^{1}

^{2}

^{3}

^{1}

^{1}

^{2}

^{3}

Traditional inventory models focus on operational decisions and inventory control. Quite few models consider the financial constraint and decision bias such as loss aversion, which are the reality in today's business environment, especially for the fashion and textiles industry. In this paper we study the inventory control problem for a loss-averse retailer with financial constraint for operations in a periodic review setting in a finite horizon. We characterize the optimal inventory control policies with self-financing or with borrowing as capital-dependent base-stock policies. We demonstrate with numerical examples that the optimal base-stock level is nonincreasing in the accumulated wealth and the loss-aversion indicator.

The fashion and textiles industry is characterized by short product lifecycle, high volatility, and low predictability, which cause financial risk to the companies in the industry. This is true especially for those start-up and growing firms in the fashion and textiles industry, who always encounter credit and capital shortage problems. According to the empirical evidence from Ayyagari et al. [

However, although there is an extensive literature on inventory control problems in both deterministic and stochastic environments, few models consider the firms’ financial decisions when characterizing the optimal inventory policy. Among them, Kogut and Kulatilaka [

The managers’ decision bias, which means that their decisions are not always consistent with optimal decisions under profit-maximizing or cost-minimizing objective, has been recognized by researchers and practitioners. For example, Kahn [

In summary, we study the inventory control and financing decision problems for a loss-averse retailer when the retailer is confronted with a financial constraint in this paper. Using a lost-sales model, we first investigate the inventory control policy for a loss-averse retailer with borrowing decisions in a finite horizon model. We find that the optimal inventory policy follows a capital-dependent base-stock policy. Then, we extend the basic model to the self-financing situation and find that the optimal inventory policy also follows a capital-dependent base-stock policy. We also study how the optimal replenishment level changes according to different parameter settings with self-financing.

The rest of this paper is organized as follows. We review the related literature in Section

Our research is mainly related to two streams of literature: the inventory models with decision criteria other than risk neutral and the inventory models with financial consideration.

Traditional inventory models consider the risk neutrality, that is, either maximizing the total profit or minimizing the total cost. However, decision makers do have risk preferences other than risk neutrality. Lau [

Financial and operational decisions are usually studied separately. The separation can be justified by the seminal work of Modigliani and Miller [

Recently, researchers in operations management field have recognized the importance of the interaction between operational and financial decisions and tried to incorporate the financial considerations into operational decisions. For the single period model setting, Gupta and Gerchak [

In this paper, we extend Chao et al.’s study [

Few traditional inventory models take financial constraint of the decision maker into consideration when deriving the optimal ordering policy. However, in practice, operations are always constrained by the decision maker’s available cash and borrowing or investing is popular. Thus, the decision maker should make a tradeoff between operating, financing, and investing when making operation decisions. To address such tradeoffs, we consider a loss-averse retailer, whose utility is measured by the loss-averse function, that makes replenishment and financial decisions over a finite horizon

We number the first period as period 1, while the last period is period

Now we focus on the investing and borrowing decisions of the retailer in our model. At the investment side, we assume that the surplus capital (after paying the procurement cost) can be invested in a savings account to earn an interest rate

Before we further analyze the different cases, we first introduce the loss-averse utility function this paper employs to measure the retailer’s utility. This utility function is first introduced by Kahneman and Tversky [

The utility function

The proof is straightforward.

In this case, the retailer can borrow from an outsider cash provider to finance his inventory decision during the planning horizon. We assume that there is no limit for borrowing for the sake of simplicity. Denote the borrowing interest rate by

Therefore, the decision problem is to choose the optimal

Denote

For any period

The result can be proved by induction. First, note that at period

The monotone property of the value function with the initial cash flow is intuitive. On one hand, the more initial capital the firm has, the better it is for the firm's terminal wealth, which leads to the better firm's terminal utility. On the other hand, the more initial capital the firm has, it provides the firm more flexibility for inventory replenishment, which makes the firm earn at least no less. However, it seems surprising to see that the more the initial inventory is, the better the retailer is. The reason lies in the assumption that there is no holding cost.

To derive the joint concave property of the value function

For any period

Note the following relationship:

It follows from Lemma

Lemma

For any period

The theorem can be proved by backward induction. Clearly,

To prove that

We have

This result establishes the concavity property of the value function, which is useful to further characterize the existence and uniqueness of the optimal policy. However, before that, we should first define the following two functions:

Then, from Lemma

For any period

This lemma implies that it is not possible that

When the state is

In this case, the ordering decision satisfies the cash-flow constraint

Then, the decision problem is the same as (

Denote

The tradeoff in the dynamic programming equation mentioned previously is between ordering inventory and putting cash in savings account. When inventory is ordered, the retailer runs the risk of not selling the inventory and therefore loses the opportunity of earning an interest. Note that the problem in the final period is effectively a newsvendor problem with order quantity limit under loss-averse utility function. To derive the optimal inventory control policy, preliminary results are needed. Similar to the analysis when there are borrowing opportunities, we neglect the proofs here.

For any period

Now, we can come up with the following policy conclusion.

When the state is

if

if

if

Refer to

We present numerical examples to demonstrate the optimal inventory policy and its dependency on wealth level

Optimal base-stock policy given target profit level.

Figure

Figure

Optimal base-stock policy given penalty coefficient.

Figure

The optimal control strategy

Figure

The optimal control strategy

Traditional inventory models usually do not take the financial constraints into consideration. Yet, in practice companies are usually constrained by the cash available for operations. This is especially true during the global financial crisis of 2008. According to the survey of 1050 Chief Financial Officers (CFOs) in the USA, Europe, and Asia, Campello et al. [

On the other hand, traditional inventory models usually assume risk neutral objective. Yet, experiments and real business practice have already demonstrated that there exists decision bias. To explain the decision bias, researchers start to consider objectives other than risk neutral objectives. Instead, they consider the risk-averse objectives and even the loss-averse objectives from prospect theory. According to the loss averse theory, people are more averse to losses than they are attracted to the same-sized gains.

In this paper, we study a periodic review inventory control and borrowing decisions for a loss-averse retailer in a finite horizon. Unsatisfied demand is lost. We find that the optimal inventory control policy is a capital-dependent base-stock policy. Moreover, we demonstrate with numerical examples that the optimal order-up-to level is nondecreasing with respect to the accumulated wealth and the loss aversion indicator. The higher the loss aversion indicator/the accumulated wealth is, the lower the base-stock level is. The theoretical results can be applied to the fashion and textiles industry. Although we have characterized the optimal inventory control policy for the financial constrained retailer in a finite horizon, we still can extend our current study in several directions. First, with financial constraint, operations management tools like inventory control and will not be enough to match supply with demand. Marketing instruments like pricing and sales effort should be incorporated in such situation. Second, we assume that the target level

The authors would like to thank the editor and two anonymous referees, whose comments improve their work significantly. L. Ma was partially supported by the National Natural Science Foundation of China (NSFC) with Grant nos. 71001073 and 71271182 and the Natural Science Foundation of SZU (201121). W. Xue was partially supported by NSFC with Grant nos. 71171105 and 70932003, the China Postdoctoral Science Foundation with Grant no. 2012M521054, and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China with Grant no. 708044. Y. Zhao was partially supported by the NSFC with Grant no. 71101028, the Program for Innovative Research Team in UIBE, and the Program for Excellent Talents, UIBE. X. Lin was partially supported by the NSFC with Grant no. 70601016.