# Optimal Decisions for Adoption of Item-Level RFID in a Retail Supply Chain with Inventory Shrinkage under CVaR Criterion

Authors: Chunming Xu; Daozhi Zhao
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

This paper investigates the effect of item-level RFID on inventory shrinkage in the retail supply chain, which consists of a risk-neutral manufacturer and a risk-averse retailer. Under conditional value-at-risk (CVaR) criterion, two different supply chain settings are discussed as follows. In the centralized setting, we develop the models in both RFID case and no RFID case, respectively. Comparisons between the two cases are made. In particular, a sufficient condition is given to judge whether to adopt item-level RFID. In the decentralized setting, we focus on discussing two different contract types including wholesale price contact and revenue sharing contract. Finally, number examples and sensitivity analysis are given to illustrate the proposed models. The results show that, for the centralized system, the sales-available rate, the recovery rate, and the tag cost are mainly the driving factors in evaluating the benefit of an item-level RFID. In particular, when the sales-available rate and the tag cost are quite small and the recovery rate is higher, the supply chain partners’ profits obtained by investment for RFID are improved significantly. For the decentralized system, under revenue sharing contract, Pareto improving outcome and coaffording risk can be achieved if the retailer sets an appropriate parameter for the manufacturer.


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## Body

## 1. Introduction

(1) Motivation. In traditional inventory system, it is often assumed that inventory record and physical inventory (actual on-hand inventory) are identical, but in the real world, the inventory record can hardly match the physical inventory; that is, inventory inaccuracy is widespread and inevitable in the store or warehouse/backroom [1, 2]. Usually, there are three different kinds of causes of inventory inaccuracy:(i)
inventory shrinkage: theft from the shelf, product spoilage, and product damage, and so forth;(ii)
misplacement: products that are placed on the wrong shelf are not accessible to customers due to improper usage of the storage area;(iii)
transaction errors: the wrong item codes are recorded at the cash register.In recent years, inventory inaccuracy is common in the retail industries and it has gained more and more attention of some researchers. Kang and Gershwin [3] noted that in a global retailer’s stores investigated only 70–75% of inventory record was accurate in the best performing retail store during its annual inventory audit. DeHoratius and Raman [4] reported that 65% of the inventory records in retail stores did not match the physical stock from about 370,000 examined SKUs (stock-keeping-units). Moreover, 20% of the inventory records differed from the physical stock by six or more items. By many investigations on how to generate errors in the retail inventory operation process, Rekik [1] concluded that inventory shrinkage is the main cause of inventory inaccuracy. Additionally, related studies also showed that the nonsale inventory shrinkage usually leads to unavailable demand for end consumers, and the magnitude of such losses stemming from inventory shrinkage is huge. Bednarz et al. [5] estimated that US retailers suffered a $31.3 billion loss due to shrinkage in 2002. According to ECR Europe [6], the value of lost inventory due to shrinkage in 2000 was 13.4 billion euros for retailers in Europe.Nowadays, however, radio frequency identification (RFID) technology can be used to track products and manage inventory based on its ability to higher visibility, and it has been seen as a promising solution for inventory shrinkage in the supply chain arena [7]. Generally, RFID has two principal advantages: high-frequency monitoring and nonline of sight reading, which yield information that is both more accurate and timely [8]. Recently, in order to characterize more practical features, Hervert-Escobar et al. [9] utilized the information obtained in consecutive read attempts to help identify a tag in RFID implementation and developed a heuristic method of selection based on Hamming distance; computer simulation is used to illustrate the validity of the proposed method. Talavera et al. [10] also studied RFID implementation in the steel industry, where RFID technology shows that the shared inventory improvement and the rate of obsolescence reduction are related to inventory management. As information technologies continue to improve and their costs continue to decrease, obtaining more accurate real-time inventory information is becoming increasingly cost-effective. As a result, more and more large companies (such as Wal-Mart, Proctor & Gamble, and Gillette) are guided by high-quality information in their daily operations and decision making [11].Motivated by the issue of RFID technology adoption in the industry (especially in supply chain management), we study the effect of item-level RFID on inventory shrinkage in the retail supply chain frame. To get more general results, we extend previous knowledge of inventory inaccuracy and item-level RFID by incorporating risk-averse consideration. As mentioned above, in reality, it is difficult for some companies to bear losses stemming from inventory shrinkage. When facing a higher inventory shrinkage rate and a greater demand uncertainty, the revenue obtained by selling products can not balance the increased loss, supply chain managers must take more risks caused by inventory shrinkage and demand uncertainty, and the results in the risk-neutral case may be considered as unrealistic. Therefore, the question we are concerned about is how the risk aversion level affects their optimal decisions in supply chain system with or without item-level RFID under conditional value-at-risk (CVaR) criterion (see Section3.1). In addition, many existing studies incorporated price-dependent stochastic demand or RFID technology into supply chain models, but few of them explored RFID technology for retail inventory shrinkage, price-dependent stochastic demand, and risk issue, simultaneously. Our paper will cover these gaps.(2) Related Literature(i) RFID and Inventory Inaccuracy. Although recent researchers have given some literature reviews on RFID technology [12, 13], the related research on RFID implementation in inventory management is relatively new. We here only focus on reviewing some recent studies on analyzing the impact of item-level RFID on the reduction of inventory inaccuracies. De Kok et al. [14] considered cost-benefit trade-off between inventory costs and the costs of RFID regarding shrinkage and proved that this break-even price is highly related to the value of the items that are lost, the shrinkage fraction, and the remaining shrinkage after employing RFID. Heese [15] studied inventory record inaccuracy in a supply chain model with a manufacturer and a retailer and analyzed the impact of inventory record inaccuracy on optimal stocking decisions and profits. By contrasting optimal decisions in a decentralized supply chain with those in an integrated supply chain, they concluded that inventory record inaccuracy exacerbates the inefficiencies resulting from double marginalization in decentralized supply chains. Rekik et al. [16] developed a newsvendor model and analyzed the RFID adoption strategy with coordination to improve the supply chain’s performance under retail inventory inaccuracy that is subject to errors stemming from execution problems. Based on imperfect inventory records and unobserved lost sales, Mersereau [17] also discussed a periodic review inventory system that explores one- and two-period versions of the problem and demonstrated several mechanisms by which the error process and associated record inaccuracy can impact optimal replenishment. Considering identical error distributions and counting costs, Kök and Shang [2] investigated how to design cycle-count policies from the perspective of the entire supply chain for a two-stage system, provided a simple recursion to evaluate the system cost, proposed a heuristic to obtain effective base-stock levels, and proved that it is more effective to conduct more frequent cycle counts at the downstream stage. Taking into account inventory inaccuracy stemming from shrinkage and delivery errors, Sarac et al. [18] investigated a simulation model with RFID technology in a three-level supply chain and evaluated the qualitative and quantitative impacts of RFID technologies on supply chain system performances and profits. The aforementioned articles all assumed demand to be a constant or a stochastic variable. To our knowledge, there has been no work on investigating the impact of inventory shrinkage on supply chain with price-dependent stochastic demand. In this paper, bearing in mind the fact that the demand of product is affected by price, we discuss a supply chain model with retail inventory shrinkage and assess the benefit of the item-level RFID implementation.(ii) Risk Aversion. Risk aversion issues in supply inventory management have been extensively studied in the past decades. There are mean variance (MV), value-at-risk (VaR), and conditional value-at-risk (CVaR) in traditional risk analysis approaches. Gan et al. [19] used MV approach to study the supply chain coordination in a two-echelon supply chain with risk-averse agents. Under MV framework, Choi et al. [20] considered two-echelon supply chain coordination problem when the partners take different/same risk attitudes; the result shows that the whole supply chain coordination depends on how different the risk related thresholds between the two supply chain agents are. Choi [21] developed the supply chain models for a multiperiod retail replenishment problem with and without RFID under the MV framework and analytically discussed the use of RFID under vendor-managed inventory (VMI) scheme in a two-echelon single-manufacturer single-retailer supply chain. Considering the application of RFID technology to eliminate the misplacement problems, Chen et al. [22] focused on analyzing how the risk attitude affects the supply chain members incentives to adopt RFID and the corresponding coordination contract, where the central semideviation is adopted to measure the retailer’s risk attitude. Özler et al. [23] utilized VaR as the risk measure in a newsvendor framework and investigated the multiproduct newsvendor problem under a VaR constraint. Based on price-dependent demands, Chen et al. [24] explored the CVaR objective as the decision criterion in the newsvendor problem. They analyzed the optimal pricing and stocking decisions and derived sufficient conditions for the existence of unique solution and further revealed the neat monotonicity properties associated with the optimal pricing and ordering decisions. Chiu and Choi [25] studied the price-dependent newsvendor problem with a VaR objective; they discussed both the linear and multiplicative price-dependent demand distributions cases and analytically derived the optimal solutions for the problem under a VaR objective. Focusing on VaR constraint and CVaR as the risk measures of the downside risk, Wu et al. [26] investigated profit maximization versus risk approaches for the standard newsvendor problem with uncertainty in demand as well as a generalized version with uncertainty in the shortage cost. Different from MV approaches, the upside of variance is not considered as the risk-averse decision-maker; in reality, the upside of variance can be viewed as the surprising gains from investment. Most risk-averse decision-makers only care about the downside losses rather than the upside gains. Thus, VaR and CVaR approaches are more intuitive and comprehensive to reflect decision-maker’s risk attitude, but, as Ahmed et al. [27] pointed out, compared to VaR approach, CVaR can be consistent with second-order stochastic dominance rules. Furthermore, Section 3.1 also shows that CVaR approach has attractive computational characteristics. However, our work is not to argue how much better CVaR is than the other approaches. Rather, we only utilize CVaR as a risk measurement. Contrary to the above papers, we just limit focusing on the effect of item-level RFID on the inventory shrinkage problem and take into account retailer’s risk caused by demand uncertainty and nonsale shrinkage in supply chain frame.(iii) RFID and Inventory Shrinkage. Inventory shrinkage is the main cause of inventory inaccuracy; there are some related studies to our paper in adopting the RFID technology for eliminating inventory shrinkage. For example, considering the holding cost, Rekik et al. [28] investigated the problem of having theft in store under a service level constraint, and they analyzed the impact of theft errors and the value of the RFID on the inventory system under the assumption of perfect RFID technology which could eliminate theft errors completely, while we suppose that only a part of shrinkage errors can be eliminated due to imperfect RFID, which is closer to reality. Although Fan et al. [29, 30] also studied the problem of RFID technology for reducing inventory shrinkage under imperfect RFID, they assumed that the demand followed a uniform distribution with known parameters when analyzing the threshold value of tag cost, while our work does not have this assumption.Under CVaR criterion, the contribution of this paper is threefold:(1) we derive optimal supply chains decisions with and without item-level RFID in centralized setting and provide a sufficient condition to judge whether to adopt item-level RFID; (2) we also derive optimal supply chains decisions with item-level RFID in decentralized setting and discuss the necessity of supply chain coordination in this case; and (3) we design wholesale price contract with revenue sharing to achieve a win-win situation for supply chain partners.The reminder of the paper is organized as follows. In the following section, model descriptions, notations, and assumptions are presented. Section3 firstly gives definition of CVaR and then focuses on analyzing supply chain’s optimal decisions in centralized case with and without item-level RFID under CVaR criterion and discusses how the optimal decisions change with model parameters; finally, we judge whether to adopt item-level RFID for assessing the benefit of the item-level RFID implementation. In Section 4, we explore the optimal policies for a decentralized supply chain with two widely used contracts including wholesale price contract and revenue sharing contract and discuss supply chain coordination for achieving the best performance of the entire supply chain. Section 5 includes number examples and the sensitivity analysis of the parameters in the proposed models. Finally, we conclude with a summary of this paper and point out the direction for future research.


## 2. Model Descriptions and Assumptions

We consider a two-echelon supply chain with one manufacturer and one retailer. A single short-life-cycle or seasonal product is provided by the manufacturer and then the retailer sells it to end consumers. Noting that there exist nonsale inventory shrinkage phenomena and demand uncertainty in many retail industries and after making the payment to the manufacturer, the retailer manages and maintains the retail inventory system alone; the manufacturer, however, has no money pumped into retail inventory; that is, the retailer bears all risk associated with nonsale inventory shrinkage and demand uncertainty, and the manufacturer has no risk. Thus, we here assume that the retailer is risk-averse and the manufacturer is risk-neutral.To model the impact of the nonsale retail inventory shrinkage problems, we defineα to be the ratio between the sales-available on-hand inventory and the total physical inventory in retail store. Related researches show that item-level RFID can achieve a higher product availability, but the effect of RFID is imperfect and inventory shrinkage problems can be only eliminated partly [31]. Similar to the research of Fan et al. [29] on item-level RFID in retail inventory, considering the inventory shrinkage problems, the following assumption is made in our research: when the retailer does not resort to a smart inventory system with item-level RFID, he or she orders Q units from the manufacturer, and only αQ units are sales-available; the other (1-α)Q units are unavailable to end customers due to inventory shrinkage errors, but when item-level RFID are used in inventory system, β(1-α)Q units for inventory shrinkage can be purchased by end customers, and the other (1-β)(1-α)Q units may remain as nonsale inventory shrinkage problems (see Figure 1).Figure 1
RFID implementation versus no RFID case in the retail inventory.We adopt the following notation throughout this paper:(1)
D(·) denotes end consumer demand during the selling season;(2)
p denotes the retailer sets retail price per unit product;(3)
w denotes the manufacturer sets wholesale price per unit product;(4)
cM denotes manufacturers marginal production cost at the production stage; that is, w>cM;(5)
t denotes RFID tag price per unit product;(6)
cR denotes operating cost per unit product at the retail stage for inventory handling, shelf-space usage, and so forth; p>cR;(7)
f(·) denotes probability density function (PDF);(8)
F(·) denotes cumulative distribution function (CDF);(9)
ε denotes random variable with PDF f(·) and CDF F(·);(10)
xh(x) denotes the general failure rate function of demand distribution;(11)
q denotes the retailers order quantity from the manufacturer during the single period;(12)
d(p) denotes deterministic and decreasing function of retail price p;(13)
l denotes stocking factor of inventory;(14)
ηR denotes the retailer’s risk aversion value; that is, ηR∈(0,1];(15)
ηC denotes the risk aversion value of the whole supply chain; that is, ηC∈0,1;(16)
πCNo(·) denotes total expected channel profit of centralized system without item-level RFID;(17)
Π′(·) denotes total expected channel profit of centralized system without item-level RFID under CVaR criterion; that is, Π′(·)=CVaR(πCNo(·));(18)
πCRF(·) denotes total expected channel profit of centralized system under item-level RFID;(19)
Π(·) denotes total expected channel profit of centralized system with item-level RFID under CVaR criterion; that is, Π(·)=CVaR(πCRF(·));(20)
πRNo(·) denotes the retailer’s expected profit without item-level RFID;(21)
ΠR′(·) denotes the retailer’s expected profit without item-level RFID under CVaR criterion; that is, ΠR′(·)=CVaR(πRNo(·));(22)
πRRF(·) denotes the retailer’s expected profit with item-level RFID;(23)
ΠR(·) denotes the retailer’s expected profit with item-level RFID under CVaR criterion; that is, ΠR(·)=CVaR(πRRF(·));(24)
πMNo(·) denotes the manufacturer’s expected profit without item-level RFID;(25)
ΠM′(·) denotes the manufacturer’s expected profit without item-level RFID under CVaR criterion; that is, ΠR(·)=CVaR(πMNo(·));(26)
πMRF(·) denotes the manufacturer’s expected profit with item-level RFID;(27)
ΠM(·) denotes the manufacturer’s expected profit with item-level RFID under CVaR criterion; that is, ΠM(·)=CVaR(πMRF(·)).In addition, we make the following assumptions.(1)
To limit the number of parameters considered in model analysis, we only consider RFID tag cost; the fixed costs of RFID implementation include reader system, infrastructure, maintenance and support, and IT investments are not part of our model. The detailed assessment of the above fixed costs is provided by several studies [32, 33].(2)
For simplicity, at the end of the selling season, any unsold retail product bears no salvage value or disposal cost in retail store; at the same time, we assume unsatisfied demand incurs no loss of goodwill cost (i.e., shortage penalty). Related studies [34, 35] show that the assumptions of zero salvage value or holding cost and zero loss of goodwill cost are appropriate reflections of reality for season or short life-cycle products.(3)
We assume that the end customer demandD(p) has the multiplicative functional form; that is, D(p)=d(p)ε, where ε is supported on [A′,B′] with B′>A′≥0.  F(·) is strictly increasing and differentiable on [A′,B′], and F(A′)=0, F(B′)=1.(4)
We consider the power form of price-dependent demand factor throughout this paper; that is,d(p)=Ap-k, where A>0, k>1; see Petruzzi and Dada [36] for an excellent review and extensions.(5)
In the power form ofd(p)=Ap-k, following Petruzzi and Dada [36], we define xh(x)≡xf(x)/[η-F(x)] that denotes the GFR (generalized failure rate) function of demand distribution under CVaR criterion; assume that it has the strictly increasing property: xhx′>0. The IGFR (increasing generalized failure rate) assumption is mild condition because it captures the most common distributions, such as the uniform, the normal, and the exponential, as well as the gamma and Weibull families, subject to parameter restrictions.


## 3. Centralized Policies under CVaR Criterion

### 3.1. Definition of CVaR

CVaR measures a conditional expectation of the realized profit when the realized profit is not more than a certain quantile of profit, which is often concerned with risk-averse decision-makers. It is a coherent risk measure with attractive computational characteristics and consequently it is widely used in the financial fields. Following Rockafellar and Uryasev [37, 38] and Wu et al. [26], CVaR maximizes the average profit of the profit falling below a certain quantile level which is defined as the maximum profit at a specified confidence level. More formally, for the given distribution of the profit function π(x,y), CVaR can be treated as follows: (1)CVaRηπx,y=Eπx,y∣πx,y≤Θηy=1η∫πx,y≤Θηπx,ygydy,where E[·] denotes expectation operator and η∈(0,1] reflects the degree of risk aversion; that is, a lower value implies a higher degree of risk aversion and η=1 implies risk neutrality; x denotes decision vector, y denotes random vector, g(y) denotes the probability density function of the random vector y, and Θη denotes η-quantile of the random vector y; that is, (2)Θηy=sup⁡υ∣Prob.πx,y≤υ≤η.In addition, a more generalized formula is introduced to compute CVaR as follows: (3)CVaRηπx,y=maxυ∈R⁡υ+1ηEmin⁡πx,y-υ,0.It is worth mentioning that Rockafellar and Uryasev [37, 38] proved that (1) and (3) are equivalent under the generalized condition, but, as compared to (1), (3) is more convenient to be used in mathematical calculation and analysis. Therefore, we will adopt (3) to model risk-averse problems with retail inventory shrinkage errors in supply chain.


### 3.2. Centralized Models under CVaR Criterion

In the centralized supply chain setting, we consider two different cases, that is, one with item-level RFID and another without item-level RFID. We first give the general expected profit as functions ofp and l and characterize the optimal decisions to the centralized system with item-level RFID, and then we explore optimal decisions to the centralized system with no RFID under CVaR criterion. Finally, for assessing the benefit of the item-level RFID implementation, we give a sufficient condition to make supply chain manager judge whether to adopt item-level RFID.


#### 3.2.1. Model with Item-Level RFID

Based on the above notations and assumptions, the expected profit function of the centralized system with item-level RFID can be written as(4)πCRFp,l=pEmin⁡q′,Dp-cM+cR+tq,where q′ is sales-available product quantity in the retail inventory; that is, q′=αq+β(1-α)q. For the end customer demand D(p)=d(p)ε, following Petruzzi and Dada [36], we define l≡q′/d(p) as stocking factor. By substituting q′=d(p)l into (4), then (4) is equivalent to (5)πCRFp,l=dppl-l-x+-cM+cR+tml,where m=1/[α+β(1-α)] and  (l-x)+=max⁡{0,(l-x)} denotes the quantity of unsold retail product due to demand uncertainty and nonsale inventory shrinkage at the end of the selling season.In what follows, consider the losses caused by demand uncertainty and nonsale inventory shrinkage may lead to the market risk, by the assumption presented in Section2. The retailer is risk-averse and the manufacturer has no any risk, so the retailer risk attitude should be viewed as the whole supply chain risk aversion level; that is, ηC=ηR. The following lemma is listed for obtaining the optimal decisions of the centralized system with item-level RFID under CVaR criterion.Lemma 1.
Under the CVaR constraint, letΠ(p,l,v)=CVaR(πCRF(p,l)); for given p,l, the unique optimal v∗(p,l)=d(p)[pl-(cM+cR+t)ml] maximizes Π(p,l,v), where A′≤l≤F-1(ηR)<B′.Proof.
ByΠ(p,l,v)=CVaR(πCRF(p,l)), from (3), the expected profit function of the centralized system under CVaR criterion is shown by (6)Πp,l,v=maxv∈R⁡v+1ηREmin⁡πCRFp,l-v,0.Substituting (5) into (6), we have (7)Πp,l,v=v-1ηR∫A′B′v-dppl-l-x+-cM+cR+tml+dFx.Equation (7) can be rewritten by (8)Πp,l,v=v-1ηR∫A′lv-dppx-cM+cR+tml+dFx-1ηR∫lB′v-dppl-cM+cR+tml+dFx.
For any givenp and l, we easily get the following.
(
1
) When v≤d(p)[pA′-(cM+cR+t)ml], then (9)Πp,l,v=v;∂Πp,l,v∂v=1.
(
2
) When d(p)[pA′-(cM+cR+t)ml]<v≤d(p)[pl-(cM+cR+t)ml], then (10)Πp,l,v=v-1ηR∫A′v+dpcM+cR+tml/pdpv-dppx-cM+cR+tmldFx,∂Πp,l,v∂v=1-1ηRFv+dpcM+cR+tmlpdp.In particular, (11)∂Πp,l,v∂vv=dppA′-cM+cR+tml=1,∂Πp,l,v∂vv=dppl-cM+cR+tml=1-1ηRFl.
(
3
) When v>d(p)[pl-(cM+cR+t)ml], then (12)Πp,l,v=v-1ηR∫A′lv-dppx-cM+cR+tmldFx-1ηR∫lB′v-dppl-cM+cR+tmldFx,∂Πp,l,v∂v=1-1ηR<0.
Based on the above cases(1)–(3), we can conclude that Π(p,l,v) is a concave function of v. Let v∗(p,l)=arg⁡maxv∈R⁡Π(p,l,v); combining cases (1), 2, and (3), it can be shown that (13)v∗p,l∈dppA′-cM+cR+tml,dppl-cM+cR+tml.Next, in order to prove (14)v∗p,l=dppl-cM+cR+tml,where A′≤l≤F-1ηR<B′, the following discussions are listed.
(a) Ifl<F-1(ηR), then (15)v∗p,l=dppl-cM+cR+tml,and we therefore have (16)Πp,l,v∗p,l=dppl-cM+cR+tml-pηR∫A′ll-xdFx.
(b) Ifl≥F-1(ηR), then (17)v∗p,l=dppF-1ηR-cM+cR+tml,and we have (18)Πp,l,v∗p,l=dppF-1ηR-cM+cR+tml-pηR∫A′F-1ηRFxdx.It follows from (18) that (19)∂Πp,l,v∗p,l∂l=-dpcM+cR+tm<0;that is, Π(p,l,v∗(p,l)) is decreasing in l∈(F-1(ηR),B′), so l=F-1(ηR) maximizes Π(p,l,v∗(p,l)).
In conclusion, combining (a) and (b), it follows from the facts that(20)v∗p,l=dppl-cM+cR+tml,where A′≤l≤F-1ηR<B′. This completes the proof.By Lemma1, we know that, for any given p,l, the unique optimal v∗(p,l)=d(p)[pl-(cM+cR+t)ml] maximizes Π1(p,l,v). By substituting v∗(p,l) into (6), we get the general expected profit function of the centralized system with item-level RFID under CVaR criterion: (21)Πp,l=dpp-cM+cR+tml-pΛlηR,where Λ(l)=∫A′l(l-x)dF(x).Now, the following theorem will give the optimal decision to the centralized system with item-level RFID under CVaR criterion.Theorem 2.
Under the CVaR constraint, for any givenl∈[A′,B′] and d(p)=Ap-k, if xh(x) is IGFR, that is, xhx′>0, the optimal stocking factor (lCRF)∗ is uniquely determined by (22)Fl=ηRl+k-1Λlkl,and the unique optimal order quantity (qCRF)∗ is listed by (23)qCRF∗=AmlCRF∗ηR-FlCRF∗mηRcM+cR+tk.Proof.
Byq=mld(p) and d(p)=Ap-k, we have p=Aml/q1/k. For any given l, q, substituting p=Aml/q1/k into (21), (21) can be written by (24)Πl,q=AqAml-1-k/kl-1ηRΛl-cM+cR+tq,and taking the first-order partial derivation of Π(l,q) with respect to l, we obtain that the necessary condition for maximizing Π(l,q) is (25)∂Πl,q∂l=A1/kq1-1/kkηRl2-1/kηRl-klFl+k-1Λl=0.Let Z(l)=ηRl-klF(l)+(k-1)Λ(l), and notice that the first factor in (25) is always positive, so first necessary condition only requires that the optimal stocking factor (lCRF)∗ satisfies Z(l)=0; solving Z(l)=0, we get the optimal (lCRF)∗ determined by F(l)=(ηRl+k-1Λl)/kl.
Next, we will prove the existence of the optimal(lCRF)∗. It is obvious that Z(l) is continuous in the support set [A′,F-1(ηR)]. After some manipulation, we get Z(A′)=A′ηR>0 and Z(F-1(ηR))=(1-k)ηRF-1(ηR)+(k-1)∫A′F-1(ηR)‍F(x)dx. Since ∫A′F-1ηR‍F(x)dx≤F(F-1(ηR))(F-1(ηR)-A′)=ηR(F-1(ηR)-A′), we have Z(F-1(ηR))≤1-kA′<0; hence, there exists the optimal (lCRF)∗ that satisfies Z(l)=0 in the support set (A′,F-1(ηR)).
Furthermore, to verify the uniqueness of the optimal(lCRF)∗, we have Z′(l)=(ηR-F(l))(1-klh(l)), and Z′′l=-Z′(l)h(l)-kηR-Fllhl′. Since lhl′>0, by Lemma 1, l≤F-1(ηR), l∈[A′,B′], we easily gain Z′′(l)|Z′(l)=0<0, which implies that Z(l) is unimodal function. Thus, the optimal (lCRF)∗ is unique.
From (22), we find that the optimal stocking factor (lCRF)∗ does not depend on the order quantity q. Substituting (22) into (24), we get (26)ΠlCRF∗,q=kA1/km-1+1/klCRF∗1+1/kq1-1/kηR-FlCRF∗k-1ηR-cM+cR+tq.In what follows, we can show that (27)dΠlCRF∗,qdq=A1/km-1+1/klCRF∗1+1/kq-1/kηR-FlCRF∗ηR-cM+cR+t,and solving dΠ((lCRF)∗,q)/dq=0, we get (qCRF)∗=Am(lCRF)∗(ηR-F(lCRF)∗)/mηRcM+cR+tk. Meanwhile, we easily gain d2Π1((lCRF)∗,q)/dq2|q=(qCRF)∗<0. According to the second-order sufficient condition, there exists the unique optimal (qCRF)∗ that maximizes Π((lCRF)∗,q). This completes the proof.The above theorem shows that it does not have any requirement on problem parameters other than the demand distribution itself to determine the optimal decisions of the centralized system under CVaR criterion. It should be pointed out that, in Theorem2, when ηR=1 and α=1, the optimal inventory factor (lCRF)∗ is the same as Wang et al. [39] and Li and Hua [40]. In addition, for the optimal order quantity (qCRF)∗, by p=Aml/q1/k, we can get the optimal retail price (28)pCRF∗=mηRcM+cR+tηR-FlCRF∗.Substituting (pCRF)∗ and (lCRF)∗ into (21), the maximum expected profit of the centralized system with item-level RFID under CVaR criterion is given by (29)ΠlCRF∗=AlCRF∗k-1mcM+cR+tk-1ηR-FlCRF∗ηRk.


#### 3.2.2. Model without Item-Level RFID

Similarly, by the assumption presented above, for the case without item-level RFID (wheret=0 and β=0), let m′=1/α; the expected profit function of the centralized system under no RFID can be written as (30)πCNop,l=dppl-l-x+-cM+cRm′l.Now, let Π′(p,l)=maxv∈R⁡CVaR(πCNo(p,l)); the optimal decision to the centralized system without item-level RFID under CVaR criterion is given by the following theorem.Theorem 3.
In the centralized system without item-level RFID, ifxh(x) is IGFR, then the decision vector ((pCNo)∗,(lCNo)∗) is the unique maximizer of Π′(p,l), where (31)pCNo∗=m′ηRcM+cRηR-FlCNo∗,and (lCNo)∗ is described by (32)Fl=ηRl+k-1Λlkl.Proof.
This proof is similar to the proof procedures of Theorem2; thus, we here omit this proof.Similarly, according to Theorem3, we can easily derive the optimal order quantity as (33)qCNo∗=Am′lCNo∗ηR-FlCNo∗m′ηRcM+cRk,and the maximum expected profit of the centralized system without item-level RFID under CVaR criterion is given by (34)Π′lCNo∗=AlCNo∗k-1m′cM+cRk-1ηR-FlCNo∗ηRk.The following proposition discusses how the optimal decisions change with model parameters in the centralized system under CVaR criterion.Proposition 4.
Ifxh(x) is IGFR, then the following hold.(1)
(lCRF)∗=(lCNo)∗.(2)
Both(lCRF)∗ and (lCNo)∗ are not affected by the sales-available proportion α, but they are increasing in ηR.(3)
Both(pCRF)∗ and (pCNo)∗ are decreasing in α; let G(x)=(xhxkFx-ηR-Fx)/(1-kxhx), and thus
(i)
whenG((lCRF)∗)>0,  (pCRF)∗ is increasing in ηR; when G((lCRF)∗)<0,  (pCRF)∗ is decreasing in ηR; when G((lCRF)∗)=0,  (pCRF)∗ is not affected by ηR;(ii)
whenG((lCNo)∗)>0,  (pCNo-RF)∗ is increasing in ηR; when G((lCNo)∗)<0,  (pCNo)∗ is decreasing in ηR; when G((lCNo)∗)=0,  (pCNo)∗ is not affected by ηR.Proof.
Part (1) Comparing (22) with (32), we can easily get the result that (lCNo)∗=(lCRF)∗.
Part (2) Since (22) and (32) do not involve the sales-available proportion α, both (lCNo)∗ and (lCRF)∗ are not affected by α. From (22), the optimal (lCRF)∗ satisfies Z((lCRF)∗)=0. By the implicit function rule, d(lCRF)∗/dηR=-(lCRF)∗/(∂Z((lCRF)∗)/∂(lCRF)∗), according to proof of Theorem 2, Z(A)=AηR>0,  Z(F-1(ηR))≤(1-k)A<0, and (lCRF)∗ solving Z((lCRF)∗)=0 is unique. It implies ∂Z((lCRF)∗)/∂(lCRF)∗<0. Thus, we get d(lCRF)∗/dηR>0. Similar to the proof procedures of d(lCRF)∗/dηR>0, we can gain d(lCNo)∗/dηR>0.
Part (3) From (28) and (31), we easily show that (pCRF)∗ is increasing in m and (pCNo)∗ is increasing in m′; in conjunction with m=1/[α+β(1-α)] and m′=1/α, they imply that m and m′ are decreasing in α; we thus have the fact that both (pCNo)∗ and (pCRF)∗ are decreasing in α.(i)
From (28), taking the first derivative of (pCRF)∗ with respect to ηR, we have d(pCRF)∗/dηR=((ηRf((lCRF)∗)(d(lCRF)∗/dηR)-F(lCRF)∗)/[ηR-F(lCRF)∗]2)m(cM+cR+t); furthermore, by the proof of Proposition 4(2), we get d(lCRF)∗/dηR=-(lCRF)∗/Z′l|l=(lCRF)∗. By simply substituting d(lCRF)∗/dηR into d(pCRF)∗/dηR, so we derive that d(pCRF)∗/dηR=(mcM+cR+t/[ηR-F(lCRF)∗]2)G((lCRF)∗), which implies that the monotone behavior of (pCRF)∗ can be determined by the sign of G((lCRF)∗). Thus, when G((lCRF)∗)>0,  (pCRF)∗ is increasing in ηR; when G((lCRF)∗)<0, (pCRF)∗ is decreasing in ηR; when G((lCRF)∗)=0,  (pCRF)∗ is not affected by ηR.(ii)
The proof is similar to the proof procedures of Part 3(i); we can gain the monotone behavior of(pCNo)∗ with respect to ηR; thus, we here omit this proof.Proposition4(1) implies that the optimal stocking factor does not depend on whether the centralized system adopts item-level RFID or not, and it seems to depend heavily on demand distribution.Proposition4(2) states that the optimal stocking factors (lCRF)∗ and (lCNo)∗ are independent of the sales-available proportion parameter; they only depend on the risk aversion value and increase with ηR increases. Because a higher value of ηR implies a lower degree of risk aversion, it implies that a lower degree of risk aversion may lead to a higher stocking factor; that is, in the centralized case, if a supply chain manager has less fear of risk, he or she always tends to order more to meet market demand no matter whether to employ item-level RFID or not.Proposition4(3) states that the optimal retail prices (pCRF)∗ and (pCNo)∗ decease with the sales-available proportion increases; it means that a higher sales-available rate may be able to make supply chain manager set a lower retail price for attracting customers to buy more, but the relationship between the optimal retail price and the risk aversion level does not absolutely increase or decrease; it depends on the sign of G((lCRF)∗) or G((lCNo)∗); that is, for adopting item-level RFID case, when G((lCRF)∗)>0, the optimal retail price increases with ηR increases, and it implies that if a supply chain manager is risk-averse enough, he or she is more likely to set a lower retail price to avoid the risk caused by market uncertainty and nonsale inventory shrinkage; when G((lCRF)∗)<0, the optimal retail price decreases with ηR increases; it means that if a supply chain manager has less fear of risk, he or she may raise his/her retail price and order less to balance the relationship between the expected benefit and the risk; when G((lCRF)∗)=0, the optimal retail price is not affected on the risk aversion level; it only depends on some special demand distributions.To assess the benefit of the item-level RFID implementation in the centralized situation under CVaR criterion, we introduce the auxiliary function asΔ=ln[Π((lCRF)∗)/Π′((lCNo)∗)], where Π((lCRF)∗) and Π′((lCNo)∗) are given by (29) and (34), respectively. Note that the auxiliary function Δ can be used to judge whether to adopt item-level RFID; that is, if Δ>0, it means an item-level RFID implementation can bring more expected profit than no RFID case in centralized system, but if Δ≤0, it means that, as compared to item-level RFID system, one case without RFID is a better choice. We will discuss how the model parameters affect item-level RFID implementation in the following.Proposition 5.
(1)Δ is independent of ηR and is decreasing in t;
(2)Δ is decreasing in α but is increasing in β.Proof.
Part (1) By (29) and (34), after some single algebra, we derive the function as (35)Δ=k-1ln⁡α+β1-αα·cM+cRcM+cR+t.From (35), Δ does not have ηR, so Δ is independent of ηR, and the conclusion that Δ decreases in t is obvious.
Part (2) Equation (35) can also be written as Δ=k-1ln⁡[[(1-β)+β/α]·((cM+cR)/(cM+cR+t))], so we easily reach the conclusion that Δ is decreasing in α. Using a similar argument, we can also gain that Δ is increasing in β.
In fact, (35) can be viewed as a sufficient condition to judge whether to adopt item-level RFID, and Proposition 5 states that, although the retailer is risk-averse, the judgment function Δ is not affected by the risk-averse level ηR; it only depends on some parameters, such as, α, β, and t. In other words, the risk-averse level is not an effective incentive for supply chain manager to adopt item-level RFID system; however, the sales-available rate and the tag cost are mainly driving factors in evaluating the benefit of an item-level RFID. In light of this, we will give the threshold values of α, β, and t in the following theorem.Theorem 6.
Under the CVaR constraint, for0<α, β≤1, if t∈(0,t¯) (t¯=β(cM+cR)(1-α)/α), then Δ>0; if t∈[t¯,+∞), then Δ≤0.Proof.
From (35), in order to show Δ>0 (≤0), we only need to show [(1-β)+β/α]·((cM+cR)/(cM+cR+t))>1 (≤1), which can be written by t<(≥)   t¯=β(cM+cR)(1-α)/α. Therefore, we have the following: if t∈(0,t¯), then Δ>0; if t∈[t¯,+∞), then Δ(α,β,t)≤0.Theorem6 gives a threshold value of t; that is, t¯=β(1-α)(cM+cR)/α, and when the RFID tag cost t is lower than threshold value t¯, item-level RFID implementation can bring more expected profit; otherwise, the supply chain will suffer losses at t≥t¯. Likewise, the threshold values of α, β are summarized in Table 1 for more details, and, furthermore, the impact of the key parameters on supply chains optimal decisions will be discussed in Example 1.Table 1
Item-level RFID implementation cases for the key parametersα,β, and t.
The parameter
The parameter’s threshold value
The parameter interval
The sign ofΔ
Use RFID?
α
α
¯
=
β
c
M
+
c
R
β
c
M
+
c
R
+
t
α
∈
0
,
α
¯ α∈α¯,+∞
>0≤0
Yes No
β
β
¯
=
α
t
1
-
α
c
M
+
c
R
β
∈
0
,
β
¯ β∈β¯,+∞
<0≥0
NoYes
t
t
¯
=
β
1
-
α
c
M
+
c
R
α
t
∈
0
,
t
¯ t∈t¯,+∞
>0≤0
YesNoFrom the discussion above, we know that the model with item-level RFID is more generalized than no RFID case in the centralized supply chain system; that is, whent=0 and m=m′, the model with item-level RFID reduces to the model without RFID system. Therefore, we only explore one case with RFID technology in the following decentralized supply chain analysis; the other scenarios are shown in Table 2.Table 2
Summary of optimal decisions and profits in different scenarios under CVaR criterion.
Decentralized systems
Centralized systems
RFID
No RFID
RFID
No RFID
Optimal sale price
k
k
-
1
m
M
N
k
k
-
1
m
′
M
′
N
′
m
M
N
m
′
M
′
N
′
Optimal order quantity
k
-
1
k
k
A
m
l
R
R
F
∗
m
M
N
k
k
-
1
k
k
A
m
′
l
R
N
o
∗
m
′
M
N
′
k
A
m
l
C
R
F
∗
m
M
N
k
A
m
′
l
C
N
o
∗
m
′
M
N
′
k
Optimal wholesale price
t
+
c
R
+
k
c
M
k
-
1
c
R
+
k
c
M
k
-
1
—
—
Retailer’s expected profit
k
-
1
k
k
-
1
Π
∗
k
-
1
k
k
-
1
Π
′
∗
—
—
Manufacturer’s expected profit
k
-
1
k
k
Π
∗
k
-
1
k
k
Π
′
∗
—
—
Supply chain expected profit
k
-
1
k
k
-
1
+
k
-
1
k
k
Π
∗
k
-
1
k
k
-
1
+
k
-
1
k
k
Π
′
∗
Π
∗
Π
′
∗
Note.
M
=
M
′
=
η
R
/
(
η
R
-
F
l
C
R
F
∗
), N=cM+cR+t, N′=cM+cR, Π∗=A(lCRF)∗/(k-1)(mN)k-1Mk, and Π′∗=A(lCRF)∗/(k-1)(m′N′)k-1Mk.


## 3.1. Definition of CVaR

CVaR measures a conditional expectation of the realized profit when the realized profit is not more than a certain quantile of profit, which is often concerned with risk-averse decision-makers. It is a coherent risk measure with attractive computational characteristics and consequently it is widely used in the financial fields. Following Rockafellar and Uryasev [37, 38] and Wu et al. [26], CVaR maximizes the average profit of the profit falling below a certain quantile level which is defined as the maximum profit at a specified confidence level. More formally, for the given distribution of the profit function π(x,y), CVaR can be treated as follows: (1)CVaRηπx,y=Eπx,y∣πx,y≤Θηy=1η∫πx,y≤Θηπx,ygydy,where E[·] denotes expectation operator and η∈(0,1] reflects the degree of risk aversion; that is, a lower value implies a higher degree of risk aversion and η=1 implies risk neutrality; x denotes decision vector, y denotes random vector, g(y) denotes the probability density function of the random vector y, and Θη denotes η-quantile of the random vector y; that is, (2)Θηy=sup⁡υ∣Prob.πx,y≤υ≤η.In addition, a more generalized formula is introduced to compute CVaR as follows: (3)CVaRηπx,y=maxυ∈R⁡υ+1ηEmin⁡πx,y-υ,0.It is worth mentioning that Rockafellar and Uryasev [37, 38] proved that (1) and (3) are equivalent under the generalized condition, but, as compared to (1), (3) is more convenient to be used in mathematical calculation and analysis. Therefore, we will adopt (3) to model risk-averse problems with retail inventory shrinkage errors in supply chain.


## 3.2. Centralized Models under CVaR Criterion

In the centralized supply chain setting, we consider two different cases, that is, one with item-level RFID and another without item-level RFID. We first give the general expected profit as functions ofp and l and characterize the optimal decisions to the centralized system with item-level RFID, and then we explore optimal decisions to the centralized system with no RFID under CVaR criterion. Finally, for assessing the benefit of the item-level RFID implementation, we give a sufficient condition to make supply chain manager judge whether to adopt item-level RFID.


### 3.2.1. Model with Item-Level RFID

Based on the above notations and assumptions, the expected profit function of the centralized system with item-level RFID can be written as(4)πCRFp,l=pEmin⁡q′,Dp-cM+cR+tq,where q′ is sales-available product quantity in the retail inventory; that is, q′=αq+β(1-α)q. For the end customer demand D(p)=d(p)ε, following Petruzzi and Dada [36], we define l≡q′/d(p) as stocking factor. By substituting q′=d(p)l into (4), then (4) is equivalent to (5)πCRFp,l=dppl-l-x+-cM+cR+tml,where m=1/[α+β(1-α)] and  (l-x)+=max⁡{0,(l-x)} denotes the quantity of unsold retail product due to demand uncertainty and nonsale inventory shrinkage at the end of the selling season.In what follows, consider the losses caused by demand uncertainty and nonsale inventory shrinkage may lead to the market risk, by the assumption presented in Section2. The retailer is risk-averse and the manufacturer has no any risk, so the retailer risk attitude should be viewed as the whole supply chain risk aversion level; that is, ηC=ηR. The following lemma is listed for obtaining the optimal decisions of the centralized system with item-level RFID under CVaR criterion.Lemma 1.
Under the CVaR constraint, letΠ(p,l,v)=CVaR(πCRF(p,l)); for given p,l, the unique optimal v∗(p,l)=d(p)[pl-(cM+cR+t)ml] maximizes Π(p,l,v), where A′≤l≤F-1(ηR)<B′.Proof.
ByΠ(p,l,v)=CVaR(πCRF(p,l)), from (3), the expected profit function of the centralized system under CVaR criterion is shown by (6)Πp,l,v=maxv∈R⁡v+1ηREmin⁡πCRFp,l-v,0.Substituting (5) into (6), we have (7)Πp,l,v=v-1ηR∫A′B′v-dppl-l-x+-cM+cR+tml+dFx.Equation (7) can be rewritten by (8)Πp,l,v=v-1ηR∫A′lv-dppx-cM+cR+tml+dFx-1ηR∫lB′v-dppl-cM+cR+tml+dFx.
For any givenp and l, we easily get the following.
(
1
) When v≤d(p)[pA′-(cM+cR+t)ml], then (9)Πp,l,v=v;∂Πp,l,v∂v=1.
(
2
) When d(p)[pA′-(cM+cR+t)ml]<v≤d(p)[pl-(cM+cR+t)ml], then (10)Πp,l,v=v-1ηR∫A′v+dpcM+cR+tml/pdpv-dppx-cM+cR+tmldFx,∂Πp,l,v∂v=1-1ηRFv+dpcM+cR+tmlpdp.In particular, (11)∂Πp,l,v∂vv=dppA′-cM+cR+tml=1,∂Πp,l,v∂vv=dppl-cM+cR+tml=1-1ηRFl.
(
3
) When v>d(p)[pl-(cM+cR+t)ml], then (12)Πp,l,v=v-1ηR∫A′lv-dppx-cM+cR+tmldFx-1ηR∫lB′v-dppl-cM+cR+tmldFx,∂Πp,l,v∂v=1-1ηR<0.
Based on the above cases(1)–(3), we can conclude that Π(p,l,v) is a concave function of v. Let v∗(p,l)=arg⁡maxv∈R⁡Π(p,l,v); combining cases (1), 2, and (3), it can be shown that (13)v∗p,l∈dppA′-cM+cR+tml,dppl-cM+cR+tml.Next, in order to prove (14)v∗p,l=dppl-cM+cR+tml,where A′≤l≤F-1ηR<B′, the following discussions are listed.
(a) Ifl<F-1(ηR), then (15)v∗p,l=dppl-cM+cR+tml,and we therefore have (16)Πp,l,v∗p,l=dppl-cM+cR+tml-pηR∫A′ll-xdFx.
(b) Ifl≥F-1(ηR), then (17)v∗p,l=dppF-1ηR-cM+cR+tml,and we have (18)Πp,l,v∗p,l=dppF-1ηR-cM+cR+tml-pηR∫A′F-1ηRFxdx.It follows from (18) that (19)∂Πp,l,v∗p,l∂l=-dpcM+cR+tm<0;that is, Π(p,l,v∗(p,l)) is decreasing in l∈(F-1(ηR),B′), so l=F-1(ηR) maximizes Π(p,l,v∗(p,l)).
In conclusion, combining (a) and (b), it follows from the facts that(20)v∗p,l=dppl-cM+cR+tml,where A′≤l≤F-1ηR<B′. This completes the proof.By Lemma1, we know that, for any given p,l, the unique optimal v∗(p,l)=d(p)[pl-(cM+cR+t)ml] maximizes Π1(p,l,v). By substituting v∗(p,l) into (6), we get the general expected profit function of the centralized system with item-level RFID under CVaR criterion: (21)Πp,l=dpp-cM+cR+tml-pΛlηR,where Λ(l)=∫A′l(l-x)dF(x).Now, the following theorem will give the optimal decision to the centralized system with item-level RFID under CVaR criterion.Theorem 2.
Under the CVaR constraint, for any givenl∈[A′,B′] and d(p)=Ap-k, if xh(x) is IGFR, that is, xhx′>0, the optimal stocking factor (lCRF)∗ is uniquely determined by (22)Fl=ηRl+k-1Λlkl,and the unique optimal order quantity (qCRF)∗ is listed by (23)qCRF∗=AmlCRF∗ηR-FlCRF∗mηRcM+cR+tk.Proof.
Byq=mld(p) and d(p)=Ap-k, we have p=Aml/q1/k. For any given l, q, substituting p=Aml/q1/k into (21), (21) can be written by (24)Πl,q=AqAml-1-k/kl-1ηRΛl-cM+cR+tq,and taking the first-order partial derivation of Π(l,q) with respect to l, we obtain that the necessary condition for maximizing Π(l,q) is (25)∂Πl,q∂l=A1/kq1-1/kkηRl2-1/kηRl-klFl+k-1Λl=0.Let Z(l)=ηRl-klF(l)+(k-1)Λ(l), and notice that the first factor in (25) is always positive, so first necessary condition only requires that the optimal stocking factor (lCRF)∗ satisfies Z(l)=0; solving Z(l)=0, we get the optimal (lCRF)∗ determined by F(l)=(ηRl+k-1Λl)/kl.
Next, we will prove the existence of the optimal(lCRF)∗. It is obvious that Z(l) is continuous in the support set [A′,F-1(ηR)]. After some manipulation, we get Z(A′)=A′ηR>0 and Z(F-1(ηR))=(1-k)ηRF-1(ηR)+(k-1)∫A′F-1(ηR)‍F(x)dx. Since ∫A′F-1ηR‍F(x)dx≤F(F-1(ηR))(F-1(ηR)-A′)=ηR(F-1(ηR)-A′), we have Z(F-1(ηR))≤1-kA′<0; hence, there exists the optimal (lCRF)∗ that satisfies Z(l)=0 in the support set (A′,F-1(ηR)).
Furthermore, to verify the uniqueness of the optimal(lCRF)∗, we have Z′(l)=(ηR-F(l))(1-klh(l)), and Z′′l=-Z′(l)h(l)-kηR-Fllhl′. Since lhl′>0, by Lemma 1, l≤F-1(ηR), l∈[A′,B′], we easily gain Z′′(l)|Z′(l)=0<0, which implies that Z(l) is unimodal function. Thus, the optimal (lCRF)∗ is unique.
From (22), we find that the optimal stocking factor (lCRF)∗ does not depend on the order quantity q. Substituting (22) into (24), we get (26)ΠlCRF∗,q=kA1/km-1+1/klCRF∗1+1/kq1-1/kηR-FlCRF∗k-1ηR-cM+cR+tq.In what follows, we can show that (27)dΠlCRF∗,qdq=A1/km-1+1/klCRF∗1+1/kq-1/kηR-FlCRF∗ηR-cM+cR+t,and solving dΠ((lCRF)∗,q)/dq=0, we get (qCRF)∗=Am(lCRF)∗(ηR-F(lCRF)∗)/mηRcM+cR+tk. Meanwhile, we easily gain d2Π1((lCRF)∗,q)/dq2|q=(qCRF)∗<0. According to the second-order sufficient condition, there exists the unique optimal (qCRF)∗ that maximizes Π((lCRF)∗,q). This completes the proof.The above theorem shows that it does not have any requirement on problem parameters other than the demand distribution itself to determine the optimal decisions of the centralized system under CVaR criterion. It should be pointed out that, in Theorem2, when ηR=1 and α=1, the optimal inventory factor (lCRF)∗ is the same as Wang et al. [39] and Li and Hua [40]. In addition, for the optimal order quantity (qCRF)∗, by p=Aml/q1/k, we can get the optimal retail price (28)pCRF∗=mηRcM+cR+tηR-FlCRF∗.Substituting (pCRF)∗ and (lCRF)∗ into (21), the maximum expected profit of the centralized system with item-level RFID under CVaR criterion is given by (29)ΠlCRF∗=AlCRF∗k-1mcM+cR+tk-1ηR-FlCRF∗ηRk.


### 3.2.2. Model without Item-Level RFID

Similarly, by the assumption presented above, for the case without item-level RFID (wheret=0 and β=0), let m′=1/α; the expected profit function of the centralized system under no RFID can be written as (30)πCNop,l=dppl-l-x+-cM+cRm′l.Now, let Π′(p,l)=maxv∈R⁡CVaR(πCNo(p,l)); the optimal decision to the centralized system without item-level RFID under CVaR criterion is given by the following theorem.Theorem 3.
In the centralized system without item-level RFID, ifxh(x) is IGFR, then the decision vector ((pCNo)∗,(lCNo)∗) is the unique maximizer of Π′(p,l), where (31)pCNo∗=m′ηRcM+cRηR-FlCNo∗,and (lCNo)∗ is described by (32)Fl=ηRl+k-1Λlkl.Proof.
This proof is similar to the proof procedures of Theorem2; thus, we here omit this proof.Similarly, according to Theorem3, we can easily derive the optimal order quantity as (33)qCNo∗=Am′lCNo∗ηR-FlCNo∗m′ηRcM+cRk,and the maximum expected profit of the centralized system without item-level RFID under CVaR criterion is given by (34)Π′lCNo∗=AlCNo∗k-1m′cM+cRk-1ηR-FlCNo∗ηRk.The following proposition discusses how the optimal decisions change with model parameters in the centralized system under CVaR criterion.Proposition 4.
Ifxh(x) is IGFR, then the following hold.(1)
(lCRF)∗=(lCNo)∗.(2)
Both(lCRF)∗ and (lCNo)∗ are not affected by the sales-available proportion α, but they are increasing in ηR.(3)
Both(pCRF)∗ and (pCNo)∗ are decreasing in α; let G(x)=(xhxkFx-ηR-Fx)/(1-kxhx), and thus
(i)
whenG((lCRF)∗)>0,  (pCRF)∗ is increasing in ηR; when G((lCRF)∗)<0,  (pCRF)∗ is decreasing in ηR; when G((lCRF)∗)=0,  (pCRF)∗ is not affected by ηR;(ii)
whenG((lCNo)∗)>0,  (pCNo-RF)∗ is increasing in ηR; when G((lCNo)∗)<0,  (pCNo)∗ is decreasing in ηR; when G((lCNo)∗)=0,  (pCNo)∗ is not affected by ηR.Proof.
Part (1) Comparing (22) with (32), we can easily get the result that (lCNo)∗=(lCRF)∗.
Part (2) Since (22) and (32) do not involve the sales-available proportion α, both (lCNo)∗ and (lCRF)∗ are not affected by α. From (22), the optimal (lCRF)∗ satisfies Z((lCRF)∗)=0. By the implicit function rule, d(lCRF)∗/dηR=-(lCRF)∗/(∂Z((lCRF)∗)/∂(lCRF)∗), according to proof of Theorem 2, Z(A)=AηR>0,  Z(F-1(ηR))≤(1-k)A<0, and (lCRF)∗ solving Z((lCRF)∗)=0 is unique. It implies ∂Z((lCRF)∗)/∂(lCRF)∗<0. Thus, we get d(lCRF)∗/dηR>0. Similar to the proof procedures of d(lCRF)∗/dηR>0, we can gain d(lCNo)∗/dηR>0.
Part (3) From (28) and (31), we easily show that (pCRF)∗ is increasing in m and (pCNo)∗ is increasing in m′; in conjunction with m=1/[α+β(1-α)] and m′=1/α, they imply that m and m′ are decreasing in α; we thus have the fact that both (pCNo)∗ and (pCRF)∗ are decreasing in α.(i)
From (28), taking the first derivative of (pCRF)∗ with respect to ηR, we have d(pCRF)∗/dηR=((ηRf((lCRF)∗)(d(lCRF)∗/dηR)-F(lCRF)∗)/[ηR-F(lCRF)∗]2)m(cM+cR+t); furthermore, by the proof of Proposition 4(2), we get d(lCRF)∗/dηR=-(lCRF)∗/Z′l|l=(lCRF)∗. By simply substituting d(lCRF)∗/dηR into d(pCRF)∗/dηR, so we derive that d(pCRF)∗/dηR=(mcM+cR+t/[ηR-F(lCRF)∗]2)G((lCRF)∗), which implies that the monotone behavior of (pCRF)∗ can be determined by the sign of G((lCRF)∗). Thus, when G((lCRF)∗)>0,  (pCRF)∗ is increasing in ηR; when G((lCRF)∗)<0, (pCRF)∗ is decreasing in ηR; when G((lCRF)∗)=0,  (pCRF)∗ is not affected by ηR.(ii)
The proof is similar to the proof procedures of Part 3(i); we can gain the monotone behavior of(pCNo)∗ with respect to ηR; thus, we here omit this proof.Proposition4(1) implies that the optimal stocking factor does not depend on whether the centralized system adopts item-level RFID or not, and it seems to depend heavily on demand distribution.Proposition4(2) states that the optimal stocking factors (lCRF)∗ and (lCNo)∗ are independent of the sales-available proportion parameter; they only depend on the risk aversion value and increase with ηR increases. Because a higher value of ηR implies a lower degree of risk aversion, it implies that a lower degree of risk aversion may lead to a higher stocking factor; that is, in the centralized case, if a supply chain manager has less fear of risk, he or she always tends to order more to meet market demand no matter whether to employ item-level RFID or not.Proposition4(3) states that the optimal retail prices (pCRF)∗ and (pCNo)∗ decease with the sales-available proportion increases; it means that a higher sales-available rate may be able to make supply chain manager set a lower retail price for attracting customers to buy more, but the relationship between the optimal retail price and the risk aversion level does not absolutely increase or decrease; it depends on the sign of G((lCRF)∗) or G((lCNo)∗); that is, for adopting item-level RFID case, when G((lCRF)∗)>0, the optimal retail price increases with ηR increases, and it implies that if a supply chain manager is risk-averse enough, he or she is more likely to set a lower retail price to avoid the risk caused by market uncertainty and nonsale inventory shrinkage; when G((lCRF)∗)<0, the optimal retail price decreases with ηR increases; it means that if a supply chain manager has less fear of risk, he or she may raise his/her retail price and order less to balance the relationship between the expected benefit and the risk; when G((lCRF)∗)=0, the optimal retail price is not affected on the risk aversion level; it only depends on some special demand distributions.To assess the benefit of the item-level RFID implementation in the centralized situation under CVaR criterion, we introduce the auxiliary function asΔ=ln[Π((lCRF)∗)/Π′((lCNo)∗)], where Π((lCRF)∗) and Π′((lCNo)∗) are given by (29) and (34), respectively. Note that the auxiliary function Δ can be used to judge whether to adopt item-level RFID; that is, if Δ>0, it means an item-level RFID implementation can bring more expected profit than no RFID case in centralized system, but if Δ≤0, it means that, as compared to item-level RFID system, one case without RFID is a better choice. We will discuss how the model parameters affect item-level RFID implementation in the following.Proposition 5.
(1)Δ is independent of ηR and is decreasing in t;
(2)Δ is decreasing in α but is increasing in β.Proof.
Part (1) By (29) and (34), after some single algebra, we derive the function as (35)Δ=k-1ln⁡α+β1-αα·cM+cRcM+cR+t.From (35), Δ does not have ηR, so Δ is independent of ηR, and the conclusion that Δ decreases in t is obvious.
Part (2) Equation (35) can also be written as Δ=k-1ln⁡[[(1-β)+β/α]·((cM+cR)/(cM+cR+t))], so we easily reach the conclusion that Δ is decreasing in α. Using a similar argument, we can also gain that Δ is increasing in β.
In fact, (35) can be viewed as a sufficient condition to judge whether to adopt item-level RFID, and Proposition 5 states that, although the retailer is risk-averse, the judgment function Δ is not affected by the risk-averse level ηR; it only depends on some parameters, such as, α, β, and t. In other words, the risk-averse level is not an effective incentive for supply chain manager to adopt item-level RFID system; however, the sales-available rate and the tag cost are mainly driving factors in evaluating the benefit of an item-level RFID. In light of this, we will give the threshold values of α, β, and t in the following theorem.Theorem 6.
Under the CVaR constraint, for0<α, β≤1, if t∈(0,t¯) (t¯=β(cM+cR)(1-α)/α), then Δ>0; if t∈[t¯,+∞), then Δ≤0.Proof.
From (35), in order to show Δ>0 (≤0), we only need to show [(1-β)+β/α]·((cM+cR)/(cM+cR+t))>1 (≤1), which can be written by t<(≥)   t¯=β(cM+cR)(1-α)/α. Therefore, we have the following: if t∈(0,t¯), then Δ>0; if t∈[t¯,+∞), then Δ(α,β,t)≤0.Theorem6 gives a threshold value of t; that is, t¯=β(1-α)(cM+cR)/α, and when the RFID tag cost t is lower than threshold value t¯, item-level RFID implementation can bring more expected profit; otherwise, the supply chain will suffer losses at t≥t¯. Likewise, the threshold values of α, β are summarized in Table 1 for more details, and, furthermore, the impact of the key parameters on supply chains optimal decisions will be discussed in Example 1.Table 1
Item-level RFID implementation cases for the key parametersα,β, and t.
The parameter
The parameter’s threshold value
The parameter interval
The sign ofΔ
Use RFID?
α
α
¯
=
β
c
M
+
c
R
β
c
M
+
c
R
+
t
α
∈
0
,
α
¯ α∈α¯,+∞
>0≤0
Yes No
β
β
¯
=
α
t
1
-
α
c
M
+
c
R
β
∈
0
,
β
¯ β∈β¯,+∞
<0≥0
NoYes
t
t
¯
=
β
1
-
α
c
M
+
c
R
α
t
∈
0
,
t
¯ t∈t¯,+∞
>0≤0
YesNoFrom the discussion above, we know that the model with item-level RFID is more generalized than no RFID case in the centralized supply chain system; that is, whent=0 and m=m′, the model with item-level RFID reduces to the model without RFID system. Therefore, we only explore one case with RFID technology in the following decentralized supply chain analysis; the other scenarios are shown in Table 2.Table 2
Summary of optimal decisions and profits in different scenarios under CVaR criterion.
Decentralized systems
Centralized systems
RFID
No RFID
RFID
No RFID
Optimal sale price
k
k
-
1
m
M
N
k
k
-
1
m
′
M
′
N
′
m
M
N
m
′
M
′
N
′
Optimal order quantity
k
-
1
k
k
A
m
l
R
R
F
∗
m
M
N
k
k
-
1
k
k
A
m
′
l
R
N
o
∗
m
′
M
N
′
k
A
m
l
C
R
F
∗
m
M
N
k
A
m
′
l
C
N
o
∗
m
′
M
N
′
k
Optimal wholesale price
t
+
c
R
+
k
c
M
k
-
1
c
R
+
k
c
M
k
-
1
—
—
Retailer’s expected profit
k
-
1
k
k
-
1
Π
∗
k
-
1
k
k
-
1
Π
′
∗
—
—
Manufacturer’s expected profit
k
-
1
k
k
Π
∗
k
-
1
k
k
Π
′
∗
—
—
Supply chain expected profit
k
-
1
k
k
-
1
+
k
-
1
k
k
Π
∗
k
-
1
k
k
-
1
+
k
-
1
k
k
Π
′
∗
Π
∗
Π
′
∗
Note.
M
=
M
′
=
η
R
/
(
η
R
-
F
l
C
R
F
∗
), N=cM+cR+t, N′=cM+cR, Π∗=A(lCRF)∗/(k-1)(mN)k-1Mk, and Π′∗=A(lCRF)∗/(k-1)(m′N′)k-1Mk.


## 3.2.1. Model with Item-Level RFID

Based on the above notations and assumptions, the expected profit function of the centralized system with item-level RFID can be written as(4)πCRFp,l=pEmin⁡q′,Dp-cM+cR+tq,where q′ is sales-available product quantity in the retail inventory; that is, q′=αq+β(1-α)q. For the end customer demand D(p)=d(p)ε, following Petruzzi and Dada [36], we define l≡q′/d(p) as stocking factor. By substituting q′=d(p)l into (4), then (4) is equivalent to (5)πCRFp,l=dppl-l-x+-cM+cR+tml,where m=1/[α+β(1-α)] and  (l-x)+=max⁡{0,(l-x)} denotes the quantity of unsold retail product due to demand uncertainty and nonsale inventory shrinkage at the end of the selling season.In what follows, consider the losses caused by demand uncertainty and nonsale inventory shrinkage may lead to the market risk, by the assumption presented in Section2. The retailer is risk-averse and the manufacturer has no any risk, so the retailer risk attitude should be viewed as the whole supply chain risk aversion level; that is, ηC=ηR. The following lemma is listed for obtaining the optimal decisions of the centralized system with item-level RFID under CVaR criterion.Lemma 1.
Under the CVaR constraint, letΠ(p,l,v)=CVaR(πCRF(p,l)); for given p,l, the unique optimal v∗(p,l)=d(p)[pl-(cM+cR+t)ml] maximizes Π(p,l,v), where A′≤l≤F-1(ηR)<B′.Proof.
ByΠ(p,l,v)=CVaR(πCRF(p,l)), from (3), the expected profit function of the centralized system under CVaR criterion is shown by (6)Πp,l,v=maxv∈R⁡v+1ηREmin⁡πCRFp,l-v,0.Substituting (5) into (6), we have (7)Πp,l,v=v-1ηR∫A′B′v-dppl-l-x+-cM+cR+tml+dFx.Equation (7) can be rewritten by (8)Πp,l,v=v-1ηR∫A′lv-dppx-cM+cR+tml+dFx-1ηR∫lB′v-dppl-cM+cR+tml+dFx.
For any givenp and l, we easily get the following.
(
1
) When v≤d(p)[pA′-(cM+cR+t)ml], then (9)Πp,l,v=v;∂Πp,l,v∂v=1.
(
2
) When d(p)[pA′-(cM+cR+t)ml]<v≤d(p)[pl-(cM+cR+t)ml], then (10)Πp,l,v=v-1ηR∫A′v+dpcM+cR+tml/pdpv-dppx-cM+cR+tmldFx,∂Πp,l,v∂v=1-1ηRFv+dpcM+cR+tmlpdp.In particular, (11)∂Πp,l,v∂vv=dppA′-cM+cR+tml=1,∂Πp,l,v∂vv=dppl-cM+cR+tml=1-1ηRFl.
(
3
) When v>d(p)[pl-(cM+cR+t)ml], then (12)Πp,l,v=v-1ηR∫A′lv-dppx-cM+cR+tmldFx-1ηR∫lB′v-dppl-cM+cR+tmldFx,∂Πp,l,v∂v=1-1ηR<0.
Based on the above cases(1)–(3), we can conclude that Π(p,l,v) is a concave function of v. Let v∗(p,l)=arg⁡maxv∈R⁡Π(p,l,v); combining cases (1), 2, and (3), it can be shown that (13)v∗p,l∈dppA′-cM+cR+tml,dppl-cM+cR+tml.Next, in order to prove (14)v∗p,l=dppl-cM+cR+tml,where A′≤l≤F-1ηR<B′, the following discussions are listed.
(a) Ifl<F-1(ηR), then (15)v∗p,l=dppl-cM+cR+tml,and we therefore have (16)Πp,l,v∗p,l=dppl-cM+cR+tml-pηR∫A′ll-xdFx.
(b) Ifl≥F-1(ηR), then (17)v∗p,l=dppF-1ηR-cM+cR+tml,and we have (18)Πp,l,v∗p,l=dppF-1ηR-cM+cR+tml-pηR∫A′F-1ηRFxdx.It follows from (18) that (19)∂Πp,l,v∗p,l∂l=-dpcM+cR+tm<0;that is, Π(p,l,v∗(p,l)) is decreasing in l∈(F-1(ηR),B′), so l=F-1(ηR) maximizes Π(p,l,v∗(p,l)).
In conclusion, combining (a) and (b), it follows from the facts that(20)v∗p,l=dppl-cM+cR+tml,where A′≤l≤F-1ηR<B′. This completes the proof.By Lemma1, we know that, for any given p,l, the unique optimal v∗(p,l)=d(p)[pl-(cM+cR+t)ml] maximizes Π1(p,l,v). By substituting v∗(p,l) into (6), we get the general expected profit function of the centralized system with item-level RFID under CVaR criterion: (21)Πp,l=dpp-cM+cR+tml-pΛlηR,where Λ(l)=∫A′l(l-x)dF(x).Now, the following theorem will give the optimal decision to the centralized system with item-level RFID under CVaR criterion.Theorem 2.
Under the CVaR constraint, for any givenl∈[A′,B′] and d(p)=Ap-k, if xh(x) is IGFR, that is, xhx′>0, the optimal stocking factor (lCRF)∗ is uniquely determined by (22)Fl=ηRl+k-1Λlkl,and the unique optimal order quantity (qCRF)∗ is listed by (23)qCRF∗=AmlCRF∗ηR-FlCRF∗mηRcM+cR+tk.Proof.
Byq=mld(p) and d(p)=Ap-k, we have p=Aml/q1/k. For any given l, q, substituting p=Aml/q1/k into (21), (21) can be written by (24)Πl,q=AqAml-1-k/kl-1ηRΛl-cM+cR+tq,and taking the first-order partial derivation of Π(l,q) with respect to l, we obtain that the necessary condition for maximizing Π(l,q) is (25)∂Πl,q∂l=A1/kq1-1/kkηRl2-1/kηRl-klFl+k-1Λl=0.Let Z(l)=ηRl-klF(l)+(k-1)Λ(l), and notice that the first factor in (25) is always positive, so first necessary condition only requires that the optimal stocking factor (lCRF)∗ satisfies Z(l)=0; solving Z(l)=0, we get the optimal (lCRF)∗ determined by F(l)=(ηRl+k-1Λl)/kl.
Next, we will prove the existence of the optimal(lCRF)∗. It is obvious that Z(l) is continuous in the support set [A′,F-1(ηR)]. After some manipulation, we get Z(A′)=A′ηR>0 and Z(F-1(ηR))=(1-k)ηRF-1(ηR)+(k-1)∫A′F-1(ηR)‍F(x)dx. Since ∫A′F-1ηR‍F(x)dx≤F(F-1(ηR))(F-1(ηR)-A′)=ηR(F-1(ηR)-A′), we have Z(F-1(ηR))≤1-kA′<0; hence, there exists the optimal (lCRF)∗ that satisfies Z(l)=0 in the support set (A′,F-1(ηR)).
Furthermore, to verify the uniqueness of the optimal(lCRF)∗, we have Z′(l)=(ηR-F(l))(1-klh(l)), and Z′′l=-Z′(l)h(l)-kηR-Fllhl′. Since lhl′>0, by Lemma 1, l≤F-1(ηR), l∈[A′,B′], we easily gain Z′′(l)|Z′(l)=0<0, which implies that Z(l) is unimodal function. Thus, the optimal (lCRF)∗ is unique.
From (22), we find that the optimal stocking factor (lCRF)∗ does not depend on the order quantity q. Substituting (22) into (24), we get (26)ΠlCRF∗,q=kA1/km-1+1/klCRF∗1+1/kq1-1/kηR-FlCRF∗k-1ηR-cM+cR+tq.In what follows, we can show that (27)dΠlCRF∗,qdq=A1/km-1+1/klCRF∗1+1/kq-1/kηR-FlCRF∗ηR-cM+cR+t,and solving dΠ((lCRF)∗,q)/dq=0, we get (qCRF)∗=Am(lCRF)∗(ηR-F(lCRF)∗)/mηRcM+cR+tk. Meanwhile, we easily gain d2Π1((lCRF)∗,q)/dq2|q=(qCRF)∗<0. According to the second-order sufficient condition, there exists the unique optimal (qCRF)∗ that maximizes Π((lCRF)∗,q). This completes the proof.The above theorem shows that it does not have any requirement on problem parameters other than the demand distribution itself to determine the optimal decisions of the centralized system under CVaR criterion. It should be pointed out that, in Theorem2, when ηR=1 and α=1, the optimal inventory factor (lCRF)∗ is the same as Wang et al. [39] and Li and Hua [40]. In addition, for the optimal order quantity (qCRF)∗, by p=Aml/q1/k, we can get the optimal retail price (28)pCRF∗=mηRcM+cR+tηR-FlCRF∗.Substituting (pCRF)∗ and (lCRF)∗ into (21), the maximum expected profit of the centralized system with item-level RFID under CVaR criterion is given by (29)ΠlCRF∗=AlCRF∗k-1mcM+cR+tk-1ηR-FlCRF∗ηRk.


## 3.2.2. Model without Item-Level RFID

Similarly, by the assumption presented above, for the case without item-level RFID (wheret=0 and β=0), let m′=1/α; the expected profit function of the centralized system under no RFID can be written as (30)πCNop,l=dppl-l-x+-cM+cRm′l.Now, let Π′(p,l)=maxv∈R⁡CVaR(πCNo(p,l)); the optimal decision to the centralized system without item-level RFID under CVaR criterion is given by the following theorem.Theorem 3.
In the centralized system without item-level RFID, ifxh(x) is IGFR, then the decision vector ((pCNo)∗,(lCNo)∗) is the unique maximizer of Π′(p,l), where (31)pCNo∗=m′ηRcM+cRηR-FlCNo∗,and (lCNo)∗ is described by (32)Fl=ηRl+k-1Λlkl.Proof.
This proof is similar to the proof procedures of Theorem2; thus, we here omit this proof.Similarly, according to Theorem3, we can easily derive the optimal order quantity as (33)qCNo∗=Am′lCNo∗ηR-FlCNo∗m′ηRcM+cRk,and the maximum expected profit of the centralized system without item-level RFID under CVaR criterion is given by (34)Π′lCNo∗=AlCNo∗k-1m′cM+cRk-1ηR-FlCNo∗ηRk.The following proposition discusses how the optimal decisions change with model parameters in the centralized system under CVaR criterion.Proposition 4.
Ifxh(x) is IGFR, then the following hold.(1)
(lCRF)∗=(lCNo)∗.(2)
Both(lCRF)∗ and (lCNo)∗ are not affected by the sales-available proportion α, but they are increasing in ηR.(3)
Both(pCRF)∗ and (pCNo)∗ are decreasing in α; let G(x)=(xhxkFx-ηR-Fx)/(1-kxhx), and thus
(i)
whenG((lCRF)∗)>0,  (pCRF)∗ is increasing in ηR; when G((lCRF)∗)<0,  (pCRF)∗ is decreasing in ηR; when G((lCRF)∗)=0,  (pCRF)∗ is not affected by ηR;(ii)
whenG((lCNo)∗)>0,  (pCNo-RF)∗ is increasing in ηR; when G((lCNo)∗)<0,  (pCNo)∗ is decreasing in ηR; when G((lCNo)∗)=0,  (pCNo)∗ is not affected by ηR.Proof.
Part (1) Comparing (22) with (32), we can easily get the result that (lCNo)∗=(lCRF)∗.
Part (2) Since (22) and (32) do not involve the sales-available proportion α, both (lCNo)∗ and (lCRF)∗ are not affected by α. From (22), the optimal (lCRF)∗ satisfies Z((lCRF)∗)=0. By the implicit function rule, d(lCRF)∗/dηR=-(lCRF)∗/(∂Z((lCRF)∗)/∂(lCRF)∗), according to proof of Theorem 2, Z(A)=AηR>0,  Z(F-1(ηR))≤(1-k)A<0, and (lCRF)∗ solving Z((lCRF)∗)=0 is unique. It implies ∂Z((lCRF)∗)/∂(lCRF)∗<0. Thus, we get d(lCRF)∗/dηR>0. Similar to the proof procedures of d(lCRF)∗/dηR>0, we can gain d(lCNo)∗/dηR>0.
Part (3) From (28) and (31), we easily show that (pCRF)∗ is increasing in m and (pCNo)∗ is increasing in m′; in conjunction with m=1/[α+β(1-α)] and m′=1/α, they imply that m and m′ are decreasing in α; we thus have the fact that both (pCNo)∗ and (pCRF)∗ are decreasing in α.(i)
From (28), taking the first derivative of (pCRF)∗ with respect to ηR, we have d(pCRF)∗/dηR=((ηRf((lCRF)∗)(d(lCRF)∗/dηR)-F(lCRF)∗)/[ηR-F(lCRF)∗]2)m(cM+cR+t); furthermore, by the proof of Proposition 4(2), we get d(lCRF)∗/dηR=-(lCRF)∗/Z′l|l=(lCRF)∗. By simply substituting d(lCRF)∗/dηR into d(pCRF)∗/dηR, so we derive that d(pCRF)∗/dηR=(mcM+cR+t/[ηR-F(lCRF)∗]2)G((lCRF)∗), which implies that the monotone behavior of (pCRF)∗ can be determined by the sign of G((lCRF)∗). Thus, when G((lCRF)∗)>0,  (pCRF)∗ is increasing in ηR; when G((lCRF)∗)<0, (pCRF)∗ is decreasing in ηR; when G((lCRF)∗)=0,  (pCRF)∗ is not affected by ηR.(ii)
The proof is similar to the proof procedures of Part 3(i); we can gain the monotone behavior of(pCNo)∗ with respect to ηR; thus, we here omit this proof.Proposition4(1) implies that the optimal stocking factor does not depend on whether the centralized system adopts item-level RFID or not, and it seems to depend heavily on demand distribution.Proposition4(2) states that the optimal stocking factors (lCRF)∗ and (lCNo)∗ are independent of the sales-available proportion parameter; they only depend on the risk aversion value and increase with ηR increases. Because a higher value of ηR implies a lower degree of risk aversion, it implies that a lower degree of risk aversion may lead to a higher stocking factor; that is, in the centralized case, if a supply chain manager has less fear of risk, he or she always tends to order more to meet market demand no matter whether to employ item-level RFID or not.Proposition4(3) states that the optimal retail prices (pCRF)∗ and (pCNo)∗ decease with the sales-available proportion increases; it means that a higher sales-available rate may be able to make supply chain manager set a lower retail price for attracting customers to buy more, but the relationship between the optimal retail price and the risk aversion level does not absolutely increase or decrease; it depends on the sign of G((lCRF)∗) or G((lCNo)∗); that is, for adopting item-level RFID case, when G((lCRF)∗)>0, the optimal retail price increases with ηR increases, and it implies that if a supply chain manager is risk-averse enough, he or she is more likely to set a lower retail price to avoid the risk caused by market uncertainty and nonsale inventory shrinkage; when G((lCRF)∗)<0, the optimal retail price decreases with ηR increases; it means that if a supply chain manager has less fear of risk, he or she may raise his/her retail price and order less to balance the relationship between the expected benefit and the risk; when G((lCRF)∗)=0, the optimal retail price is not affected on the risk aversion level; it only depends on some special demand distributions.To assess the benefit of the item-level RFID implementation in the centralized situation under CVaR criterion, we introduce the auxiliary function asΔ=ln[Π((lCRF)∗)/Π′((lCNo)∗)], where Π((lCRF)∗) and Π′((lCNo)∗) are given by (29) and (34), respectively. Note that the auxiliary function Δ can be used to judge whether to adopt item-level RFID; that is, if Δ>0, it means an item-level RFID implementation can bring more expected profit than no RFID case in centralized system, but if Δ≤0, it means that, as compared to item-level RFID system, one case without RFID is a better choice. We will discuss how the model parameters affect item-level RFID implementation in the following.Proposition 5.
(1)Δ is independent of ηR and is decreasing in t;
(2)Δ is decreasing in α but is increasing in β.Proof.
Part (1) By (29) and (34), after some single algebra, we derive the function as (35)Δ=k-1ln⁡α+β1-αα·cM+cRcM+cR+t.From (35), Δ does not have ηR, so Δ is independent of ηR, and the conclusion that Δ decreases in t is obvious.
Part (2) Equation (35) can also be written as Δ=k-1ln⁡[[(1-β)+β/α]·((cM+cR)/(cM+cR+t))], so we easily reach the conclusion that Δ is decreasing in α. Using a similar argument, we can also gain that Δ is increasing in β.
In fact, (35) can be viewed as a sufficient condition to judge whether to adopt item-level RFID, and Proposition 5 states that, although the retailer is risk-averse, the judgment function Δ is not affected by the risk-averse level ηR; it only depends on some parameters, such as, α, β, and t. In other words, the risk-averse level is not an effective incentive for supply chain manager to adopt item-level RFID system; however, the sales-available rate and the tag cost are mainly driving factors in evaluating the benefit of an item-level RFID. In light of this, we will give the threshold values of α, β, and t in the following theorem.Theorem 6.
Under the CVaR constraint, for0<α, β≤1, if t∈(0,t¯) (t¯=β(cM+cR)(1-α)/α), then Δ>0; if t∈[t¯,+∞), then Δ≤0.Proof.
From (35), in order to show Δ>0 (≤0), we only need to show [(1-β)+β/α]·((cM+cR)/(cM+cR+t))>1 (≤1), which can be written by t<(≥)   t¯=β(cM+cR)(1-α)/α. Therefore, we have the following: if t∈(0,t¯), then Δ>0; if t∈[t¯,+∞), then Δ(α,β,t)≤0.Theorem6 gives a threshold value of t; that is, t¯=β(1-α)(cM+cR)/α, and when the RFID tag cost t is lower than threshold value t¯, item-level RFID implementation can bring more expected profit; otherwise, the supply chain will suffer losses at t≥t¯. Likewise, the threshold values of α, β are summarized in Table 1 for more details, and, furthermore, the impact of the key parameters on supply chains optimal decisions will be discussed in Example 1.Table 1
Item-level RFID implementation cases for the key parametersα,β, and t.
The parameter
The parameter’s threshold value
The parameter interval
The sign ofΔ
Use RFID?
α
α
¯
=
β
c
M
+
c
R
β
c
M
+
c
R
+
t
α
∈
0
,
α
¯ α∈α¯,+∞
>0≤0
Yes No
β
β
¯
=
α
t
1
-
α
c
M
+
c
R
β
∈
0
,
β
¯ β∈β¯,+∞
<0≥0
NoYes
t
t
¯
=
β
1
-
α
c
M
+
c
R
α
t
∈
0
,
t
¯ t∈t¯,+∞
>0≤0
YesNoFrom the discussion above, we know that the model with item-level RFID is more generalized than no RFID case in the centralized supply chain system; that is, whent=0 and m=m′, the model with item-level RFID reduces to the model without RFID system. Therefore, we only explore one case with RFID technology in the following decentralized supply chain analysis; the other scenarios are shown in Table 2.Table 2
Summary of optimal decisions and profits in different scenarios under CVaR criterion.
Decentralized systems
Centralized systems
RFID
No RFID
RFID
No RFID
Optimal sale price
k
k
-
1
m
M
N
k
k
-
1
m
′
M
′
N
′
m
M
N
m
′
M
′
N
′
Optimal order quantity
k
-
1
k
k
A
m
l
R
R
F
∗
m
M
N
k
k
-
1
k
k
A
m
′
l
R
N
o
∗
m
′
M
N
′
k
A
m
l
C
R
F
∗
m
M
N
k
A
m
′
l
C
N
o
∗
m
′
M
N
′
k
Optimal wholesale price
t
+
c
R
+
k
c
M
k
-
1
c
R
+
k
c
M
k
-
1
—
—
Retailer’s expected profit
k
-
1
k
k
-
1
Π
∗
k
-
1
k
k
-
1
Π
′
∗
—
—
Manufacturer’s expected profit
k
-
1
k
k
Π
∗
k
-
1
k
k
Π
′
∗
—
—
Supply chain expected profit
k
-
1
k
k
-
1
+
k
-
1
k
k
Π
∗
k
-
1
k
k
-
1
+
k
-
1
k
k
Π
′
∗
Π
∗
Π
′
∗
Note.
M
=
M
′
=
η
R
/
(
η
R
-
F
l
C
R
F
∗
), N=cM+cR+t, N′=cM+cR, Π∗=A(lCRF)∗/(k-1)(mN)k-1Mk, and Π′∗=A(lCRF)∗/(k-1)(m′N′)k-1Mk.


## 4. Decentralized Policies under CVaR Criterion

In this section, we explore the optimal policies for a decentralized supply chain with a separate manufacturer and a separate retailer, and then we discuss a wholesale price contract. Furthermore, we study a revenue sharing contract for coordinating the supply chain, which concentrates on the allocation of the expected sale revenue between the manufacturer and the retailer.


### 4.1. Wholesale Price Contract

We here consider that, facing nonsale inventory shrinkage phenomena and demand uncertainty in the retail setting, the retailer (like Wal-Mart, Target, etc.) takes the initiative in employing RFID for achieving a higher product availability and bears all of the RFID tags cost. The manufacturer needs to decide wholesale price contract parameters to achieve his/her performance. The order quantity is delivered to the retailer before the selling season, and transfer payments are made between supply chain players based on the agreed contract.In decentralized supply chain system with item-level RFID under CVaR criterion, the retailer’s expected profit function is similar to the function of centralized system in Section3, so we here directly give the following: (36)πRRFp,l=dppl-l-x+-w+cR+tml,and substituting (35) into (3), the retailer’s expected profit function with item-level RFID under CVaR criterion is shown by (37)CVaRπRRFp,l=maxvR∈R⁡vR+1ηR·Emin⁡πRRFp,l-vR,0.Let ΠR(p,l,v(p,l))=CVaR(πRRF(p,l)); similar to the proof of Lemma 1, there also exists the unique optimal v∗(p,l)=Ap-kl[p-(w+cR+t)m] which maximizes ΠR(p,l,v(p,l)), and the retailer’s expected decision function becomes (38)ΠRp,l=Ap-kp-w+cR+tml-pΛlηR.In what follows, we provide a theorem for getting the retailer’s optimal decision.Theorem 7.
In decentralized setting, for the retailer, ifxh(x) is IGFR, then the decision vector ((pRRF)∗,(lRRF)∗) is the unique maximizer of ΠR(p,l), where (lRRF)∗ is determined by (39)Fl=ηRl+k-1Λlkl,pRRF∗=mηRw+cR+tηR-FlRRF∗.Proof.
Similar to the proof of Theorem2, thus we here omit this proof.According to Theorem7, by p=Aml/q1/k, we can derive the retailer’s optimal order quantity as (40)qRRF∗=AmlRRF∗ηR-FlRRF∗mηRw+cR+tk,and the maximum expected profit is given by (41)ΠRlRRF∗=AlRRF∗k-1mw+cR+tk-1ηR-FlRRF∗ηRk.From Theorems2, 3, and 7, we easily find that the retailers optimal stocking factor is always equal to that of the centralized system; that is, (lRRF)∗=(lCRF)∗=(lCNo)∗; it seems to depend heavily on demand distribution and risk-averse level and does not depend on some parameters, such as t, α, and β.Knowing the retailer’s order quantity(qRRF)∗, the manufacturer’s expected profit function is easily written as (42)ΠMw=w-cMAmlRRF∗ηR-FlRRF∗mηRw+cR+tk.For obtaining the manufacturer’s optimal decision, we show the following theorem.Theorem 8.
The optimal wholesale price for manufacturer is unique and is given byw∗=(t+cR+kcM)/(k-1).Proof.
Recall that(lRRF)∗ chosen by the retailer does not depend on w; from (41), we can gain that the necessary condition for the maximum of ΠM(w) is(43)dΠMwdw=AmlRRF∗ηR-FlRRF∗mηRw+cR+tk·1-kw-cMw+cR+t=0,and notice that the first four terms in the left part of (42) are each positive, so it only requires the optimal wholesale price w∗ which satisfies 1-k(w-cM)/(w+cR+t)=0. After simple manipulation, we give w∗=(t+cR+kcM)/(k-1). Furthermore, (44)dΠMwdww=w∗=-AmlRRF∗ηR-FlRRF∗mηRw+cR+tk·kt+cM+cRw+cR+t2<0,and we therefore conclude that ΠM(w) is strictly concave in w and the optimal wholesale price w=w∗ is unique.Remark 9.
In fact, substitutingw=w∗ into (41) and (42), both the optimal retailer’s expected profit and the optimal manufacturer’s expected profit are shown, respectively; that is, (45)ΠRlRRF∗=AlRRF∗k-1mcM+cR+tk-1ηR-FlRRF∗ηRk·k-1kk-1,(46)ΠMw∗=AlRRF∗k-1mcM+cR+tk-1ηR-FlRRF∗ηRk·k-1kk.Under a wholesale price contract, it is not difficult to find that (47)ΠRlRRF∗+ΠMw∗=k-1kk-1+k-1kk·AlRRF∗k-1mcM+cR+tk-1·ηR-FlRRF∗ηRk<2e·ΠlCRF∗<ΠlCRF∗,which shows that the total of decentralized supply chain profit is always lower than the centralized case. More specifically, it means that more than 26 percent ((e-2/e)∗100%) of the whole supply chain’s profit is lost due to double marginalization; meanwhile, it also implies that there exists a potential incentive to coordinate between supply chain players.In what follows, we discuss supply chain coordination for achieving the best performance of the entire supply chain. Following Cachon [41] and He et al. [42], a contract designed by the manufacturer is said to coordinate the supply chain if it satisfies the first-order condition of centralized supply chain’s profit function at (lCRF) and (qCRF). By the above discussion, we know (lRRF)∗=(lCRF)∗, so there is only a need to satisfy (qRRF)∗=(qCRF)∗. After simple manipulation, we can see that there exists w=cM which is required to coordinate the retailer’s order quantity. However, it will directly lead to zero profit for the manufacturer, so the wholesale price contract cannot coordinate the supply chain.


### 4.2. Wholesale Price Contract with Revenue Sharing

In this section, we further discuss supply chain coordination under wholesale price contract with revenue sharing. The manufacturer offers a lower wholesale price to the retailer and shares a fraction of sale revenue earned by the retailer. The retailer decides sale price and order quantity. No money changes hands unless an item is sold. Letγ be a proportion of sale revenue kept by the retailer, and the other 1-γ is shared by the manufacturer.Under wholesale price contract with revenue sharing, using similar arguments, the retailer’s general expected profit function under CVaR criterion is(48)ΠRp,l,γ=dpγp-cM+cR+tml-γpΛlηR,and the manufacturer’s general expected profit function is (49)ΠMw,γ=dp1-γp+w-cMml-1-γpΛlηR.Theorem 10.
Under the wholesale price contract with revenue sharing, for givenγ, if contract parameters satisfy the condition w=γcM+(γ-1)(cR+t) and d(p)=Ap-k, then the vector ((pCRF)∗,(lCRF)∗) is also optimal decision in decentralized case.Proof.
Substitutingw=γcM+(γ-1)(cR+t) and d(p)=Ap-k into (48) and (49), we have (50)ΠRp,l,γ=γΠp,l,ΠMw,γ=1-γΠp,l.
For a givenγ, it means that when Π(p,l) reaches its maximum value in ((pCRF)∗,(lCRF)∗), ΠR(p,l,γ) and ΠM(w,γ) can also achieve their optimal expected profits, respectively; that is, the vector ((pCRF)∗,(lCRF)∗) is also optimal decision in that case.From Theorem10, we can see that, under the wholesale price contract with revenue sharing, an arbitrary allocation of the optimal centralized supply chain profit between the manufacturer and the retailer can be achieved by changing the proportion parameter γ.Further, from (29), (45), and (46), combined with Theorem 10, we have the next corollary.Corollary 11.
If the revenue sharing proportion parameterγ satisfies the fact that w=γcM+(γ-1)(cR+t) and γ∈(((k-1)/k)k-1,1-((k-1)/k)k), then (51)γΠlCRF∗>ΠRlRRF∗,1-γΠlCRF∗>ΠMw∗.The above corollary shows that, setting appropriate contract parameters, the wholesale price contract with revenue sharing can coordinate the supply chain, and both the manufacturer and the retailer get expected profits higher than their respective reservation expected profits. It is implied that if properly designed it is attractive for both parties of supply chain to accept the coordination contract, and the contract parameterγ depends on the partners’ bargaining power.In fact, from the above discussion, we can also see that while gaining more profit from sale revenue, the manufacturer takes a part of risk caused by demand uncertainty and nonsale inventory shrinkage; in return, for the retailer, he/she bears less risk and earns more profit from the coordination contract, which can achieve a win-win situation for supply chain partners.


## 4.1. Wholesale Price Contract

We here consider that, facing nonsale inventory shrinkage phenomena and demand uncertainty in the retail setting, the retailer (like Wal-Mart, Target, etc.) takes the initiative in employing RFID for achieving a higher product availability and bears all of the RFID tags cost. The manufacturer needs to decide wholesale price contract parameters to achieve his/her performance. The order quantity is delivered to the retailer before the selling season, and transfer payments are made between supply chain players based on the agreed contract.In decentralized supply chain system with item-level RFID under CVaR criterion, the retailer’s expected profit function is similar to the function of centralized system in Section3, so we here directly give the following: (36)πRRFp,l=dppl-l-x+-w+cR+tml,and substituting (35) into (3), the retailer’s expected profit function with item-level RFID under CVaR criterion is shown by (37)CVaRπRRFp,l=maxvR∈R⁡vR+1ηR·Emin⁡πRRFp,l-vR,0.Let ΠR(p,l,v(p,l))=CVaR(πRRF(p,l)); similar to the proof of Lemma 1, there also exists the unique optimal v∗(p,l)=Ap-kl[p-(w+cR+t)m] which maximizes ΠR(p,l,v(p,l)), and the retailer’s expected decision function becomes (38)ΠRp,l=Ap-kp-w+cR+tml-pΛlηR.In what follows, we provide a theorem for getting the retailer’s optimal decision.Theorem 7.
In decentralized setting, for the retailer, ifxh(x) is IGFR, then the decision vector ((pRRF)∗,(lRRF)∗) is the unique maximizer of ΠR(p,l), where (lRRF)∗ is determined by (39)Fl=ηRl+k-1Λlkl,pRRF∗=mηRw+cR+tηR-FlRRF∗.Proof.
Similar to the proof of Theorem2, thus we here omit this proof.According to Theorem7, by p=Aml/q1/k, we can derive the retailer’s optimal order quantity as (40)qRRF∗=AmlRRF∗ηR-FlRRF∗mηRw+cR+tk,and the maximum expected profit is given by (41)ΠRlRRF∗=AlRRF∗k-1mw+cR+tk-1ηR-FlRRF∗ηRk.From Theorems2, 3, and 7, we easily find that the retailers optimal stocking factor is always equal to that of the centralized system; that is, (lRRF)∗=(lCRF)∗=(lCNo)∗; it seems to depend heavily on demand distribution and risk-averse level and does not depend on some parameters, such as t, α, and β.Knowing the retailer’s order quantity(qRRF)∗, the manufacturer’s expected profit function is easily written as (42)ΠMw=w-cMAmlRRF∗ηR-FlRRF∗mηRw+cR+tk.For obtaining the manufacturer’s optimal decision, we show the following theorem.Theorem 8.
The optimal wholesale price for manufacturer is unique and is given byw∗=(t+cR+kcM)/(k-1).Proof.
Recall that(lRRF)∗ chosen by the retailer does not depend on w; from (41), we can gain that the necessary condition for the maximum of ΠM(w) is(43)dΠMwdw=AmlRRF∗ηR-FlRRF∗mηRw+cR+tk·1-kw-cMw+cR+t=0,and notice that the first four terms in the left part of (42) are each positive, so it only requires the optimal wholesale price w∗ which satisfies 1-k(w-cM)/(w+cR+t)=0. After simple manipulation, we give w∗=(t+cR+kcM)/(k-1). Furthermore, (44)dΠMwdww=w∗=-AmlRRF∗ηR-FlRRF∗mηRw+cR+tk·kt+cM+cRw+cR+t2<0,and we therefore conclude that ΠM(w) is strictly concave in w and the optimal wholesale price w=w∗ is unique.Remark 9.
In fact, substitutingw=w∗ into (41) and (42), both the optimal retailer’s expected profit and the optimal manufacturer’s expected profit are shown, respectively; that is, (45)ΠRlRRF∗=AlRRF∗k-1mcM+cR+tk-1ηR-FlRRF∗ηRk·k-1kk-1,(46)ΠMw∗=AlRRF∗k-1mcM+cR+tk-1ηR-FlRRF∗ηRk·k-1kk.Under a wholesale price contract, it is not difficult to find that (47)ΠRlRRF∗+ΠMw∗=k-1kk-1+k-1kk·AlRRF∗k-1mcM+cR+tk-1·ηR-FlRRF∗ηRk<2e·ΠlCRF∗<ΠlCRF∗,which shows that the total of decentralized supply chain profit is always lower than the centralized case. More specifically, it means that more than 26 percent ((e-2/e)∗100%) of the whole supply chain’s profit is lost due to double marginalization; meanwhile, it also implies that there exists a potential incentive to coordinate between supply chain players.In what follows, we discuss supply chain coordination for achieving the best performance of the entire supply chain. Following Cachon [41] and He et al. [42], a contract designed by the manufacturer is said to coordinate the supply chain if it satisfies the first-order condition of centralized supply chain’s profit function at (lCRF) and (qCRF). By the above discussion, we know (lRRF)∗=(lCRF)∗, so there is only a need to satisfy (qRRF)∗=(qCRF)∗. After simple manipulation, we can see that there exists w=cM which is required to coordinate the retailer’s order quantity. However, it will directly lead to zero profit for the manufacturer, so the wholesale price contract cannot coordinate the supply chain.


## 4.2. Wholesale Price Contract with Revenue Sharing

In this section, we further discuss supply chain coordination under wholesale price contract with revenue sharing. The manufacturer offers a lower wholesale price to the retailer and shares a fraction of sale revenue earned by the retailer. The retailer decides sale price and order quantity. No money changes hands unless an item is sold. Letγ be a proportion of sale revenue kept by the retailer, and the other 1-γ is shared by the manufacturer.Under wholesale price contract with revenue sharing, using similar arguments, the retailer’s general expected profit function under CVaR criterion is(48)ΠRp,l,γ=dpγp-cM+cR+tml-γpΛlηR,and the manufacturer’s general expected profit function is (49)ΠMw,γ=dp1-γp+w-cMml-1-γpΛlηR.Theorem 10.
Under the wholesale price contract with revenue sharing, for givenγ, if contract parameters satisfy the condition w=γcM+(γ-1)(cR+t) and d(p)=Ap-k, then the vector ((pCRF)∗,(lCRF)∗) is also optimal decision in decentralized case.Proof.
Substitutingw=γcM+(γ-1)(cR+t) and d(p)=Ap-k into (48) and (49), we have (50)ΠRp,l,γ=γΠp,l,ΠMw,γ=1-γΠp,l.
For a givenγ, it means that when Π(p,l) reaches its maximum value in ((pCRF)∗,(lCRF)∗), ΠR(p,l,γ) and ΠM(w,γ) can also achieve their optimal expected profits, respectively; that is, the vector ((pCRF)∗,(lCRF)∗) is also optimal decision in that case.From Theorem10, we can see that, under the wholesale price contract with revenue sharing, an arbitrary allocation of the optimal centralized supply chain profit between the manufacturer and the retailer can be achieved by changing the proportion parameter γ.Further, from (29), (45), and (46), combined with Theorem 10, we have the next corollary.Corollary 11.
If the revenue sharing proportion parameterγ satisfies the fact that w=γcM+(γ-1)(cR+t) and γ∈(((k-1)/k)k-1,1-((k-1)/k)k), then (51)γΠlCRF∗>ΠRlRRF∗,1-γΠlCRF∗>ΠMw∗.The above corollary shows that, setting appropriate contract parameters, the wholesale price contract with revenue sharing can coordinate the supply chain, and both the manufacturer and the retailer get expected profits higher than their respective reservation expected profits. It is implied that if properly designed it is attractive for both parties of supply chain to accept the coordination contract, and the contract parameterγ depends on the partners’ bargaining power.In fact, from the above discussion, we can also see that while gaining more profit from sale revenue, the manufacturer takes a part of risk caused by demand uncertainty and nonsale inventory shrinkage; in return, for the retailer, he/she bears less risk and earns more profit from the coordination contract, which can achieve a win-win situation for supply chain partners.


## 5. Number Examples and Sensitivity Analysis

In this section, we implement sensitivity analysis of the key parametersα, β, and t to illustrate the impact of changes of parameters on supply chain’s optimal decisions under CVaR criterion, and then we further explore the impact of the retailer’s risk attitude, inventory shrinkage rate, demand uncertainty, and price-elasticity index of the demand on the total supply chain’s profit with coordination for getting more insights. In our number examples, the base values of the parameters are listed as follows: α=0.6, β=0.8, t=0.2, k=3, A′=0, A=500, ηC=ηR=0.2, cM=0.6, and cR=0.6; the random component of the demand ε is assumed to follow the normal distribution with μ=50 and σ2=52.Example 1 (the impact ofα, β, and t).
In this subsection, we calculate supply chain’s optimal solutions with three possible cases:(1) different values of α and different values of β; (2) different values of α and different values of t; (3) different values of β and different values of t. The sensitivity analysis is performed by, respectively, changing the value of two parameters but keeping other parameters constant. For notational convenience, we do not distinguish between RFID case and no RFID case; that is, we generally use qR∗ to represent the retailer’s optimal order quantity with RFID or without RFID under CVaR criterion, and the similar notations will be used in pR∗, w∗, ΠR∗, ΠM∗, pC∗, qC∗, and Π∗. The corresponding results are shown in Tables 3–5.
From Table3, no matter which case happens, the centralized system or the decentralized system, we can observe the following: (1) the centralized supply chain’s optimal profits Π∗, the retailer’s optimal profits ΠR∗, and the manufacturer’s optimal profits ΠM∗ all increase as the sales-available proportion α and the recovery rate β increase; (2) the centralized optimal order quantity qC∗ and the retailer’s optimal order quantity qR∗ all increase as α and β increase; (3) the centralized optimal retail price pC∗ and the retailer’s optimal retail price pR∗ all decrease as α and β increase, but the manufacturer’s wholesale price w∗ is not influenced by the sales-available proportion α and the positive value of the recovery rate β. Table 3 also illustrates that, compared to no RFID case (such as α=0.6, β=0), if item-level RFID performs quite well (such as β=0.7,0.8,0.9), the benefits from RFID is larger than the costs incurred by adopting RFID, and the higher the recovery rate, the more superior the item-level RFID in improving the supply chain performance.
From Table4, when the recovery rate is constant and item-level RFID is employed, Π∗, ΠR∗, ΠM∗, qC∗, and qR∗ all decrease as the tag cost t increases, but pC∗, pR∗, and w∗ all increase as the tag cost t increases. In other words, for the retailer, the manufacturer, and the centralized system, when the recovery rates remain unchanged, their benefits will be reduced due to higher tag price. In fact, by comparing different cases (such as α=0.9, t=0.3 and α=0.6, t=0.1), it is not difficult to find that when sales-available proportion is fairly small and the tag price is lower, the effect of adopting item-level RFID is much better; but when the tag price is higher, the benefits from RFID could not compensate for the costs incurred by using RFID, and no RFID case is a better choice (such as α=0.6, t=0 and α=0.6, t=0.5).
From Table5, when the tag prices remain unchanged, as the recovery rate increases, Π∗, ΠR∗, ΠM∗, qC∗, and qR∗ increase and pC∗ and pR∗ decrease; but when the recovery rates remain unchanged, as the tag price increases, the opposite of the aforementioned results is true. In addition, we notice that although item-level RFID performs quite well (such as t=0.7 and β=0.9), as compared to no RFID case (such as t=0 and β=0), it does not offer a better decision due to higher tag cost, and the higher the tag cost, the greater the harm to the supply chain performance.Table 3
The optimal decisions and profits with varyingα and β.
α
β
q
R
∗
p
R
∗
w
∗
Π
R
∗
Π
M
∗
p
C
∗
q
C
∗
Π
∗
0.6
0
638.398
3.799
1.15
526.678
351.119
2.533
2154.592
1185.025
0.6
0.7
831.959
3.062
1.25
811.160
540.774
2.041
2807.863
1852.111
0.8
909.311
2.928
1.25
886.578
590.052
1.952
3068.924
1994.801
0.9
990.100
2.806
1.25
965.348
643.565
1.871
3341.588
2172.033
0.7
0.7
889.651
2.961
1.25
867.409
578.273
1.974
3002.571
1951.671
0.8
949.276
2.866
1.25
925.544
617.029
1.911
3203.806
2082.474
0.9
1010.835
2.778
1.25
985.564
657.043
1.852
3411.567
2217.519
0.8
0.7
949.276
2.866
1.25
925.544
617.029
1.911
3203.806
2082.474
0.8
990.100
2.806
1.25
965.348
643.565
1.871
3341.588
2172.033
0.9
1031.784
2.749
1.25
1005.990
670.660
1.833
3482.272
2263.477
0.9
0.7
1010.835
2.778
1.25
985.564
657.043
1.852
3411.567
2217.519
0.8
1031.784
2.749
1.25
1005.990
670.660
1.833
3482.272
2263.477
0.9
1052.948
2.721
1.25
1026.625
684.416
1.814
3553.701
2309.906Table 4
The optimal decisions and profits with varyingα and t.
α
t
q
R
∗
p
R
∗
w
∗
Π
R
∗
Π
M
∗
p
C
∗
q
C
∗
Π
∗
0.6
0
638.398
3.799
1.15
526.678
351.119
2.533
2154.592
1185.025
0.6
0.1
1156.109
2.703
1.20
1040.498
693.665
1.802
3901.867
2341.120
0.3
728.045
3.154
1.30
764.447
509.632
2.102
2457.152
1720.007
0.5
487.773
3.604
1.40
584.280
390.187
2.403
1646.100
1316.880
0.7
0.1
1206.921
2.646
1.20
1086.229
724.152
1.764
4073.357
2444.014
0.3
760.043
3.087
1.30
798.046
532.030
2.058
2565.146
1795.602
0.5
509.17
3.528
1.40
611.004
407.336
2.352
1718.448
1374.758
0.8
0.1
1258.825
2.591
1.20
1132.943
755.295
1.727
4248.536
2549.121
0.3
792.73
3.022
1.30
832.366
554.911
2.015
2675.463
1872.824
0.5
531.067
3.454
1.40
637.280
424.854
2.303
1792.351
1433.881
0.9
0.1
1311.823
2.538
1.20
1180.641
787.094
1.692
4427.402
2656.441
0.3
826.104
2.961
1.30
867.409
578.273
1.974
2788.102
1951.671
0.5
553.425
3.384
1.40
664.110
442.740
2.256
1867.810
1494.248Table 5
The optimal decisions and profits with varyingt and β.
t
β
q
R
∗
p
R
∗
w
∗
Π
R
∗
Π
M
∗
p
C
∗
q
C
∗
Π
∗
0
0
638.398
3.799
1.15
526.678
351.119
2.533
2154.592
1185.025
0.1
0.7
1057.763
2.826
1.20
951.987
634.658
1.884
3569.950
2141.870
0.8
1156.109
2.703
1.20
1040.498
693.665
1.802
3901.867
2341.120
0.9
1258.825
2.591
1.20
1132.943
755.295
1.727
4248.536
2549.121
0.3
0.7
666.113
3.297
1.30
699.419
466.279
2.198
2248.132
1573.692
0.8
728.045
3.154
1.30
764.447
509.632
2.102
2457.152
1720.007
0.9
792.730
3.022
1.30
832.366
554.911
2.015
2675.463
1872.824
0.5
0.7
446.244
3.768
1.40
535.493
356.995
2.512
1506.073
1204.858
0.8
487.733
3.604
1.40
585.280
390.187
2.403
1646.100
1316.880
0.9
531.067
3.454
1.40
637.280
424.854
2.303
1792.351
1433.881
0.7
0.7
313.411
4.239
1.50
423.015
282.070
2.826
1057.763
951.987
0.8
342.551
4.055
1.50
462.444
308.296
2.703
1156.109
1040.498
0.9
372.985
3.886
1.50
503.530
335.687
2.591
1258.825
1132.943From the above analysis, we find that the tag cost, the recovery rate, and the sales-available proportion can influence supply chain performance; to be specific, when the sales-available proportion is of lower level (higher shrinkage rate), tag cost is quite small, and when the recovery rate is of higher level, it is very important for adopting the item-level RFID to improve supply chain’s performance. Therefore, in practice, there is a need to exactly assess the threshold values of the tag cost, the recovery rate, and the sales-available proportion to make better choices.Example 2 (the impact ofσ, 1-α, and k).
In this subsection, we firstly calculate the total supply chain’s profit by varyingσ from 0 to 10 in different risk settings (see Figure 2). Next, for the given σ=5 and k=2, we vary 1-α from 0.05 to 0.4 to study the impacts of the shrinkage rate on the total supply chain’s profit with coordination in different risk settings (see Figure 3). Finally, we change the price-elasticity index from 2.1 to 2.9 in different RFID tag costs to explore the impact of the price-elasticity index on the total supply chain’s profit with coordination (see Figure 4).
In Figure2, interestingly, we find that, for each risk setting, as the standard deviation σ increases, the total supply chain’s profit with coordination firstly increases and then decreases, but when σ is smaller, the more the supply chains are risk-averse, the more all the supply chains get profit from coordination; when σ is greater, the opposite of the above result is true. An intuitive explanation for this fact is that a larger value of the standard deviation σ means a higher uncertainty in demand; in early stage, for the risk-averse supply chain, the superiority of improving performance is obvious, but, in later stage, for the risk-averse supply chain with more fear of risk, the more dramatic the uncertainty in demand, the more dramatic the decrease in total supply chain sales revenue. They care about their expected profits in a conservative attitude.Figure 2
Impact of changes ofσ on the total supply chain’s profit with coordination in different ηR.Figure 3
Impact of changes of1-α on the total supply chain’s profit with coordination in different ηR.Figure 4
Impact of changes ofk on the total supply chain’s profit with coordination in different t.Recall the above discussion;1-α can be viewed as shrinkage rate, which is often concerned with risk-averse supply chain in the retail setting. In Figure 3, we find that, regardless of risk neutrality (ηC=1) or risk aversion (ηC<1), the total supply chain’s profit decreases as the shrinkage rate increases, and, meanwhile, the more the supply chains are risk-averse, the more the total supply chain’s profit is for coordination. Moreover, we also find that, compared to risk neutrality case, the risk-averse supply chain always gets more profits.For the power form of price-dependent demand factord(p)=Ap-k, the parameter k is the price-elasticity index, and the larger the value of k is, the more sensitive the demand is to a change in price. In Figure 4, for the case with or without item-level RFID, the total supply chain’s profit decreases as the price-elasticity index increases, and, as the tag cost increases, the total supply chain’s profit decreases. That is because the larger the price-elasticity index, the more dramatic the decrease in demand with an increase in retail price. The supply chain only gets less profits from the less end customer demand and meanwhile, with price-elasticity index increasing, the supply chain with RFID always gets more profits than the one without RFID.In fact, from Figures2, 3, and 4, the total supply chain performance from coordination not merely depends heavily on the isoprice-elastic demand function form; we also find that, for different risk levels, the effect of the demand uncertainty and the shrinkage rate on the total supply chain’s profit is very significant; therefore, it is necessary for supply chain manager to exactly evaluate these parameters in the real world.


## 6. Summary and Conclusions

Most of the literatures on the application of RFID in improving product availability assumed that the reliability of RFID is very perfect and shrinkage errors can be eliminated completely. However, in practice, nonsale inventory shrinkage problems can only be eliminated partly due to RFID misreading. In this paper, we further explore supply chain optimization and coordination from imperfect RFID perspective. We develop supply chain models with price-dependent stochastic demand in both centralized scenario and decentralized scenario, under a conditional value-at-risk (CVaR) criterion, and we analyze the optimal supply chain’s decisions in the two different scenarios.In centralized scenario, we give the optimal supply chain’s decisions under item-level RFID or no RFID case (see Table2); worthwhile to mention is that, regardless of item-level RFID case or no RFID case, the optimal stocking factors have no any requirement on model parameters other than the demand distribution itself and the risk aversion level, and the optimal stocking factor does not depend on whether to adopt item-level RFID or not; in particular, setting ηR=1 and α=1, the optimal inventory factor is the same as the one proposed by Wang et al. [39] and Li and Hua [40]; the incentives of the centralized system to employ RFID are not affected by the risk aversion level; they mainly depend on the relative values of the sales-available rate, the recovery rate of RFID, and the tag cost, respectively.In decentralized scenario, we only explore one case with RFID technology and consider two widely used contracts including wholesale price contract and revenue sharing contract. Unfortunately, we find that the total of decentralized supply chain profit is no more than 74% ((2/e)∗100%) of the centralized supply chain profit due to double marginalization under wholesale price contract. Furthermore, we discuss supply chain coordination for achieving the best performance of the entire supply chain under revenue sharing contract; we find that Pareto improving outcome will be achieved if the retailer sets an appropriate revenue sharing proportion parameter for the manufacturer; interestingly, under this contract, for the manufacturer and the retailer, while getting expected profits higher than their respective reservation expected profits, they actually achieve coaffording risk which could have been taken by the retailer alone.Finally, we implement sensitivity analysis of the key parameters to illustrate the impact of changes of parameters on supply chain performance and the total supply chain’s profit with coordination; number results show that especially if the sales-available proportion is of lower level (higher shrinkage rate), tag cost is quite small, and when the recovery rate is of higher level, supply chain partners’ profits obtained by investment in RFID are improved significantly, and, for different risk levels, the effects of the demand uncertainty and the shrinkage rate on supply chain’s profit with coordination are very obvious.There are several possible extensions for further research. In this paper, we only consider and analyze the supply chain including a risk-averse retailer and a risk-neutral manufacturer on this topic. A natural extension is to assume that supply chain composed of a manufacturer and a retailer is risk-averse. In addition, it is worth consideration that one manufacturer sells to two or multiple competing retailers. What strategy can be used to coordinate these agents’ performance when the retailers face inventory shrinkage and demand uncertainty? It would also be interesting to investigate how different risk attitudes affect supply chain performance.



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*Source: 7834751-2016-03-17.xml*