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This paper investigates the ordering decisions and coordination mechanism for a distributed short-life-cycle supply chain. The objective is to maximize the whole supply chain's expected profit and meanwhile make the supply chain participants achieve a Pareto improvement. We treat lead time as a controllable variable, thus the demand forecast is dependent on lead time: the shorter lead time, the better forecast. Moreover, optimal decision-making models for lead time and order quantity are formulated and compared in the decentralized and centralized cases. Besides, a three-parameter contract is proposed to coordinate the supply chain and alleviate the double margin in the decentralized scenario. In addition, based on the analysis of the models, we develop an algorithmic procedure to find the optimal ordering decisions. Finally, a numerical example is also presented to illustrate the results.

Along with the development of new technology, increasingly diversified customers’ demand, and higher competition, the lifespan of some products are getting shorter and shorter (so are called short-life-cycle products). Short-life-cycle products, such as toys, sunglasses, high-technology products, and fashion clothing, which are characterized by short sales season, high demand variety, and long replenishment lead time. In addition, for short-life-cycle products, both overstock and understock could result in huge loss, thus reducing demand uncertainty is the main task in managing this kind of supply chain (see Lowson et al. [

In practice, QR (quick response) with forecast updates is a method for reducing demand uncertainty (see, e.g., Tersine and Hummingbird [

Concerning forecast updates, the Bayesian approach is a well-known technique to use new information to revise the demand forecast. The procedure of forecast updates using Bayesian approach is as follows. First, a demand distribution is chosen from a family of distributions whose parameters are partly or both stochastic with given prior distributions. Second, if new information becomes available, Bayesian formula is used to calculate posterior distributions of the parameters. Last, the posterior distributions of the parameters are used to update demand distribution.

When lead time can be shortened by extra investment (when it is controllable), there are two problems in ordering decision making. The first question is when to order and how much to order from the stand point of a buyer. This question involves two tradeoffs: longer versus shorter lead time, more versus less ordering quantity. A shorter lead time leads to lower demand uncertainty, more crashing cost, and vice versa. On the other hand, the more ordering quantity results in less cost of understock, more cost of overstock, and vice versa. The second question is how to motivate the supply chain participants to cooperate and achieve a win-win situation with contract from the stand point of the supply chain. This is because decisions made from the perspective of the whole supply chain may not always make both supply chain participants better off.

Lead time is an important factor that affects the operational cost, service level and forecast uncertainty in a short-life-cycle-product supply chain. However, most of the literatures on short-life-cycle products focused on how much to order (decision on ordering quantity) and neglected the decision on when to order (decision on lead time). Because of the complexity of the decision based on both the lead time and ordering quantity, the optimal ordering policy for short-life-cycle products with controllable lead time has not been fully studied.

The purpose of this paper is to take lead time decision making into account in a distribution supply chain, which consists of one retailer and one manufacturer. The supply chain operates a single short-life-cycle product. Optimal ordering policies with lead time decision in two scenarios are considered. In the decentralized decision-making scenario, the retailer makes decisions on lead time and ordering quantity to maximize his own profit. In the centralized decision scenario, decisions on lead time and ordering quantity are made to maximize the whole supply chain’s profit. In addition, a supply chain contract is developed to coordinate the supply chain, and the conditions that the supply chain contract works are investigated.

The remainder of the paper is organized as follows. Section

There are three categories of literature related to our research. The first is on QR policy with forecast updates, the second is on inventory management with controllable lead time, and the third is on supply coordination with contracts. We briefly review them, respectively.

This kind of the literature utilizes the Bayesian theorem to update forecast for short-life-cycle products, which is the same as our work. However, the literature focuses on the concept of lead time reduction other than decision on lead time. In addition, the cost of lead time reduction and the relationship between lead time and forecast error have not been clearly considered. In this section, we review some of the works especially related to our study.

Among them, Iyer and Bergen [

Wu [

Similarly, without concerning crashing cost, Choi [

Fisher and Raman [

Similarly, Sethi et al. [

The research of the paper is very similar to that of Choi et al. [

This research explicitly takes the relationship between lead time and the forecast error, the relationship between lead time and crashing cost, into account, simultaneously addressing the two problems mentioned in Section

The literature falling into this category mainly focus on ordering quantity, lead time, service level, and so forth. The treatment of lead time as a deterministic decision variable within inventory models began with Liao and Shyu [

This paper is similar to the above-mentioned literatures with the concept that lead time is a decision variable and can be reduced at an added crashing cost. However, there are three differences between this paper and the above-mentioned studies. Firstly, the above-mentioned studies dealt with multiperiod inventory problems for durable goods; this paper dealt with single-period ordering problem for short-life-cycle products, whose demands have high uncertainty. Secondly, neither the relationship between lead time and the forecast error nor forecast updates are taken into account in the above-mentioned studies. Thirdly, the above-mentioned literature investigated an optimal decision from a stand point of the single supply chain participant or from a standpoint of the whole supply chain, and the coordination was not considered.

Due to the conflicts of interests and the distributed nature of the decision structure in supply chain, there exists double marginalization (Spengler [

Each kind of contract has its limitations and strengths as Arshinder et al. [

In fact, to develop a contract needs to orient the specific problem and the potential risk. In our work, the retailer’s decisions on lead time and ordering quantity can bring crashing cost, overstock loss, or understock loss. Neither single contract above mentioned can coordinate the two risks. Therefore, we develop a three-parameter supply contract to coordinate these risks simultaneously. Further, the contract can flexibly allocate the profit of the supply chain according to the participants’ value-added capabilities, and it is easy for practitioners to adopt in practice.

We will begin the modeling process by presenting the following notations and assumptions.

The notations employed in the paper are as follows:

The assumptions employed in this paper are stated as follows.

A single-buyer-single-vendor supply chain that manufactures and sells a single short-life-cycle product. The retailer and the manufacturer are both risk neutral. Viewing the wholesale price as a given parameter, the retailer makes decisions on when to order and how many to order. This paper assumes that the retailer only has one ordering opportunity in the entire selling season, which is based on long replenishment lead time and other situations (see, Choi et al. [

For the sake of simplicity, we assume that the selling season begins at time zero, see Figure

To facilitate our discussion,

The maximum lead time, all whose components are equal to regular time, is denoted by

The minimum lead time, all whose components are equal to the minimum time, is denoted by

If the components from 1 to

It is easy to know that

Given lead time

Process of demand forecast updates.

In this paper, we extend the demand uncertainty structure in Iyer and Bergen [

In the traditional system, the order is placed at time

In a QR system, an order is placed at time

Thus, the pdf for

This update can be shown in Figure

The essential benefit from QR for fashion products is that the information gathered regarding the sales of related items can be used to reduce forecast error of the expected demand. The magnitude of this reduction in forecast error can be substantial (see, Iyer and Burger [

Let

Here, parameters

Relationship between the lead time and the forecast error of the expected demand (see Lowson et al. [

In this section, a decentralized decision-making scenario to maximize the retailer’s profit and an integrated decision-making scenario to maximize the system-wide supply chain’s profit are formulated. Furthermore, we discuss the existence and attributes of the optimal solutions to the above-mentioned models and develop the efficient algorithms to find the optimal solutions.

Under the assumptions in Section

If the market demand and ordering quantity are

Simplifying (

The expected profit of the retailer is

The expected profit of the manufacturer is

The expected profit of the whole supply chain is

Thus, the decision-making model from the view of the retailer is as follows.

Model (I):

The optimal solution of Model (I) is denoted by

In order to illustrate the effects of different decision-making scenarios on the profits of the supply chain participants and the whole supply chain, we assume that there would exist a virtual center whose decision objective is to maximize the profit of the whole supply chain channel. We call this situation as a centralized decision-making scenario and denote it as model (II).

Model (II):

The optimal solution to the model (II) is written as

In order to solve the model (I) and model (II), some analysis needs to be done firstly. Since the two models are similar, we only analyze model (I). As far as model (II) is concerned, it may be deduced by analogy. For Model (I), we have the following conclusions.

Denote the standard variance of demand after updates as

For the sake of simplicity,

Substituting (

For fixed

Two identical equations used are presented as follows.

Taking the second partial derivative of

Setting the RHS of (

It is clearly seen from (

Denote

From (

Taking the partial derivative of

The optimal solution to model (I) is in existence and unique.

For fixed

Therefore, the Hessian Matrix of

Then, we proceed by evaluating the principal minor determinants of

The first-order principal minor determinant of

The second-order principal minor determinant of

Thus,

Since Model (I) is similar to Model (II), we analyze the Model (I) only. From Theorem

The optimal

The optimal

If

If

Since

If

If

If

According to Theorem

Compute the values of

If

If

If

For

If a non-positive number occurs at

If a non-positive number occurs at

A Golden section method or two-section method can be used to search the optimal solution to the equation

Compute the optimal

The algorithm for solving model (II) is similar to that of the model (I) as mentioned above, so we skip it here.

In a distributed supply chain, since the supply chain participants are independent entities; thus, decisions are made to maximize the decision maker’s own profit. To some extent, the double marginalization (e.g., Spengler [

In apparel industries, the manufacturer usually uses buy-back contract to enable retailers to enlarge ordering quantity (such as Padmanabhan and Png [

In order to coordinate the supply chain, a three-parameter contract which combines buy-back and risk-sharing contract is developed in this paper.

Let

In this scenario, the expected profit of the retailer is

The expected profit of the manufacturer is

The expected profit of the whole supply chain is

Under the cooperative circumstances with the three-parameter contract, in order to get a win-win situation, the optimal decision from the respective of the retailer (denoted by Model (III)) should be described as follows

Model (III):

Equations (

The optimal solution to the Model (III) is denoted by

For the three-parameter contract

If the three-parameter contract

It is obvious that

Since

Substituting (

Substituting (

Substituting (

Because

Substituting (

Substituting (

For a contract to be implementable, it is necessary that the contract can arbitrarily allocate the supply-chain profit between the retailer and supplier (see, e.g., Chen et al. [

The profit of the whole supply chain system is divided into two parts by the three-parameter contract

Substituting (

Here,

If the wholesale price changes from

To illustrate the above models and algorithms, we consider a supply chain: the retail price

Lead time data.

Component element of lead time ( | Regular time | Minimum time | Crash costs |
---|---|---|---|

1 | 50 | 30 | |

2 | 40 | 24 | |

3 | 40 | 20 | |

4 | 40 | 26 | |

5 | 30 | 20 |

Using the algorithms introduced in Section

The optimal solution to Model (I) and (II) with different values of

Decentralized decision-making model | Centralized decision-making model | ||||||||||

Model (I) | Model (II) | ||||||||||

Optimal solution | Profit allocation | Optimal solution | Profit allocation | ||||||||

0.10 | 120.00 | 2057.6 | 18260 | 10288 | 28548 | 120.00 | 2306.4 | 17570 | 11532 | 29103 | 555 |

0.50 | 122.58 | 2058.0 | 18134 | 10290 | 28424 | 144.00 | 2328.5 | 17363 | 11643 | 29006 | 582 |

1.00 | 144.00 | 2061.7 | 18037 | 10309 | 28345 | 164.00 | 2348.2 | 17223 | 11741 | 28964 | 619 |

2.00 | 164.00 | 2065.4 | 17955 | 10327 | 28282 | 180.00 | 2364.7 | 17106 | 11824 | 28929 | 647 |

5.00 | 180.00 | 2068.5 | 17867 | 10343 | 28210 | 200.00 | 2386.6 | 16975 | 11933 | 28908 | 698 |

6.00 | 185.15 | 2069.6 | 17847 | 10348 | 28195 | 200.00 | 2386.6 | 16975 | 11933 | 28908 | 713 |

8.00 | 200.00 | 2072.6 | 17845 | 10363 | 28208 | 200.00 | 2386.6 | 16975 | 11933 | 28908 | 700 |

10.00 | 200.00 | 2072.6 | 17845 | 10363 | 28208 | 200.00 | 2386.6 | 16975 | 11933 | 28908 | 700 |

As shown in Table

It is obvious that the risk for the retailer to order more is much higher than that for the supply chain. Therefore, the retailer’s decision on order quantities tends to be conservative.

As far as lead time is concerned, the retailer can benefit from the demand forecast updates by reducing the lead time while it can bring more crashing cost to the whole supply chain. Thus, the decentralized decision from the retailer cannot be consistent with that of the centralized decision from the supply chain. Column 1 of Table

Hence, in order to coordinate the supply chain, risk should be distributed and shared among supply chain partners. This idea is embodied in the three-parameters contract proposed in Section

Contract and supply chain coordination.

Contract parameters | Retailer | Manufacturer | Supply chain | |||||||

10.3025 | 0.8229 | 0.6465 | 18815 | 555 | 100% | 10288 | 0 | 0% | 29103 | 555 |

10.4 | 0.827 | 0.64 | 18626 | 366 | 66% | 10477 | 189 | 34% | 29103 | 555 |

10.5 | 0.8311 | 0.6333 | 18432 | 172 | 31% | 10671 | 383 | 69% | 29103 | 555 |

10.5884 | 0.8346 | 0.6274 | 18260 | 0 | 0% | 10843 | 555 | 100% | 29103 | 555 |

10.3213 | 0.8237 | 0.6452 | 18716 | 582 | 100% | 10290 | 0 | 0% | 29006 | 582 |

10.4 | 0.827 | 0.64 | 18564 | 430 | 74% | 10442 | 152 | 26% | 29006 | 582 |

10.5 | 0.8311 | 0.6333 | 18370 | 237 | 41% | 10636 | 345 | 59% | 29006 | 582 |

10.6 | 0.8351 | 0.6267 | 18177 | 43 | 7% | 10829 | 539 | 93% | 29006 | 582 |

10.6223 | 0.836 | 0.6252 | 18134 | 0 | 0% | 10872 | 582 | 100% | 29006 | 582 |

10.4 | 0.827 | 0.64 | 18537 | 500 | 81% | 10427 | 118 | 19% | 28964 | 619 |

10.5 | 0.8311 | 0.6333 | 18344 | 307 | 50% | 10620 | 311 | 50% | 28964 | 619 |

10.6 | 0.8351 | 0.6267 | 18151 | 114 | 18% | 10813 | 505 | 82% | 28964 | 619 |

10.4 | 0.827 | 0.64 | 18515 | 559 | 86% | 10414 | 87 | 14% | 28929 | 647 |

10.5 | 0.8311 | 0.6333 | 18322 | 366 | 57% | 10607 | 280 | 43% | 28929 | 647 |

10.6 | 0.8351 | 0.6267 | 18129 | 174 | 27% | 10800 | 473 | 73% | 28929 | 647 |

10.4 | 0.827 | 0.64 | 18501 | 634 | 91% | 10407 | 64 | 9% | 28908 | 698 |

10.5 | 0.8311 | 0.6333 | 18308 | 441 | 63% | 10600 | 257 | 37% | 28908 | 698 |

10.6 | 0.8351 | 0.6267 | 18116 | 249 | 36% | 10792 | 450 | 64% | 28908 | 698 |

10.4 | 0.827 | 0.64 | 18501 | 654 | 92% | 10407 | 59 | 8% | 28908 | 713 |

10.5 | 0.8311 | 0.6333 | 18308 | 461 | 65% | 10600 | 252 | 35% | 28908 | 713 |

10.6 | 0.8351 | 0.6267 | 18116 | 268 | 38% | 10792 | 445 | 62% | 28908 | 713 |

10.7 | 0.839 | 0.62 | 17923 | 76 | 11% | 10985 | 637 | 89% | 28908 | 713 |

10.4 | 0.827 | 0.64 | 18501 | 656 | 94% | 10407 | 44 | 6% | 28908 | 700 |

10.5 | 0.8311 | 0.6333 | 18308 | 463 | 66% | 10600 | 236 | 34% | 28908 | 700 |

10.6 | 0.8351 | 0.6267 | 18116 | 270 | 39% | 10792 | 429 | 61% | 28908 | 700 |

10.7 | 0.839 | 0.62 | 17923 | 78 | 11% | 10985 | 622 | 89% | 28908 | 700 |

10.4 | 0.827 | 0.64 | 18501 | 656 | 94% | 10407 | 44 | 6% | 28908 | 700 |

10.5 | 0.8311 | 0.6333 | 18308 | 463 | 66% | 10600 | 236 | 34% | 28908 | 700 |

10.6 | 0.8351 | 0.6267 | 18116 | 270 | 39% | 10792 | 429 | 61% | 28908 | 700 |

10.7 | 0.839 | 0.62 | 17923 | 78 | 11% | 10985 | 622 | 89% | 28908 | 700 |

For fixed

For fixed

In Table

From Table

In this paper, we investigate two questions for a short-life-cycle-product supply chain, in which the forecast error of expected demand is a log-linear function of lead time, and crashing cost is a stepwise linear function of the lead time. Under this assumption and accounting for the considerations performed for the single-buyer-single-vendor supply chain, we model decisions on lead time and ordering quantity in a decentralized decision-making scenario and a centralized decision-making scenario, respectively. Then, the attributes and existence of the optimal solutions to the models are analyzed. Based on the analysis, the algorithms to solve the models are developed. To validate the algorithms and models, a numerical example is presented. The numerical result shows that the algorithms are effective. Furthermore, in order to alleviate the double margin in the decentralized decision-making scenario, a three-parameter supply chain contract is proposed to coordinate the supply chain. Through the three-parameter contract, the whole supply chain’s profit is divided into two parts. Each of them represents the single supply chain participant’s value-added capacities. The contract can arbitrarily allocate the supply chain profit between the manufacturer and the retailer by tuning the value of parameters. For this reason, it is easy for practitioners to adopt in practical.

In this study, only single product and single ordering are considered. Moreover, multiple-product supply chain is universal in practice and, with shorter lead time, more than one order can be placed. Therefore, it would be interesting to study multiple products’ scenario or multiple-order policies scenario in the future research.

As shown in Figure

Values of

12 | 9 | 6 | 3 | 0 | |

40 | 23 | 19 | 15 | 10 | |

3.6889 | 3.1355 | 2.9444 | 2.7081 | 2.3026 |

Computing the coefficient between

The coefficient matrix of

The result (

Hence, denoting

This work is supported by the Natural Science Foundation of China under Grant no. 70872047. The authors gratefully acknowledge the helpful comments and suggestions of the editor and the anonymous reviewers which led to a substantial improvement of this paper. The authors also thank Yiliu (Paul) Tu, the professor of The Schulich School of Engineering, University of Calgary, and Ding Zhang, the professor of the School of Business, the State University of New York at Oswego, for their helpful works which have contributed to improving the readability of the paper.