Discrete Dynamics in Nature and Society

Volume 2018, Article ID 7840264, 14 pages

https://doi.org/10.1155/2018/7840264

## Study on Single Cycle Production Allocation and Supply Strategy for DCEs Based on the CVaR Criterion

^{1}College of Economics and Management, Zhejiang University of Technology, Liuxia, Hangzhou 210023, China^{2}Zhejiang University of Finance and Economics, Xiasha, Hangzhou 310018, China^{3}Keyi College of Zhejiang Sci-Tech University, Shaoxing, Zhejiang 312369, China

Correspondence should be addressed to Zhiqing Meng; nc.ude.tujz@gniqihzgnem

Received 29 August 2018; Accepted 13 November 2018; Published 2 December 2018

Academic Editor: Juan L. G. Guirao

Copyright © 2018 Leiyan Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Direct chain enterprises (DCEs) face a decision-making issue as to how to allocate and supply their products to their stores for sales with the minimum losses and maximum profits for the manufacturers. This paper presents a single-cycle optimal allocation model for DCEs under the given total production amount and conditional value at risk loss. The optimal strategy for production allocation and supply is derived. Subsequently, an approximate algorithm for solving the optimal total production amount is presented. The optimal allocation and supply strategy, the minimum total production amount, the minimum allocation strategy, and the discount pricing strategy are obtained for the single cycle. Finally, with the sales data of a food DCE, numerical results corroborate that adopting different production and supply strategies reduces the risk of expected losses and increases the expected return. It is of an important theoretical significance in guiding the production and operation of direct chain enterprises.

#### 1. Introduction

A direct chain enterprise (DCE) is the one whose chain stores are directly operated by the head office of the DCE; that is, the head office implements the unified management of its staff, finances, materials, business, logistics, and information flows for each of its chain stores. Wal-Mart, Carrefour, Starbucks, Uniqlo, Zara, Casamiel, and other companies are typical direct chain enterprises. The production allocation and supply of DCEs has always been a key and difficult problem that must be solved especially for clothing and food enterprises. The phenomenon of oversupply or insufficient production is particularly common. How to provide scientific production allocation and supply strategies has become urgent for the survival of chain enterprises.

Few theoretical researches on the production allocation and supply model for DCEs are available, although there are similar studies on centralized inventory problem =for suppliers and retailers. The idea of the centralized inventory originated in the 1960s. Centralized inventory refers to the fact of a company to provide services from one inventory point to its multiple markets with uncertain demands. Moreover, centralized inventory can effectively reduce the volatility of demands, reduce operating costs, and increase profits. The current research mainly focuses on the affirmation that suppliers provide the centralized supply to several retailers. The relationship between suppliers and retailers is hardly a chain relationship, and the sales and inventory levels among retailers affect each other. The research on the centralized production supply model of DCEs is almost nonexistent. Given that the centralized supply plays an important role in the rapid development of e-commerce or offline direct-operated stores, its impact on enterprises’ supply chain management and operation management has earned considerable attention from the academic community. Therefore, studying the production allocation and supply strategy for DCEs’ the centralized supply is necessary.

The research methods used in studying the centralized inventory are generally divided into two categories. One is quantitative research methods, such as mathematical optimization algorithms, Lagrangian relaxation algorithms, stochastic programming algorithms, and etc. As the model becomes more complex, scholars continue to develop new methods with the aid of computer software. The other is empirical and case study methods used to verify whether the companies’ inventory performances, inventory turnover, and costs are improved. For examples, Eppen (1979) [1] compared the centralized inventory with the decentralized inventory for the first time, where it is concluded that the expected inventory holding costs in the centralized inventory model are lower than those of the decentralized inventory system when the demand is normally distributed. After that, Chang and Lin (1991) [2] concluded that the inventory and out-of-stock costs of a centralized inventory are lower than those of a decentralized inventory by different methods where demands follow any distribution. Chen and Lin (1989) [3] considered the centralized inventory problem of Poisson distribution. Cai and Du (2009) [4] considered the centralized inventory problem when the demand is of any distribution. Kevin (1999) [5] proved that the operation costs can be reduced by a centralized inventory when demands are uncertain. Cherikh (2000) [6] studied the centralized inventory system for single-period, single-product, multi-inventory points and demonstrated that centralized inventory can increase the expected profits and reduce the risk caused by uncertain demand. Netessin and Rudi (2003) [7] discussed the substitutability of products and the centralized inventory problem in the case of random demands. Benjaafar (2011) [8] studied the problem of the centralized inventory efficiency. Kemahlīogluzīya and Bartholdi (2011) [9] studied the profit distribution in a centralized system by cooperative game theory, proved that the centralized supply chain can reduce the cost, and analyzed the impact of demand changes on retailers’ profits. Swiney (2012) [10] studied the purchasing behavior in centralized inventory, where customers usually buy the discounted products at the end of season and give up the purchase at high price such that companies can choose a centralized or a decentralized inventory. Yang et al. (2014) [11] studied joint inventory and pricing strategies in case of high volume orders. Hossinifard and Abbasi (2018) [12] studied the application of the centralized inventory management in hospital blood management, where they validated that centralized inventory is a key factor in the blood supply chain and can improve the sustainability of the blood supply chain. Shivagi Viral Thakker (2018) [13] proved that centralized inventory is better than decentralized inventory in FFCG industry and that centralized inventory can reduce the bullwhip effect and achieve the maximum profits. Christoph Weskamp (2018) [14] proposed a model of two-stage random integer programming and analyzed the centralized production allocation plan to reduce risk. In sum, the above literatures affirmed that centralized inventory management can not only reduce the risk for enterprises or supply chain, but also increase the profits of enterprises. However, research on centralized inventory mainly focuses on non-DCEs; i.e., research on the centralized supply production and allocation strategies of DCEs has not been seen before. Only part of the research considers the interacting allocation problem, and the centralized supply model [1–6] is based on allocation to retailers. Models focus on centralized inventory management, and some model assumptions are not applicable to production allocation and supply strategy for modern DCEs.

The production allocation and supply of the DCE is a risk decision problem. The expected mean loss model is a decision model usually used for risk neutral situation. When the change in demand is relatively large, the expected risk loss decision model cannot measure very well the loss of demand fluctuations. In the supply chain risk ordering decision problem, many people adopt a risk approach, that is, establishing the ordering model of supply chain based on conditional value at risk (CVaR) (2000) [15]. If a fat tail occurs in radon risk distribution, the accumulated losses are extremely high. Hence, the expected mean loss model cannot reflect such distribution, while CVaR can solve this problem. For instance, the oversupply of the market is the main reason for the huge losses of DCE due to insufficient demands, which makes avoiding such losses difficult for DCEs. Therefore, solving the production and supply problems of DCE by CVaR—a risk criteria— is necessary. Alexander et al. (2004) [16] compared VaR with CVaR in portfolio investment. They also verified that CVaR is more effective than VaR and mean model in measuring high risk. CVaR has been widely used in risk decision models since 2007. Many studies have applied conditional value at risk to supply chain ordering research. For instance, Gotoh and Takano (2007) [17] established a single-cycle newsboy CVaR model to determine the best ordering strategy. Zhou and Chen (2008) [18] constructed a single-cycle multiproduct CVaR ordering model. Wu et al. (2013,2014) [19, 20] studied the risk-averse CVaR newsboy model. Xu and Li (2010) [21] established the CVaR newsboy model. Xu and Meng (2015) [22] proposed a risk-averse CVaR loss newsboy model. Xue et al. (2015) [23] established the CVaR model. Xu and Meng (2016) [24] considered the CVaR newsboy ordering model with profit losses. The above literatures mainly considered newsboy models with orders from retailers to suppliers where wholesale prices apply. These models are not suitable for production and supply decision-making problems of DCEs. DCEs must decide the product supply that should be distributed to their chain stores for sales, and it is their goals to achieve an expected sales the same as the production volume with a maximum total profit or a minimum loss, where wholesale prices and recovery price do not apply. Several studies on the previous newsboy models are order decision problems for single retailer, which are rarely in the DCEs. In recent year, market competition has been fierce, and product oversupply has been widespread. Hence, the risk decision-making problem of production allocation and supply for DCEs is urgently needed, which avoids the loss of the companies due to the inconsistency between production and sales.

Therefore, in order to solve the risk loss problem of DCEs’ production allocation and supply, we establish a single-cycle optimal production allocation and supply model based on CVaR losses which derive an optimal allocation strategy with the total production volume for DCEs and obtain the minimum allocation strategy under the maximum CVaR risk loss. By the sales data of a food direct chain company, the numerical results corroborate that using the optimal supply strategy and discount strategy can reduce the expected risk loss and increase the expected profit. This paper is of important theoretical significance on guiding the centralized production and operation of DCEs.

The rest of this paper is organized as follows. Section 2 presents a new model of optimal production allocation and supply for DCEs and the optimal production allocation solution. Section 3 deals with the sensitivity analysis of the optimal production allocation and supply strategy. Section 4 proposes an approximate algorithm of the production allocation and supply strategy based on the above model. Section 5 presents a numerical example, which affirms that the optimal production allocation and supply strategy for the food DCE is efficient. Section 6 concludes the paper.

#### 2. Model of Optimal Production Allocation and Supply

The fact that every chain store faces a supply loss must be taken into consideration by DECs. Given its capacity, the DCE must decide the production volume that should be produced and supplied to all its direct chain stores to reach an optimal allocation and the minimum total loss, which is very difficult.

It is assumed that the number of DCES’ chain store is limited and every store sells the identical products. All products are supplied once in a single-cycle, and no supplementary is provided in the single period. A sale cycle is divided into two periods: the normal price period and the discount period. Hence, products are sold at a uniform retail price during the normal price period. After the normal price sale period, each retail store checks the inventory of its remaining products and decides whether the discount price will be used according to the inventory. When the inventory is smaller than the given value (minimum inventory), the remaining products are not discounted for sale. Otherwise, the remaining products are sold at the discounted price, which is determined according to the number of the remaining products.

After our field survey, the above description is consistent with the business model of many DCEs. Hence, assume: be the retail price of the product; the cost price (including the transportation, operation, and storage fee), where ; the allocation assigned to the -th store; the unit discount price of the remaining unsold products, where indicates that disposal fee is required and products have no residual value, means that products have the residual value and are sold at discount price during the discount sales period, or indicates that the value of remaining unsold products is zero. Let be the total production supply volume; hence, we have . Let be the random demand from the -th chain store, with corresponding probability density and probability distribution function , where , . The loss function of the -th chain store can be expressed as follows:where is the loss of oversupply and is the loss of short supply, .

Based on CVaR [15], we establish a single-cycle CVaR loss model of production allocation for the DCEs. Let the probability distribution where the loss function of the -th store is less than the given loss :Let be the given confidence level, then we define the VaR loss value of the -th store: is maximum possible loss of supply to the -th store under the confidence level . Hence, the CVaR loss at is defined by the following: is the cumulative loss of supply to the -th store under the confidence level . According to [15], (4) can be solved by the following loss function:If there is only one optimal solution to , then .

Let () be the loss weight of the -th store. Then the expected loss based on CVaR of the -th store can be expressed asHence, the total loss model based on CVaR of a DCE can be expressed aswhere , . We prove the following.

Theorem 1. *If there is a satisfying: and for , then the optimal allocation and supply strategies to () are*

*Proof. *The Lagrangian function of (*DCEP*) can be expressed aswhere is Lagrangian parameter. Then, with and , we can get , soThree situations are discussed as follows.

(1) When , (10) can be expressed asSo, and . We know that is monotonic decrease with , and . So, we have(2) When , (10) can be expressed asSolve the first derivative of :From and , we haveSolve the second derivative of :thenTherefore, is positive definite.

(3) When , (10) can be expressed asSolve the first derivative of : is monotonic decrease by , and .

According to above, is a convex on . Evidently with (16) and (17), for the fixed , we have a unique optimal solution to . Sothus , , where must satisfy the following inequalities:Let , , and , and (22) and (23) have solutions. Then the theorem conclusion is established.

Corollary 2. *When , DCEs are of decentralized supply, and the optimal allocation amount is expressed aswhere total supply .*

According to Theorem 1, set in (16), the optimal supply for a chain store is (26). If in (16), we get Theorem 3.

Theorem 3. *If there is a satisfying , and , and set in ( DECP), then the optimal allocation strategy based on the excepted loss is*

Equation (8) is the optimal solution of the production allocation of the DCE based on CVaR. Equation (26) is the optimal solution of the production allocation under decentralized supply. Equation (27) is the optimal solution of production allocation based on the excepted loss.

#### 3. Sensitivity Analysis of the Optimal Strategy

This section gives sensitivity analysis of the optimal strategy (8), i.e., when any of the parameters changes, how the optimal allocation strategy (8) changes accordingly. According to (8), let

Then the optimal allocation and supply strategy (8) is as follows:

Theorem 4. *Under the optimal allocation strategy, if , and are fixed, then the optimal allocation amount increases with the confidence level , and the optimal allocation amount decreases when the confidence level increases.*

*Proof. *According to (28) and (29)Evidently, the conclusion is true.

From Theorem 1, we know that the following conclusions are clearly true.

Theorem 5. *Under the optimal allocation strategy, if , and are fixed, then the optimal allocation amount decreases when increases.*

*Proof. *According to (28) and (29)Therefore, ; the conclusion is true.

Theorem 6. *Under the optimal allocation strategy, if , and are fixed, then the optimal allocation amount and decrease when the cost increases.*

*Proof. *According to (28) and (29)Therefore, the conclusion is true.

Theorem 7. *Under the optimal allocation supply strategy, when , and are fixed, if , then the optimal allocation amount decreases when the weight increases; or if , then the optimal allocation amount increases with the weight .*

*Proof. *According to (28) and (29)Therefore, the conclusion is true according to (30).

Theorem 8. *Under the optimal allocation strategy, if , and are fixed, the optimal allocation amounts and decrease when discount price increases.*

*Proof. *According to (28) and (29)Therefore, the conclusion is true.

Theorem 9. *Under the optimal allocation strategy, if , and are fixed then the optimal allocation amounts and increase with the retailer price .*

*Proof. *According to (28) and (29)Therefore, the conclusion is true.

We give the following numerical examples to illustrate Theorems 4, 5, and 8.

*Example 10. *Let , and in (28) and (29). Suppose that the demand probability density of a product satisfies a normal distribution: N (300,50^{2}).

(1) Figure 1(a) shows the different allocation amount at different confidence levels and multipliers , where disposal price and satisfies . From Figure 1(a), it is understood that the allocation amount increases with the confidence level when optimal multipliers ; and when the optimal multipliers the allocation amount decreases when the confidence level increases.