An Optimization Model of the Single-Leg Air Cargo Space Control Based on Markov Decision Process

Abstract

Based on the single-leg air cargo issues, we establish a dynamic programming model to consider the overbooking and space inventory control problem. We analyze the structure of optimal booking policy for every kind of booking requests and show that the optimal booking decision is of threshold type (known as booking limit policy). Our research provides a theoretical support for the air cargo space control.

1. Introduction

For the air cargo carrier, the main purpose of implementing seat inventory control in the actual operation is to avoid cargo space being occupied by too many low-value goods, causing the lack of timely transportation of high-value goods and resulting in the loss of some potential gains. In fact, some customers (FITs or agents) who order the space through telephone or network temporarily cancel the booking or directly do not appear while the aircraft is taking off. It brings a lot of losses to the air cargo carrier. In order to reduce the losses caused by the empty cabin, decision makers tend to overbook. However, overbooking too much may lead to greater economic losses. For this reason, during the air cargo overbooking, it is necessary to consider freight revenue, but at the same time decision makers have to consider the cost of overbooking as well. Above all, the decision makers’ aim is to obtain the maximum profit of freight. Usually, most of the goods are transported by single legs (single routes). For some relatively tight (the market demand is greater than supply) segments (routes), it is more suitable for scientific space control and proper overbooking. The problem can be described in detail as follows: in some tight legs, if there are some booking requests, decision makers will decide whether to accept or not. If it is accepted, decision makers will have to set aside proper space to its customers in accordance with the cargo information. Simply put, in the case of implementing overbooking, the goal is to maximize air cargo revenue by properly accepting customers' booking requests.

2. Related Literature Review

Based on the analysis of the characteristics of the products, Kasilingam [1] establishes the air cargo overbooking decision-making model on the case of space capacity random, discusses the production capacity in discrete case and continuous case, and identifies the optimal overbooking for each situation. Kleywegt and Papastavrou [2] use dynamic random backpack model to discuss air cargo revenue management problem. The above-mentioned method can be used to describe the multidimensional problems, but the model is too difficult to solve easily. Luo et al. [3] establish a two-dimensional air cargo overbooking model and then determine the approximate optimal overbooking level. In this model, both the weight attributes and volume attributes are taken into account, and the objective is to minimize freight costs. Moussawi and Cakanyildrima [4] further study the two-dimensional air cargo overbooking model. The objective is to maximize freight revenue. What they do is different from Luo et al. [3]. Considering the benefits of passengers and cargo, Sandhu and Klabjan [5] establish space inventory control model with the static method. On the assumption of fixed volume, using the method of reducing dimension, they get the model’s approximate solution and gain the maximum revenue. Considering the freight forwarders and the delay of cargo transportation, Chew et al. [6] establish a stochastic dynamic programming model of short-term space inventory control, and the objective is to minimize transportation costs. Amaruchkul et al. [7] consider a single-leg air cargo space control problem. Under the random cargo volume and weight, they build a Markov decision process model and take a heuristic algorithm to analyze the model. Considering protocol sale customers and free sale customers, Levin and Nediak [8] build a space inventory control model by using dynamic programming methods, and the purpose is to make maximum total income rooted in the receipt of the goods. Under uncertain environment, Wang and Kao [9] establish an air cargo overbooking model and solve it by the fuzzy systems approach. Taking into account the two different demands of customers, Modarres and Sharifyazdi [10] build a random capacity space inventory control model and get the optimal decision. By analyzing the expected revenue function in dynamic programming model, combining the randomness of cargo volume and weight, Huang and Chang [11] establish a more efficient algorithm in air cargo revenue management problem. Supposing the cargo volume, weight and the yields of the air cargo are random and the cargo space booking process has no aftereffect, Han et al. [12] set up a single-leg air cargo revenue management space allocation model and take the bid control strategy to determine the goods receiving. Amaruchkul and Lorchirachoonkul [13] study the allocation of air transport capacity for the number of freight forwarders, get the probability distribution of the actual use of the transport capacity through discrete Markov chain, and solve the model by dynamic programming methods.

By contrast, our work is to consider a single-leg air cargo overbooking and space inventory control problem by dynamic programming. We analyze the structure of optimal booking policy for every kind of booking requests and show that the optimal booking decision is of booking limit policy.

3. Model Description

Next we will study the single-leg air cargo space inventory control and overbooking under the conditions of shipping season (the demand is greater than supply). Because of the complexity of the actual situation, in order to abstract practical problems to theoretical issues, we need some basic assumptions.(i)Aircraft total capacity is fixed (cargo size is unchanged);(ii)customers’ booking requests are sufficient, namely, the supply is adequate;(iii)the booking requests are divided into multiple classes;(iv)each class’s arrival is independent;(v)there is at most one booking request at any time;(vi)on the condition that the aircraft is due to take off, whether or not to show up for each booking request is independent;(vii)each booking request’s weight (volume) is identically independently distributed;(viii)during whole booking period, accepted booking requests will not be free to cancel.

Based on the above assumptions, we will establish mathematical model and make decisions on air cargo. In our model, the main parameters and variables are as follows: 𝑡: decision time, 0≤𝑡≤𝑇, where 𝑇 is a booking period; 𝑗: 𝑗th class booking request, 𝑗=1,2,…,𝑛, where 𝑛 is the number of booking classes; 𝑝_𝑗𝑡: the arrival probability of 𝑗th class booking request in time 𝑡; 𝑝_0𝑡=1−∑^𝑛_𝑗=1𝑝_𝑗𝑡: probability of no booking request in time 𝑡; 𝑟_𝑗: unit tariff of 𝑗th class booking request; 𝜌: penalty cost because of overbooking resulting in denying a booking request;  𝑦_𝑗: cumulative number of the accepted 𝑗th class booking request in time 𝑡; 𝑦=∑^𝑛_𝑗=1𝑦_𝑗: cumulative number of all accepted booking requests in time 𝑡;  𝜃: the show-up rate of each accepted booking request near the takeoff; 𝐶_𝑤: the maximum available load of the aircraft;  𝐶_𝑣: the maximum available volume of the aircraft; 𝑊: weight of each booking request in time 𝑡; 𝑉: volume of each booking request in time 𝑡; 𝑤: expected weight of each booking request in time 𝑡, namely, 𝐸(𝑊); 𝑣: expected volume of each booking request in time 𝑡, namely, 𝐸(𝑉); 𝜆: IATA required standard density, value of 0.006m³/kg.

In order to meet up only one request in a period, we will divide booking lead time into a number of small discrete booking periods. 𝑡=0 means that air cargo carrier begins to accept the booking; 𝑡=𝑇 marks the end of booking, then the air cargo carrier should consider the DB (denying booking) problem.

When 0<𝑡<𝑇, 𝑈_𝑡(𝑦) is the maximum total expected profit given state 𝑦 in time 𝑡. Given 𝑡 and 𝑦, the dynamic programming optimality equation can be written as 𝑈_𝑡𝑦=^𝑛_𝑗=1𝑝_𝑗𝑡max𝑈_𝑡+1𝑦+1+𝑟_𝑗max𝑤,𝑣𝜆,𝑈_𝑡+1𝑦+𝑝_0𝑡𝑈_𝑡+1𝑦=𝑈_𝑡+1𝑦+^𝑛_𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦,0,(3.1)^′ (3.1) where Δ𝑈_𝑡(𝑦)=𝑈_𝑡+1(𝑦)−𝑈_𝑡+1(𝑦+1) is the opportunity cost of accepting some class request in time 𝑡. From (3.1)’, it is optimal to accept 𝑗th class booking request in time 𝑡 if 𝑟_𝑗max{𝑤,𝑣/𝜆}−Δ𝑈_𝑡(𝑦)≥0.

Obviously, 𝑈₀(0) is the maximum total expected profit from the beginning of booking to the end of booking. In 𝑡=𝑇, the penalty cost of overbooking is 𝑈_𝑇𝑦=−𝜌𝐸𝑦𝜃𝑊−𝐶_𝑤⁺+𝑦𝜃𝑉−𝐶_𝑣𝜆⁺∀𝑦. (3.2)

The boundary condition (3.2) shows that because of overbooking and relatively fixed capacity, air cargo carrier needs to deny some accepted requests at the end of booking.

4. Structural Properties

In (3.2), there exist continuous random variables 𝑊 and 𝑉. The cumulative distribution function and probability density function are 𝐹_𝑊(⋅), 𝐹_𝑉(⋅) and 𝑓_𝑊(⋅), 𝑓_𝑉(⋅), respectively.

Next, we will prove that 𝑈_𝑡(𝑦) satisfies the following first-order and second-order properties as follows.

Property 1. Δ𝑈_𝑡(𝑦)≥0, namely, 𝑈_𝑡(𝑦) is decreasing in 𝑦.

Proof. When 𝑡=𝑇, 𝑈_𝑇𝑦=−𝜌𝐸𝑦𝜃𝑊−𝐶_𝑤⁺+𝑦𝜃𝑉−𝐶_𝑣𝜆⁺=−𝜌_𝐶^+∞_{𝑤/𝑦𝜃}𝑦𝜃𝑤−𝐶_𝑤𝑓_𝑊(𝑤)𝑑𝑤+_𝐶^+∞_{𝑣/𝑦𝜃}𝑦𝜃𝑣−𝐶_𝑣𝜆𝑓_𝑉(𝑣)𝑑𝑣=−𝜌𝑦𝜃_𝐶^+∞_{𝑤/𝑦𝜃}𝑤𝑓_𝑊(𝑤)𝑑𝑤−𝐶_𝑤𝐶^+∞_{𝑤/𝑦𝜃}𝑓_𝑊(𝑤)+𝑑𝑤𝑦𝜃𝜆_𝐶^+∞_{𝑣/𝑦𝜃}𝑣𝑓_𝑉(𝑣)𝐶𝑑𝑣−^𝑣𝜆_𝐶^+∞_{𝑣/𝑦𝜃}𝑓_𝑉(𝑣),∴𝑑𝑣𝑑𝑈_𝑇𝑦𝑑𝑦=−𝜌𝜃_𝐶^+∞_{𝑤/𝑦𝜃}𝑤𝑓_𝑊(𝑤)1𝑑𝑤+𝜆_𝐶^+∞_{𝑣/𝑦𝜃}𝑣𝑓_𝑉(𝑣)𝑑𝑣≤0. (4.1) Then, 𝑈_𝑇(𝑦) is decreasing in 𝑦.
Suppose that 𝑈_𝑡+1(𝑦) is decreasing in 𝑦. Next, we will show that 𝑈_𝑡(𝑦) is decreasing in 𝑦.
From (3.1), we can get the conclusion easily. This completes the proof.

Property 2. 𝑈_𝑡(𝑦) is concave in 𝑦, that is, for each given 𝑡, Δ𝑈_𝑡(𝑦) is increasing in 𝑦.

Proof. When 𝑡=𝑇, 𝑑²𝑈_𝑇𝑦𝑑𝑦²𝐶=−𝜌𝜃^𝑤𝑓𝑦𝜃^𝑊𝐶_𝑤𝐶𝑦𝜃𝜃^𝑤1𝑦²+1𝜆𝐶_𝑣𝑓𝑦𝜃^𝑊𝐶_𝑣𝐶𝑦𝜃𝜃^𝑣1𝑦²≤0. (4.2)
Then Δ𝑈_𝑇(𝑦) is increasing in 𝑦, namely, 𝑈_𝑇(𝑦) is concave in 𝑦.
Suppose that Δ𝑈_𝑡(𝑦) is increasing in 𝑦. Next, we will show that Δ𝑈_𝑡−1(𝑦) is increasing in 𝑦.
By (3.1)’, we have Δ𝑈_𝑡−1𝑦=Δ𝑈_𝑡𝑦+^𝑛_𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦−,0_𝑛𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦+1,0. (4.3) Let 𝐴=Δ𝑈_𝑡𝑦+^𝑛_𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦,0,𝐵=−^𝑛_𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦+1,0,∴Δ𝑈_𝑡−1𝑦=𝐴+𝐵. (4.4)
For 𝐵, by the above assumption, Δ𝑈_𝑡(𝑦+1) is increasing in 𝑦, then 𝐵 is increasing in 𝑦.
For 𝐴, suppose that there are 𝑚=0,1,…,𝑛 classes booking requests satisfying 𝑟_𝑗max{𝑤,𝑣/𝜆}≥ Δ𝑈_𝑡(𝑦), 𝑛−𝑚 classes booking requests meeting 𝑟_𝑗max{𝑤,𝑣/𝜆}≤Δ𝑈_𝑡(𝑦). We may assume that the first 𝑚 classes in total 𝑛 satisfy 𝑟_𝑗max{𝑤,𝑣/𝜆}≥Δ𝑈_𝑡(𝑦), and the last 𝑛−𝑚 classes satisfy 𝑟_𝑗max{𝑤,𝑣/𝜆}≤Δ𝑈_𝑡(𝑦). Then, 𝐴=Δ𝑈_𝑡𝑦+^𝑚_𝑗=1𝑝_𝑗𝑡𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦=Δ𝑈_𝑡𝑦+^𝑚_𝑗=1𝑝_𝑗𝑡𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦^𝑚_𝑗=1𝑝_𝑗𝑡=1−^𝑚_𝑗=1𝑝_𝑗𝑡Δ𝑈_𝑡𝑦+^𝑚_𝑗=1𝑝_𝑗𝑡𝑟_𝑗max𝑤,𝑣𝜆. (4.5)
By above hypothesis, we know that Δ𝑈_𝑡(𝑦) is increasing in 𝑦, then 𝐴 is increasing in 𝑦.
From above, we can get: Δ𝑈_𝑡−1(𝑦) is increasing in 𝑦. This completes the proof.

Property 3. Δ𝑈_𝑡(𝑦) is decreasing in 𝑡, that is, Δ𝑈_𝑡−1(𝑦)≥Δ𝑈_𝑡(𝑦).

Proof. From (4.3), we have Δ𝑈_𝑡−1𝑦−Δ𝑈_𝑡𝑦=^𝑛_𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦−,0_𝑛𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡=𝑦+1,0_𝑛𝑗=1𝑝_𝑗𝑡max𝑟_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦𝑟,0−max_𝑗max𝑤,𝑣𝜆−Δ𝑈_𝑡𝑦+1,0. (4.6)
By Property 2, we have Δ𝑈_𝑡(𝑦)≤Δ𝑈_𝑡(𝑦+1), then Δ𝑈_𝑡−1(𝑦)≥Δ𝑈_𝑡(𝑦). This completes the proof.

Define 𝑛^∗_𝑗𝑡=max{𝑦|𝑟_𝑗max{𝑤,𝑣/𝜆}≥Δ𝑈_𝑡(𝑦)}, combining Property 3, we have at any time 𝑡, as long as 𝑦≤𝑛^∗_𝑗𝑡, air cargo carrier will accept 𝑗th class booking request. And then, we have the following optimal control policy.

Theorem 4.1. The optimal control policy of the 𝑛 classes booking requests is a booking limit policy: it is optimal to accept 𝑗th class booking request in time 𝑡 if 𝑦≤𝑛^∗_𝑗𝑡, or reject it. Furthermore, the threshold 𝑛^∗_𝑗𝑡 has the following properties:(1)If 𝑟₁≥𝑟₂≥⋯≥𝑟_𝑛, 𝑛^∗_𝑗𝑡 is decreasing in 𝑗, that is, 𝑛^∗_1𝑡≥𝑛^∗_2𝑡≥⋯≥𝑛^∗_𝑛𝑡; (2)𝑛^∗_𝑗𝑡 is increasing in 𝑡, that is, 𝑛^∗_𝑗1≤𝑛^∗_𝑗2≤⋯≤𝑛^∗_𝑗𝑛.

Proof. From Properties 2 and 3, we can have the conclusions easily. This theorem is verified.

The theorem shows that the optimal booking limits are time-dependent and nested in classes. Furthermore, the unit tariff is higher, the optimal booking control policy is more relaxed; the optimal booking limit policy of each class is increasing in time.

5. Conclusions

In this paper, we consider a single-leg air cargo overbooking and space inventory control problem. Based on actual problem, we discrete the booking time into small pieces and establish the dynamic space inventory control model of considering overbooking. After some powerful proofs, we get the optimal booking-limit policy for each class of goods.

Acknowledgments

The authors would like to express their thanks to the referees for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (Grants nos. 71131006, 71172197, and 70771068) and the Central University Fund of Sichuan University (no. skgt201202).