α}^{U} w]}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}} .$ (20) provided that $4 {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} > δ ({b_{2}}_{α}^{L})^{2}$ .

Proof. According to (5) in the Definition 1, for each fixed feasible solution (w, p, s), the value of VaR_α (π_r) is the opposite number of the α-pessimistic value of π_r. That is ${VaR}_{α} (π_{r}) = - {π_{r}}_{α}^{L}$ . Similarly ${VaR}_{α} (π_{m}) = - {π_{m}}_{α}^{L}$ . It follows from (2) and (5), model (18) can be shown to be equivalent to

$\begin{matrix} min_{w} {VaR}_{α} [π_{m} (w, p^{*} (w), s^{*} (w))] \\ = - (w - {c_{m}}_{α}^{U} - {c_{t}}_{α}^{U}) (a_{α}^{L} - {b_{1}}_{α}^{U} p^{*} (w) + {b_{2}}_{α}^{L} s^{*} (w)) \\ s . t . (p^{*}, s^{*}) solves \\ min_{(p, s)} {VaR}_{α} (π_{r}) = {c_{s}}_{α}^{U} s^{2} \\ - (δ p - w) (a_{α}^{L} - {b_{1}}_{α}^{U} p + {b_{2}}_{α}^{L} s) . \end{matrix}$ (21)

By denoting VaR_α (π_r) as ${VaR}_{α}^{r}$ . Similarly as (15), the first-order and the second-order partial derivatives of ${VaR}_{α}^{r}$ with respect to (p, s) can be derived as $\frac{\partial {VaR}_{α}^{r}}{\partial p} = 2 δ {b_{1}}_{α}^{U} p - δ {b_{2}}_{α}^{L} s - δ a_{α}^{L} - {b_{1}}_{α}^{L} w,$ (22) $\frac{\partial {VaR}_{α}^{r}}{\partial s} = - δ {b_{2}}_{α}^{L} p + 2 {c_{s}}_{α}^{U} s + {b_{2}}_{α}^{L} w,$ (23) $\begin{matrix} \frac{\partial^{2} {VaR}_{α}^{r}}{\partial p^{2}} & = & 2 δ {b_{1}}_{α}^{U}, \frac{\partial^{2} {VaR}_{α}^{r}}{\partial s^{2}} = 2 {c_{s}}_{α}^{U}, \\ \frac{\partial^{2} {VaR}_{α}^{r}}{\partial p \partial s} & = & \frac{\partial^{2} {VaR}_{α}^{r}}{\partial s \partial p} = - δ {b_{2}}_{α}^{L} . \end{matrix}$

Note that ${VaR}_{α}^{r}$ is a convex function of (p, s). By letting (22) and (23) equal to zero, we can obtain (19) and (20) as proposed.

Proposition 3. Under the M model, the optimal wholesale price $w_{m}^{*}$ is $w_{m}^{*} = \frac{δ a_{α}^{L} + {b_{1}}_{α}^{U} C_{α}^{U}}{2 {b_{1}}_{α}^{U}} .$ (24)

Proof. From Equations (7), (8), (19) and (20), it can be obtained that $\begin{matrix} {VaR}_{α} [π_{m} (w, p^{*} (w), s^{*} (w))] \\ = - C_{α}^{U} (a_{α}^{L} - {b_{1}}_{α}^{U} p^{*} (w) + {b_{2}}_{α}^{L} s^{*} (w)) . \end{matrix}$

Denote VaR_α [π_m (w, p^* (w), s^* (w))] as ${VaR}_{α}^{m}$ , the first-order and second-order derivatives of which with respect to w are $\begin{matrix} \frac{d {VaR}_{α}^{m}}{d w} & = & \frac{4 {({b_{1}}_{α}^{U})}^{2} {c_{s}}_{α}^{U} w - 2 {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} [δ a_{α}^{L} + {b_{1}}_{α}^{U} C_{α}^{U}]}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}}, \end{matrix}$ (25) $\frac{d^{2} {VaR}_{α}^{m}}{d w^{2}} = \frac{4 {({b_{1}}_{α}^{U})}^{2} {c_{s}}_{α}^{U}}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}} > 0 .$

Thus, ${VaR}_{α}^{m}$ is a convex function with respect to w. Setting (25) to zero for an optimum solution and solve for w, we can obtain (24).

Proposition 4. The retailer’s optimal decisions of $p_{m}^{*}$ and $s_{m}^{*}$ under the M model are $\begin{matrix} p_{m}^{*} \\ = \frac{δ a_{α}^{L} [6 {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}] + C_{α}^{U} [2 {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}]}{2 {b_{1}}_{α}^{U} [4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}]}, \end{matrix}$ (26) $s_{m}^{*} = \frac{δ {b_{2}}_{α}^{L}}{2} \cdot \frac{[δ a_{α}^{L} - {b_{1}}_{α}^{U} C_{α}^{U}]}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}} .$ (27)

Proof. Substituting (24) into (19) and (20), then (26) and (27) can be obtained.

Corollary 2. If the RFID is not provided, the optimal wholesale price $w_{m}^{* *}$ in the absence of RFID corresponding to $w_{m}^{*}$ in (24), and retail price $p_{m}^{* *}$ in the absence of RFID corresponding to $p_{m}^{*}$ in (26) are $w_{m}^{* *} = \frac{γ a_{α}^{L} + {b_{1}}_{α}^{U} {c_{m}}_{α}^{U}}{2 {b_{1}}_{α}^{L}},$ and $p_{m}^{* *} = \frac{3 γ a_{α}^{L} + {b_{1}}_{α}^{U} {c_{m}}_{α}^{U}}{4 γ {b_{1}}_{α}^{L}} .$

3.3 Scenario R-retailer leader system

Under this scenario, the retailer has larger scale and bigger market/bargaining power. So, the retailer will first set the retail price p and shelf-space decision variable s, which the manufacturer observes. Then, the manufacturer makes a wholesale price w according her observation of (p, s). Finally, the retailer and the manufacturer bear their VaRs. Consequently, the Retailer-leader (R) optimal model can be established as

$\begin{matrix} min_{(p, s)} Va R_{α} (π_{r} (w^{*} (p, s), p, s)) \\ = - inf {λ | Cr {π_{r} \leq λ} \geq α} \\ s . t . w^{*} solves \\ min_{w} Va R_{α} (π_{m}) = - inf {λ | Cr {π_{m} \leq λ} \geq α} \end{matrix}$ (28)

Proposition 5. In model (28), the optimal response decision of the manufacture to the retail price p and shelf-decision variable s, denoted by w^* (p, s) can be obtained as $w^{*} (p, s) = \frac{a_{α}^{L} + {b_{2}}_{α}^{L} s}{{b_{1}}_{α}^{U}} + C_{α}^{U} - p .$ (29)

Proof. Model (28) is equivalent to $\begin{matrix} min_{(p, s)} {VaR}_{α} (π_{r} (w^{*} (p, s), p, s)) \\ = - [(δ p - w^{*} (p, s)) (a - b_{1} p + b_{2} s) - c_{s} s^{2}]_{α}^{L} \\ s . t . w^{*} solves \\ min_{w} {VaR}_{α} (π_{m}) \\ = - [(w - c_{m} - c_{t}) (a - b_{1} p + b_{2} s)]_{α}^{L} . \end{matrix}$

Let ${VaR}_{α}^{m} = {VaR}_{α} (π_{m})$ , then ${VaR}_{α}^{m} = {c_{s}}_{α}^{U} s^{2} - (w - C_{α}^{U}) (a_{α}^{L} - {b_{1}}_{α}^{U} p + {b_{2}}_{α}^{L} s) .$ (30)

Given a wholesale price w and a retail price p, the marginal revenue of the retailer is defined to be m = p - w. Then, the first-order and second-order derivatives of (30) with respect to w can be obtained as $\begin{matrix} \frac{d {VaR}_{α}^{m}}{d w} & = & {b_{1}}_{α}^{U} w - a_{α}^{U} - {b_{2}}_{α}^{L} s - {b_{1}}_{α}^{U} C_{α}^{U} + {b_{1}}_{α}^{U} p, \end{matrix}$ (31) $\frac{d^{2} {VaR}_{α}^{m}}{d w^{2}} = 2 {b_{1}}_{α}^{U} .$

Hence, ${VaR}_{α}^{m}$ is a convex function with respect to w, and (29) can be obtained by letting (31) equal to zero and solving for w.

Proposition 6. In the R case, the optimal strategy of the retail price $p_{r}^{*}$ and the shelf-space $s_{r}^{*}$ are $p_{r}^{*} = \frac{2 (2 + δ) a_{α}^{L} {c_{s}}_{α}^{U} + [2 {b_{1}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}] C_{α}^{U}}{4 (1 + δ) {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}},$ (32) $s_{r}^{*} = \frac{δ^{2} a_{α}^{L} {b_{2}}_{α}^{L} - δ {b_{1}}_{α}^{U} {b_{2}}_{α}^{L} C_{α}^{U}}{4 (1 + δ) {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}},$ (33) provided that $4 (1 + δ) {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} > δ ({b_{2}}_{α}^{L})^{2} .$

Proof. Let ${VaR}_{α}^{r} = {VaR}_{α} [π_{r} (w^{*} (p, s), p, s))]$ . From (7), (9) and (29), it can be obtained that $\begin{matrix} {VaR}_{α}^{r} & = & {c_{s}}_{α}^{U} s^{2} - [δ p + p - \frac{a_{α}^{L} + {b_{2}}_{α}^{L} s}{{b_{1}}_{α}^{U}} \\ - C_{α}^{U}] (a_{α}^{L} - {b_{1}}_{α}^{U} p + {b_{2}}_{α}^{L} s) . \end{matrix}$

The first-order, second-order partial derivatives of ${VaR}_{α}^{r}$ with respect to (p, s) can be obtained as $\begin{matrix} \frac{\partial {VaR}_{α}^{r}}{\partial p} & = & 2 (1 + δ) {b_{1}}_{α}^{U} p \\ - (2 + δ) {b_{2}}_{α}^{L} s - 2 a_{α}^{L} - {b_{1}}_{α}^{U} C_{α}^{U}, \end{matrix}$ (34) $\begin{matrix} \frac{\partial {VaR}_{α}^{r}}{\partial s} & = & {b_{2}}_{α}^{L} [\frac{2 a_{α}^{L}}{{b_{1}}_{α}^{U}} + C_{α}^{U}] - 2 (1 + δ) {b_{2}}_{α}^{L} p \\ + 2 [\frac{{({b_{2}}_{α}^{L})}^{2}}{{b_{1}}_{α}^{U}} + {c_{s}}_{α}^{U}] s, \end{matrix}$ (35) $\begin{matrix} \frac{\partial^{2} {VaR}_{α}^{r}}{\partial p^{2}} & = & 2 (1 + δ) {b_{1}}_{α}^{U}, \\ \frac{\partial^{2} {VaR}_{α}^{r}}{\partial s^{2}} & = & 2 [\frac{{({b_{2}}_{α}^{L})}^{2}}{{b_{1}}_{α}^{U}} + {c_{s}}_{α}^{U}], \end{matrix}$ and $\begin{matrix} \frac{\partial^{2} {VaR}_{α}^{r}}{\partial p \partial s} = \frac{\partial^{2} {VaR}_{α}^{r}}{\partial s \partial p} = - (2 + δ) {b_{2}}_{α}^{L} . \end{matrix}$

The Hessian matrix is positive definite. Let (34) and (35) equal to zero, we can obtain (32) and (33).

Proposition 7. The optimal wholesale price $w_{r}^{*}$ under the R model can be obtained as $\begin{matrix} w_{r}^{*} & = & C_{α}^{U} \\ + \frac{2 δ a_{α}^{L} {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} + (δ^{2} - δ) a_{α}^{L} {({b_{2}}_{α}^{L})}^{2} - 2 C_{α}^{U} {({b_{1}}_{α}^{U})}^{2} {c_{s}}_{α}^{U}}{{b_{1}}_{α}^{U} [4 (1 + δ) {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}]} . \end{matrix}$ (36)

Proof. The above wholesale price (36) can be obtained by substituting (32) and (33) into (29), simultaneously.

Corollary 3. If the RFID is not provided, the optimal wholesale price $w_{r}^{* *}$ in the absence of RFID corresponding to $w_{r}^{*}$ in (36), and retail price $p_{r}^{* *}$ in the absence of RFID corresponding to $p_{r}^{*}$ in (32) are $w_{r}^{* *} = \frac{γ a_{α}^{L} + (1 + 2 γ) {b_{1}}_{α}^{U} {c_{m}}_{α}^{U}}{2 (1 + γ) {b_{1}}_{α}^{L}},$ and $p_{r}^{* *} = \frac{(2 + γ) a_{α}^{L} + {b_{1}}_{α}^{U} {c_{m}}_{α}^{U}}{2 (1 + γ) {b_{1}}_{α}^{L}} .$

3.4 Scenario N-simultaneous decision system

Under this circumstance, the manufacture and the retailer have same scale and the same equal market/bargaining power or so. Therefore, they will make their own optimal decisions, simultaneously. Hence, our Model N (Nash Game) can be formulated as follows ${\begin{matrix} min_{w} Va R_{α} (π_{m}) = - inf {λ | Cr {π_{m} \leq λ} \geq α} \\ min_{(p, s)} Va R_{α} (π_{r} (p, s)) = - inf {λ | Cr {π_{r} \leq λ} \geq α} . \end{matrix}$ (37)

Proposition 8. Under the N Model, the respective optimal strategies of $w_{n}^{*}$ made by the manufacturer and $(p_{n}^{*}, s_{n}^{*})$ made by the retailer are $w_{n}^{*} = \frac{2 δ a_{α}^{L} {c_{s}}_{α}^{U} + C_{α}^{U} [4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}]}{2 (1 + 2 δ) {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}},$ (38) $p_{n}^{*} = \frac{2 δ a_{α}^{L} {c_{s}}_{α}^{U} + [2 {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ {({b_{2}}_{α}^{L})}^{2}] w_{n}^{*}}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}},$ (39) and $s_{n}^{*} = \frac{δ {b_{2}}_{α}^{L} [δ a_{α}^{L} - {b_{1}}_{α}^{U} w_{n}^{*}]}{4 δ {b_{1}}_{α}^{U} {c_{s}}_{α}^{U} - δ^{2} {({b_{2}}_{α}^{L})}^{2}} .$ (40)

Proof. Similar to the models of M and R, (37) is equivalent to the following optimization ${\begin{matrix} min_{w} {VaR}_{α} [π_{m} (w)] \\ = - [(w - c_{m} - c_{t}) (a - b_{1} p + b_{2} s)]_{α}^{L} \\ min_{(p, s)} {VaR}_{α} [π_{r} (p, s)] \\ = - [(δ p - w) (a - b_{1} p + b_{2} s) - c_{s} s^{2}]_{α}^{L} . \end{matrix}$

Equations (19) and (20) provide the optimal strategy of (p, s) as a function of w for the R model, while equation (29) gives the optimal strategy w as a function of p and s for the M model, the solving of (19), (20) and (29) simultaneously yields the Nash decision solution (38)-(40) when both optimization functions are optimized simultaneously.

Corollary 4. When RFID is not considered, the optimal wholesale price $w_{n}^{* *}$ in the absence of RFID corresponding to $w_{n}^{*}$ in (38), the retail price $p_{n}^{* *}$ in case of no RFID corresponded to $p_{n}^{*}$ in (39) are $w_{n}^{* *} = \frac{γ (a_{α}^{L} + {b_{1}}_{α}^{U} {c_{m}}_{α}^{U})}{2 (1 + γ) {b_{1}}_{α}^{L}},$ and $p_{n}^{* *} = \frac{(1 + γ) (a_{α}^{L} + {b_{1}}_{α}^{U} {c_{m}}_{α}^{U})}{(1 + 2 γ) {b_{1}}_{α}^{L}} .$

4 Numerical study

In this section, a numerical example is presented to compare the results acquired in the above four different decision scenarios and to analyze the effects of the market/bargaining power on the channel prices and the shelf-space decision.

Here, we focus on a newborn supply chain without any historical data such that no probability density and distribution functions can be obtained. Under this circumstance, the fuzzy set theory [14, 15] is an appropriate modeling tool, where the uncertain parameters are able to be approximately estimated by experts’ judgments and experiences. These estimate values contain subjective uncertainty, and should be described by the method of linguistic hedge [26], e.g. the phrase “a small level” to describe a size that can be regarded as a fuzzy variable [15].

The relationship between linguistic hedges and triangular fuzzy variables are often determined by experts’ experience [27, 28]. In our case, we have used experiences and some subjective judgements to arrive at some relationship between linguistic hedges and triangular fuzzy variables at different levels for manufacturing cost c_m, tag cost of RFID c_t, market base a, price elasticity b₁, shelf-space elasticity b₂ and smart shelf management cost c_s. These are shown in Table 1 below.

Table 1
Relationship between Linguistic Hedge and Triangular Fuzzy Variable

Linguistic expression Triangular fuzzy variable

c _m Low (about 1) (0.5, 1, 1.5)

Medium (about 2) (1, 2, 3)

High (about 3) (2.5, 3, 3.5)

c _t Low (about 0.01) (0, 0.01, 0.02)

Medium (about 0.02) (0.01, 0.02, 0.03)

High (about 0.03) (0.02, 0.03, 0.04)

a Large (about 700) (600, 700, 800)

Small (about 500) (400, 500, 600)

b ₁ Very sensitive (about 40) (30, 40, 50)

Sensitive (about 20) (10, 20, 30)

b ₂ Very sensitive (about 3) (2, 3, 4)

Sensitive (about 2) (1, 2, 3)

c _s Very sensitive (about 0.6) (0.5, 0.6, 0.7)

Sensitive (about 0.4) (0.3, 0.4, 0.5)

	Linguistic expression	Triangular fuzzy variable
c _m	Low (about 1)	(0.5, 1, 1.5)
	Medium (about 2)	(1, 2, 3)
	High (about 3)	(2.5, 3, 3.5)
c _t	Low (about 0.01)	(0, 0.01, 0.02)
	Medium (about 0.02)	(0.01, 0.02, 0.03)
	High (about 0.03)	(0.02, 0.03, 0.04)
a	Large (about 700)	(600, 700, 800)
	Small (about 500)	(400, 500, 600)
b ₁	Very sensitive (about 40)	(30, 40, 50)
	Sensitive (about 20)	(10, 20, 30)
b ₂	Very sensitive (about 3)	(2, 3, 4)
	Sensitive (about 2)	(1, 2, 3)
c _s	Very sensitive (about 0.6)	(0.5, 0.6, 0.7)
	Sensitive (about 0.4)	(0.3, 0.4, 0.5)

To initiate computation, we start the manufacturing cost c_m and RFID cost c_t at a medium level, the market base a at small level, while the price elastic coefficient b₁, the shelf-space elastic coefficient and the shelf-space cost coefficient b₂ for the retailer are in very sensitive level. We therefore obtain from Table 1 the triangular fuzzy parameters c_m = (1, 2, 3), c_t = (0.01, 0.02, 0.03), a = (400, 500, 600), b₁ = (30, 40, 50), b₂ = (2, 3, 4), and c_s = (0.5, 0.6, 0.7). Following the procedure in [20], the respective α-optimistic or α -pessimistic critical values can be obtained as $a_{α}^{L} = 400 + 200 α$ , ${c_{m}}_{α}^{U} = 3 - 2 α$ , ${c_{t}}_{α}^{U} = 0.03 - 0.02 α$ , ${b_{1}}_{α}^{U} = 50 - 20 α$ , ${b_{2}}_{α}^{L} = 2 + 2 α$ , and ${c_{s}}_{α}^{U} = 0.7 - 0.2 α$ .

These expressions are common to the models of C, M, R and N for different confidence threshold α, and from them, one can obtain the optimal retail prices, wholesale prices, shelf-space, of the system (i.e., ${VaR}_{α}^{c}$ ), of the manufacturer (i.e., ${VaR}_{α}^{m}$ ) and of the retailer (i.e., ${VaR}_{α}^{r}$ ) under different sales-available rate with RFID, the recovery rate β and the sales-available rate with RFID δ = γ + β (1 - γ). In the following, Table 2 tabulates the results for (γ, β, δ) = (0.8, 0.5, 0.9), Table 3 for (γ, β, δ) = (0.9, 0.5, 0.95), and Table 4 for (γ, β, δ) = (0.9, 0.8, 0.98) under different threshold values α.

Table 2

Optimal Decisions with γ = 0.8, β = 0.5

Model	α	p ^*	p ^**	w ^*	w ^**	s ^*	${VaR}_{α}^{c^{*}}$	${VaR}_{α}^{c * *}$	${VaR}_{α}^{m *}$	${VaR}_{α}^{m * *}$	${VaR}_{α}^{r^{*}}$	${VaR}_{α}^{r * *}$
C	0.4	7.27	7.09			9.79	-813.73	-632.67
	0.5	7.84	7.50			12.63	-1033.82	-800.00
	0.6	8.5329	7.97			16.20	-1299.06	-993.63
	0.7	9.38	8.50			20.74	-1621.50	-1216.80
M	0.4	9.35	9.26	6.25	5.67	4.97	-611.25	-460.73	-407.82	-316.33	-318.66	-158.17
	0.5	10.17	10.00	6.64	6.00	6.44	-777.28	-571.47	-518.84	-400.00	-377.83	-200.00
	0.6	11.1086	10.83	7.07	6.37	8.31	-978.09	-693.62	-653.35	-496.82	-429.03	-248.41
	0.7	12.19	11.75	7.56	6.80	10.73	-1223.58	-825.11	-818.28	-608.4	-455.14	-304.20
R	0.4	9.94	9.50	4.01	4.13	4.47	-492.16	-437.40	-134.06	-156.21	-358.10	-281.19
	0.5	10.82	10.28	4.13	4.22	5.72	-632.36	-553.09	-178.57	-197.53	-453.79	-355.56
	0.6	11.82	11.14	4.30	4.34	7.26	-799.54	-686.96	-233.18	-245.34	-566.56	-441.62
	0.7	12.98	12.11	4.50	4.49	9.17	-999.13	-841.24	-300.14	-300.44	-698.98	-540.80
N	0.4	8.83	9.44	5.23	3.03	6.12	-700.27	-447.76	-380.78	-69.37	-319.49	-378.39
	0.5	9.60	10.04	5.51	3.22	7.84	-886.39	-593.80	-485.94	-120.34	-400.45	-473.46
	0.6	10.51	10.72	5.84	3.44	9.98	-1108.63	-763.28	-613.89	-184.85	-494.74	-578.43
	0.7	11.57	11.49	6.24	3.69	12.64	-1375.57	-958.93	-771.11	-263.78	-604.46	-695.15

Table 3

Optimal Decisions with γ = 0.9, β = 0.5

Model	α	p ^*	p ^**	w ^*	w ^**	s ^*	${VaR}_{α}^{c^{*}}$	${VaR}_{α}^{c * *}$	${VaR}_{α}^{m *}$	${VaR}_{α}^{m * *}$	${VaR}_{α}^{r^{*}}$	${VaR}_{α}^{r * *}$
C	0.4	7.23	6.94			10.53	-887.62	-762.75
	0.5	7.82	7.36			13.55	-1122.28	-950.69
	0.6	8.5289	7.84			17.36	-1405.46	-1167.25
	0.7	9.40	8.39			22.23	-1750.57	-1416.10
M	0.4	9.33	9.18	6.54	6.24	5.29	-666.24	-545.99	-444.34	-381.38	-222.90	-190.69
	0.5	10.16	9.93	6.95	6.63	6.83	-842.77	-668.65	-562.21	-475.35	-280.57	-237.67
	0.6	11.1066	10.76	7.41	7.06	8.79	-1056.19	-802.80	-704.83	-583.63	-351.30	-291.81
	0.7	12.19	11.69	7.90	7.55	11.29	-1317.04	-945.65	-879.42	-708.05	-437.62	-354.03
R	0.4	9.84	9.30	4.14	4.33	4.93	-558.06	-551.46	-153.89	-190.16	-404.17	-361.30
	0.5	10.72	10.07	4.28	4.43	6.29	-712.65	-687.34	-203.60	-237.02	-509.04	-450.33
	0.6	11.71	10.92	4.46	4.57	7.96	-897.38	-843.91	-264.91	-291.00	-632.48	-552.91
	0.7	12.87	11.87	4.69	4.73	10.04	-1118.75	-1023.83	-340.71	-353.04	-778.14	-670.78
N	0.4	8.75	9.25	5.35	3.23	6.70	-771.44	-560.79	-409.89	-94.13	-361.55	-466.66
	0.5	9.53	9.84	5.63	3.43	8.57	-971.78	-729.61	-520.95	-152.64	-450.83	-576.96
	0.6	10.45	10.51	5.97	3.67	10.89	-1211.28	-924.35	-656.46	-225.44	-554.82	-698.91
	0.7	11.52	11.26	6.40	3.93	13.79	-1499.64	-1148.22	-823.69	-313.56	-675.94	-834.66

Table 4

Optimal Decisions with γ = 0.9, β = 0.8

Model	α	p ^*	p ^**	w ^*	w ^**	s ^*	${VaR}_{α}^{c^{*}}$	${VaR}_{α}^{c * *}$	${VaR}_{α}^{m *}$	${VaR}_{α}^{m * *}$	${VaR}_{α}^{r^{*}}$	${VaR}_{α}^{r * *}$
C	0.4	7.21	6.94			10.97	-932.39	-762.75
	0.5	7.81	7.36			14.11	-1175.92	-950.69
	0.6	8.52	7.84			18.07	-1470.06	-1167.25
	0.7	9.42	8.39			23.13	-1829.13	-1416.10
M	0.4	9.32	9.18	6.71	6.24	5.49	-699.52	-547.89	-466.43	-381.38	-233.10	-190.69
	0.5	10.15	9.93	7.14	6.63	7.07	-882.40	-669.82	-588.42	-475.35	-293.98	-237.67
	0.6	11.11	10.76	7.61	7.06	9.07	-1103.44	-802.74	-735.92	-583.63	-367.52	-291.81
	0.7	12.21	11.69	8.16	7.55	11.63	-1373.59	-943.58	-916.31	-708.05	-457.28	-354.03
R	0.4	9.79	9.37	4.21	4.33	5.21	-598.74	-551.46	-166.13	-190.16	-432.61	-361.30
	0.5	10.66	10.07	4.36	4.43	6.64	-762.18	-687.34	-219.07	-237.02	-543.11	-450.33
	0.6	11.65	10.92	4.55	4.57	8.40	-957.78	-843.91	-284.57	-291.00	-673.21	-552.91
	0.7	12.81	11.87	4.80	4.73	10.58	-1192.86	-1023.83	-365.96	-353.04	-826.90	-670.78
N	0.4	8.71	9.25	5.41	3.23	7.06	-814.77	-560.79	-427.09	-94.13	-387.68	-466.66
	0.5	9.50	9.84	5.70	3.43	9.02	-1023.76	-729.61	-541.68	-152.64	-482.08	-576.96
	0.6	10.41	10.51	6.05	3.67	11.45	-1273.81	-924.35	-681.75	-225.44	-592.06	-698.91
	0.7	11.50	11.26	6.49	3.93	14.50	-1575.34	-1148.22	-855.08	-313.56	-720.56	-834.66

The following observations can be made from Tables 2, 3 and 4:

Under all models, the VaRs of the system, the manufacturer, and the retailer (namely, ${VaR}_{α}^{c *}$ , ${VaR}_{α}^{m *}$ and ${VaR}_{α}^{r *}$ in case with RFID, and ${VaR}_{α}^{c * *}$ , ${VaR}_{α}^{m * *}$ and ${VaR}_{α}^{r * *}$ in case without RFID) are all negative. It shows that our supply chain has no absolute lost meaning that the revenues of the system, the manufacturer, and the retailer are always positive.

For fixed α, β and γ, ${VaR}_{α}^{c *}$ has the smallest value in case of C, followed by N, M and R (having the biggest value). ${VaR}_{α}^{m *}$ has the smallest value in case of M, followed by N, and then R. On the other hand, ${VaR}_{α}^{r *}$ has the smallest value in R case, followed by N and then M. The following insights can be obtained. Compared with the decentralized cases, the whole system suffers the least loss in the centralized scenario. Moreover, both the manufacturer and the retailer have an incentive aspiration to play the leader’s role for that the leader holds advantage in suffering the relatively less loss.

For fixed α, β and γ, we have ${VaR}_{α}^{i *} \leq {VaR}_{α}^{i * *}$ , (i = c, m and r) with exceptions of ${VaR}_{α}^{m *}$ in case of R and ${VaR}_{α}^{r *}$ in case of N. These mean that using RFIDs can always achieve lower VaR values for the system, the manufacturer, and the retailer than not using RFIDs. The exception of ${VaR}_{α}^{m *}$ in case R means that the manufacturer may have less intention to adopt RFID when the retailer is the Stackelberg leader, and the exception of ${VaR}_{α}^{r *}$ in case N means that the retailer may have less intention to adopt RFID in the N case. In fact, a similar result has been obtained by Szmerekovsky et al. [12], where criteria of maximizing revenue was applied. Szmerekovsky et al. [12] showed that in case of manufacture-leader, the retailer may be forced to adopt RFID even if it does not maximize the system revenue.

For fixed α, β and γ, using RFIDs always results in higher wholesale price than not using RFID. The exception is the R model, which leads to the more loss of manufacturer in case of adopting RFID (i.e., ${VaR}_{α}^{m *} \geq {VaR}_{α}^{m * *}$ in the case of R). This phenomenon means that the manufacturer might bear more loss (that is to say, might not derive more economic benefits) from adopting RFID provided that the retailer is the leader. In fact, a relatively similar result has been obtained in [12], as mentioned in the above paragraph. Similarly, using RFIDs always result in higher retail price than not using RFID. The exception is the N model with some α values. This exception leads to more loss of the retailer in case of adopting RFID (i.e., ${VaR}_{α}^{r *} \geq {VaR}_{α}^{r * *}$ in the case of N). This phenomenon means that in case of Nash game, the retailer may also be forced to adopt RFID even if it does not minimize its loss.

The wholesale price, the retail price and the shelf-space are increasing functions about the threshold value α. This means that an increase in the threshold value always enhance the confidences of the decision makers, independent of β and γ. These may explain the fact that with the increase of threshold value α, the VaR is decreasing.

For fixed α and γ, VaRs are decreasing, the wholesale prices and the shelf-space are increasing, while the retail prices are decreasing first and then increasing functions about the recovery rate β. The insight that higher recovery rate leads to much more RFIDs and much lower losses is intuitively true. The explanation for the fact that the retail price are decreasing first and then increasing maybe the retailer’s strategy for attracting more purchase.

5 Conclusion

This paper has introduced a new evaluation criterion called Value-at-Risk (VaR) and formulated an optimal decision problem using a new supply chain model and accounting for demand shrinkage and fuzzy uncertainty. To reduce demand shrinkage Radio Frequency Identification Device (RFID) was employed, and to describe the fuzzy uncertainties, the method of linguistic hedges was adopted. With the assumption that the demand is sensitive to both retail price and shelf-space, four optimal programming models were developed under the VaR criterion for supply chain systems with perishable products and fuzzy demand, hence extending the classical supply chain models with no shrinkage or stochastic demand given in the past. By using fuzzy theory and game theory, the analytical equilibrium solutions of the pricing and shelf-space strategies of both the manufacturer and the retailer were derived by an analytical method. Analysis of our new supply chain model was carried out and performance was performed. Our results give insights into the economic behavior of supply chain, and can be served as the basis for empirical study in the future.

Our results are based upon some simplistic assumptions for analytical convenience. For example, the demand is a linear function, the decision makers are risk neutral, and the information is complete. Therefore, possible extensions include exponential form or more general forms for the demand function, risk-awareness of the decision makers, asymmetric information for the supply chain, and a system with more than one manufacturer and/or one retailer over multiple periods.

Footnotes

Acknowledgments

This work was partly supported by an National Key R&D Program of China under Grant No. 2016YFB0800805, Tianjin Key Lab of Intelligent Computing and Novel Software Technology, Key Laboratory of Computer Vision and System, Ministry of Education, the National Science Foundation of China under Grants Nos. 11426163, 61202381, Tianjin Nature Science Foundation under Grant No.15JCQNJC00500 and Projects of Science and Technology Service Industry in Tianjin under Grants No. ZXFWGX00140. The authors would like to thank the anonymous referee experts for the amendments and valuable advice to perfect the work.

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