We study the capacity allocation policies of a thirdparty 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.
In today’s business world, a large number of companies outscore their warehouse functions to the thirdparty 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 thirdparty 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 [
Several researchers have worked at thirdparty warehousing. Gong and de Koster [
Robust optimization is a useful method to solve stochastic programming with unknown probability. Soyster [
The rest of this paper is organized as follows. In Section
In this paper, we consider such a 3PW company which provides several different levels of warehousing service for customers, with fixed capacity
Hypotheses are made as the following:
: total capacity of the 3PW company;
The TPW is a unitload 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
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.
There exists the optimum solution in formula (
Analyzing formula (
And the secondorder derivative is
Now we know that the expected revenue function is a concave function about variable
However, in practice we can hardly know the cumulative distribution function
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 [
In case that the demand realizations for item
Analyzing the objective function, we can get
Thus, the absolute robust allocation
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.
There exists an optimal solution
With this theorem, we can get the equivalent form of formula (
The optimal solution of (
The deviation robustorder 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 minimax regret approach where the regret is captured by the difference
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
The third robust formulation is called relative robustness and the corresponding formulation is given by
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
In this section, we adapt the continuous knapsack procedure to the three robust formulations.
Define the weight
If
Otherwise identify the critical item
As the algorithm for DR model is the same as the above algorithm, we do not show it again here.
Otherwise identify the critical item
There is a thirdparty warehouse company
Each cost of warehouse
Level 








1  10  4  4  500  700  10  0.00238 
2  8  2  3  600  700  9  0.00214 
3  12  5  5  200  300  12  0.00686 
4  14  4  3  300  450  13  0.00289 
By Table
The capacity allocation policies of 3 types of robust model.
Level  Prior (A/DR)  Prior (RR) 







1  3  3  557  557  643  643  628  628 
2  4  4  627  388  682  171  679  207 
3  2  1  229  229  271  271  262  262 
4  1  2  326  326  415  415  403  403 
Table
As a result, for any budget level and for every one of the three objectives, our formulations result to order the maximum possible number of units starting with high priority items and continue on with items of lower priority. For example, in the relative robust policy, we should first satisfy the first three highest priority levels: level 4, level 3, and level 1 with the capacity allocations 326, 529, and 557. For level 4, the only left capacity is 388. Similarly, we can get the deviation robust policy and relative robust policy, which are shown as the seventh and ninth column in Table
In this section, we analyze the effect of these three robust policies on the company revenue. In general, we use three scenarios to simulate the demand market: scenario 1, the lowest situation, the demand of each level is the minimum demand, that is, 500, 600, 200, and 300; scenario 2, the highest situation, the demand of each level is the maximum demand, that is, 700, 700, 300, and 450; scenario 3, the middle situation, the demand of each level is 600, 650, 250, and 375. Using the three robust policies in Table
The revenue of each policy.
Robust policy  Scenario 1  Scenario 2  Scenario 3  Average revenue 

AR  8615  8298  9323  8745.3 
DR  5752  8866  8594  7737.3 
RR  6229  8776  8729  7911.3 
From Table
The revenue of each policy under each scenario.
By Figure
The results make sense for the 3PW company holder. On one hand, if the demand market is not so high, that is, the demand for each level is lower, he should choose the absolute robust policy and avoid deviation robust policy. Otherwise, he should choose deviation robust or relative robust policy when the demand for each level is high. On the other hand, from the perspective of average revenue, the absolute robust policy is the best policy for the conservative holders, whose managements are risk aversion.
In this section, we extend this capacity allocation problem into multiperiod condition. The following are the new hypotheses which are used in this section, and the remaining parameters are the same as last section.
We assume there are no goods staying before period 0 and all the goods have to be retrieved on or before the last period. The 3PW is a unitload warehouse; that is, all goods in this warehouse need to occupy the same storage space (one pallet); split of the pallet does not exist.
We consider a particular period
The first part of this equation stands for the number of goods which stay over period
With the fixed capacity, we have the following constraints for period
Particularly, on the period 0, we have
The objective function is to maximize the total revenue of all period and all service level. The third constraint condition is the total capacity constraint; it means that in every time period the sum of capacity allocations of all service level can not be larger than the total capacity of the 3PW. The last constraint condition stands for the fact that the capacity allocation variable
The problem looks like a linear integer programming problem. Unfortunately, the parameters
According to Mulvey et al. [
An optimal solution is solution robust with respect to optimality if it remains “close” to being optimal for any scenario
An optimal solution is model robust with respect to feasibility if it remains “almost” feasible for any scenario
Consider such a stochastic programming:
There are several different forms of
Under this form and the above robust model, we can get a robust formulation of model (
In model (
We can use a similar method to deal with the other constraint containing
The prominent feature of formulation (
Consider such a 3PW company which can provide three different levels of storage service for the storage customers. According to the history data, there are three main demand scenarios
Demands of customers (scenario
ST: 
RT: 
RT: 
RT: 


1  2  3  4  1  2  3  4  1  2  3  4  
0  200  180  270  150  120  130  190  140  100  180  70  30 
1  0  350  280  120  0  200  150  135  0  95  110  50 
2  0  0  280  100  0  0  160  150  0  0  130  230 
3  0  0  0  200  0  0  0  120  0  0  0  50 
Demands of customers (scenario
ST: 
RT: 
RT: 
RT: 


1  2  3  4  1  2  3  4  1  2  3  4  
0  100  140  290  100  90  100  120  110  120  100  70  40 
1  0  320  180  120  0  150  200  120  0  80  100  40 
2  0  0  220  90  0  0  120  100  0  0  100  150 
3  0  0  0  150  0  0  0  100  0  0  0  70 
Demands of customers (scenario
ST: 
RT: 
RT: 
RT: 


1  2  3  4  1  2  3  4  1  2  3  4  
0  130  200  250  120  100  120  170  120  90  150  90  50 
1  0  270  310  100  0  180  180  150  0  100  75  50 
2  0  0  240  120  0  0  150  120  0  0  110  200 
3  0  0  0  180  0  0  0  120  0  0  0  60 
In these tables, the “ST” means the storage time and “RT” means the retrieval time of the goods. The first number in Table
Capacity allocation policy with multiple demands.
ST: 
RT: 
RT: 
RT: 


1  2  3  4  1  2  3  4  1  2  3  4  
0  170  180  270  110  120  120  185  125  120  80  90  50 
1  0  270  250  100  0  195  180  140  0  120  110  50 
2  0  0  200  110  0  0  150  135  0  0  130  230 
3  0  0  0  200  0  0  0  120  0  0  0  70 
The total revenue of the 3PW is 219357 dollars. According to solution the linear programming model, the capacity allocation for level 1 is 951 units, the capacity allocation for level 2 is 612 units, and the capacity allocation for level 3 is 437 units. The optimal capacity allocation policies are summarized in Table
According to Table
In this paper, we consider the capacity allocation problem in 3PW company which provides several different level storage services in different price under uncertain market demand. On the revenue management perspective, we propose the mathematical formulations of this problem for both single and multiple periods condition. For the single period situation, as the demand is uncertain, we use three robust methods, absolute robust, deviation robust, and relative robust, to maximize the whole revenue. Based on the analysis of the optimal solution in each situation, we adapt continuous knapsack method to give the corresponding algorithm. Then we use some numerical examples to verify the practical applicability of our method. And we find that the 3PW company managers should provide the maximum possible units of the storage service level with high priority. As the objective function of each method is different, these three methods do not perform the same under the same market scenario. We find that the absolute robust method performs better than the other two methods in most situations. For the multiperiod situation, 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. The major contribution of this paper is that we use robust optimization to deal with the uncertainty of market demand in 3PW industry. In many of existing references of 3PW revenue management, authors consider revenue optimization under deterministic demand or suppose stochastic demand with known distribution such as Poisson process. In this paper, we do not know the distribution of market demand in 3PW industry and linearize the uncertain mathematical programming by different robust methods.
There remain several limitations in our work. First, we consider the demand of each level and each period is independent. Actually, the demands between different levels may affect each other in some conditions and the demand in one period may be affected by its demand in last period. In our future research, we can analyze the affecting factors of demand and characterize the form of demand to improve the match degree between capacity allocation and demand. Secondly, we set the price of each level as exogenous variables in this paper. In the following research, we can combine the dynamic pricing policy and capacity allocation to improve the revenue of 3PW more efficiently.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the Associate Editor and anonymous referees for their constructive comments and suggestions, which have led to significant improvements of the paper. This research is partially supported by National Natural Science Foundation of China (nos. 71131004, 71471071) and Humanities and Social Sciences Foundation of Chinese Ministry of Education (no. 12YJC630149).