Mathematical Problems in Engineering

Volume 2015, Article ID 810798, 10 pages

http://dx.doi.org/10.1155/2015/810798

## A Robust Optimization of Capacity Allocation Policies in the Third-Party Warehouse

^{1}School of Management, Huazhong University of Science and Technology, Wuhan 430074, China^{2}School of Business, Hubei University, Wuhan 430062, China

Received 10 October 2014; Accepted 15 January 2015

Academic Editor: John Gunnar Carlsson

Copyright © 2015 Xu Xian-hao 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

We study the capacity allocation policies of a third-party warehouse center, which supplies several different level services on different prices with fixed capacity, on revenue management perspective. For the single period situation, we use three different robust methods, absolute robust, deviation robust, and relative robust method, to maximize the whole revenue. Then we give some numerical examples to verify the practical applicability. For the multiperiod situation, as the demand is uncertain, we propose a stochastic model for the multiperiod revenue management problem of the warehouse. A novel robust optimization technique is applied in this model to maximize the whole revenue. Then we give some numerical examples to verify the practical applicability of our method.

#### 1. Introduction and Literature Review

In today’s business world, a large number of companies outscore their warehouse functions to the third-party Warehouse (3PW) company in order to minimize their operation costs and focus on their core competencies. Therefore, warehousing industry becomes a booming business all over the world. According to the survey data from National Bureau of Statistics (NBS) of China, the national warehousing investment in fixed assets amounted to 69.20 billion dollars in 2013, increasing 32.7% over 2012. With the fast development of third-party warehousing industry, the revenue problem has received considerable attentions from both 3PW practitioners and researchers. 3PW company can provide storage services to different customers with fixed storage capacity and then capacity allocation policy plays an important role in revenue management.

The aim of capacity allocation in 3PW is to pursue a better fit between storage capacity allocation and market demand for each level in order to improve the expected revenue. In addition, customer demands for each level are uncertain. In this paper, we focus on the capacity allocation policy of a 3PW company for both single storage period and multiperiod with a revenue management perspective and robust optimization method.

Revenue management (RM) is a useful tool to help companies sell their products or services to right customers at right price and right time and make greatest revenue [1]. The field of revenue management is originated in the airline industry as a way to efficiently allocate fixed capacity to different classes of customers [2]. Talluri and Van Ryzin [3] discussed the network RM problems extensively, and their work was based on the independent demand model. The success of airline revenue management was widely reported, and this stimulated development of revenue management systems for other transportation sectors and in other areas of the services sectors, such as automobile rental [4], broadcasting [5], and hospitality [6]. Chiang et al. [7] provided an overview paper on revenue management. The solution methods that are mentioned in these papers are mainly mathematical programming, dynamic programming, and heuristics such as genetic algorithm. Revenue management is applied into many industries but not warehouse. In this paper, we study the capacity allocation problem with the perspective on revenue management.

Several researchers have worked at third-party warehousing. Gong and de Koster [8] gave a review on stochastic models and analysis on warehouse operations. Analysis of third-party warehousing contracts with commitments was studied without revenue consideration with capacity allocation in Chen et al. [9]. Lin [10] studied capacity allocation policy of third-party warehouse with dynamic optimization in revenue management perspective. Zhang et al. [11] provided the mathematical model of allocating customers to different warehouse spaces with deterministic demand and they solved this problem with a scheduling approach. Gong et al. [12] proposed a facility design method to improve the warehouse revenue with the consideration of stochastic market demand and the customers arrived according to a Poisson process. However, it is difficult to characterize the distribution of the uncertain demand, and robust optimization with uncertain demand is not included in these papers.

Robust optimization is a useful method to solve stochastic programming with unknown probability. Soyster [13] first proposed factor uncertainty in the field of optimization and gave its robust method. Ben-Tal and Nemirovski [14–16] proposed several uncertainties regarding the form and its applications and analyzed the robust methods of the linear programming and general convex programming. El Ghaoui et al. [17, 18] derived a similar conclusion and gave the robust methods to solve optimization problems in practical application. Bertsimas et al. [19, 20] did a further research on the basis of their work; they relaxed the conservation of the robust optimization and introduced the concept of the price of robust. Vairaktarakis [21] presented robust newsboy models with uncertain demand and provided an alternative approach using deterministic optimization models which could be solved by dynamic programming. Their work studied the single period problem, and our paper can solve the capacity allocation problem for both single period and multiple periods.

The rest of this paper is organized as follows. In Section 2, we describe the capacity allocation problem of the 3PW company and the corresponding mathematical model. Then we propose three different robust models of single period problem, the corresponding algorithms, and we give some numerical examples in Section 3. Section 4 presents the robust capacity allocation policies for multiperiod situation and then gives some illustrative examples. Finally, Section 5 concludes the paper with a short summary and future direction.

#### 2. Problem Description

In this paper, we consider such a 3PW company which provides several different levels of warehousing service for customers, with fixed capacity . The demand in each level is uncertain. The service price or cost for each storage level is different; therefore the unit revenue for each level is quite different. Facing such condition, this company should decide the storage capacity for each level. As the total capacity is fixed, if we increase one level’s capacity, the opportunity cost of other levels maybe occur; if not, satisfaction degree of customer in this level may decrease and give orders to other competitors. In order to improve the revenue of this warehouse center and use the storage capacity more appropriately, they should provide an appropriate capacity allocation policy which can meet the market demand more accurately. Therefor, this capacity allocation problem is how this 3PW company will allocate its limited warehousing capacity to each storage service level; then it can maximize the total revenue.

Hypotheses are made as the following:: total capacity of the 3PW company; : number of the service levels; : capacity number of th level, ; : the unit price of th level per period; : the unit cost of th level per period, and ; : the unit cost of lost sales of th level per period; : the demand of th level capacity, it is an interval uncertain variable with the probability density function and cumulative distribution function ; : the revenue of th level capacity with capacity and demand .

The TPW is a unit-load warehouse; that is, all goods in this warehouse need to occupy the same storage space (one pallet); split of the pallet does not exist.

The profit function for the th item is given by

In a similar way as stochastic knapsack method, dynamic warehousing capacity allocation model is obtained as follows:

The first constraint is the total capacity constraint, and the second one implies capacity of each level cannot be negative.

Theorem 1. *There exists the optimum solution in formula (2) without constraint.*

*Proof. *Analyzing formula (2), we can get the first-order derivative as follows:

And the second-order derivative is

Now we know that the expected revenue function is a concave function about variable , so there exists the optimum solution if there is no constraint condition, and it should satisfy the following condition:

However, in practice we can hardly know the cumulative distribution function of the demand variable . On the contrary, the maximum and minimum value of can easily be got. In the rest of this paper, we discuss the interval demand condition with .

#### 3. Robust Optimization for Single Period

There exist many methods to describe the uncertainty in management optimization problems. One of the most classic versions is the assumption that the probability distribution of the random variable is known. However, it is always not realistic in the actual problem. Robust optimization is a useful method to solve stochastic programming with unknown probability.

According to Vairaktarakis [21], there are three different types of robust methods: absolute robust, deviation robust, and relative robust method. Applying these three robust methods in warehousing capacity allocation, we can get the following three models.

##### 3.1. Absolute Robust Model

In case that the demand realizations for item take values from the interval , our absolute robust formulation with a budget constraint becomes

Analyzing the objective function, we can get

Thus, the absolute robust allocation should satisfy the following equation:

That implies

Now, we can get the absolute robust allocation model with uncertain interval demand as follows:

The following observations can be made for model AR.

Theorem 2. *There exists an optimal solution for this AR model, and , .*

With this theorem, we can get the equivalent form of formula (10):

The optimal solution of (11) maximizes the quantity , and therefore AR model can be reduced to a continuous knapsack problem. The corresponding algorithm will be introduced in the next section.

##### 3.2. Deviation Robust Model

The deviation robust-order quantity is the solution of

This formulation provides a solution that minimizes over all choices of order quantities the maximum profit loss due to demand uncertainty. This is a mini-max regret approach where the regret is captured by the difference . The profit could be realized if there is no demand uncertainty in which case we would order .

That equals

Thus, the deviation robust allocation should satisfy the following equation:

Now, we can get the absolute robust allocation model with uncertain interval demand as follows:

Just as AR model, the objective function equals

It makes us maximize . Therefore DR model reduces to a continuous knapsack problem. The corresponding algorithm will be introduced in the next section.

##### 3.3. Relative Robust Model

The third robust formulation is called relative robustness and the corresponding formulation is given bywhich minimizes the relative profit loss per unit of profit that could be made if there was no demand uncertainty. Note that the relative profit loss measures the lost profit as a percentage of the profit that could be made if we knew the actual demand.

In the rest of our analysis it will become clear that the three objectives result in very different choices of order quantities. Similar formulations can be written for the case of interval scenarios. The only difference in modeling the continuous case is that there is a constraint .

As we have analyzed above, it equals

The last equation has the optimum solution if and only if .

Finally, we can get the RR model:

That equals

##### 3.4. Algorithm and Illustrative Examples

In this section, we adapt the continuous knapsack procedure to the three robust formulations.

*Algorithm for AR Model *

*Step 1. *Define the weight , , and index the items such that .

*Step 2. *If , then , .

Otherwise identify the critical item such that

*Step 3.* If , ; if , ; if , .

As the algorithm for DR model is the same as the above algorithm, we do not show it again here.

##### 3.5. Algorithm for RR Model

*Step 1.* Define the weight , , and index the items such that .

*Step 2.* If , then , .

Otherwise identify the critical item such that

*Step 3.* For , ; for , ; for , .

##### 3.6. Illustrative Examples

There is a third-party warehouse company , the total capacity is 1500, and the service price and cost for each level are shown as Table 1. Assume that, for each level , we are given a scenario of demand quantities that may be realized. The number of likely minimum and maximum demand of is and . Then we can calculate the weights and , which are shown in the last two columns.