A revenue management model for sea cargo

A REVENUE MANAGEMENT MODEL FOR SEA CARGO SIM MONG SOON NATIONAL UNIVERSITY OF SINGAPORE 2005 A REVENUE MANAGEMENT MODEL FOR SEA CARGO SIM MONG SOON (B.Eng.(Hons.),NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Contents Contents i Acknowledgment v Abstract vi Nomenclature viii List of Tables xii List of Figures xiii Introduction 1.1 Organization of the thesis . . . . . . . . . . . . . . . . . . . . Literature Review 2.1 What is revenue management? . . . . . . . . . . . . . . . . . . 2.2 Its applications and successful stories . . . . . . . . . . . . . . 2.3 Various applications of revenue management . . . . . . . . . . i 2.3.1 Airlines . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Hotels . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.3 Cargo transportation industry . . . . . . . . . . . . . . 12 2.3.4 Restaurant, internet service provider and manufacturing plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 The single leg seat inventory control problem . . . . . . . . . . 18 2.5 Extensions of the single leg seat inventory control problem . . 22 The Proposed Sea Cargo Revenue Management Model 32 3.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.2 The mathematical model . . . . . . . . . . . . . . . . . 40 3.3 Remarks on the solution techniques . . . . . . . . . . . . . . . 42 Some Structural Properties of the Sea Cargo Revenue Management Model 44 t,k 4.1 The optimal βAC . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Threshold policy . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2.1 Structural Conditions . . . . . . . . . . . . . . . . . . . 49 4.2.2 The implication of the structural conditions . . . . . . 51 ii 4.2.3 The existence of structural conditions . . . . . . . . . . 58 4.2.4 The optimality of the threshold policy . . . . . . . . . 73 4.2.5 The monotonic property of the threshold values . . . . 77 4.2.6 The stationary threshold policy . . . . . . . . . . . . . 79 Implementation of the Stationary Threshold Policy 5.1 The stationary threshold problem . . . . . . . . . . . . . . . . 5.1.1 81 82 A mixed integer programming reformulation . . . . . . 85 5.2 The proposed perturbation approach . . . . . . . . . . . . . . 87 5.2.1 The general idea . . . . . . . . . . . . . . . . . . . . . 88 5.2.2 How the idea is applied to the problem? . . . . . . . . 89 5.2.3 When should δt,k and δt,k be changed? . . . . . . . . . 92 5.2.4 The general algorithm . . . . . . . . . . . . . . . . . . 100 5.2.5 The shadow price approximation . . . . . . . . . . . . 102 5.3 Solving the stationary threshold problem by meta-heuristics . 105 5.3.1 Genetic algorithms . . . . . . . . . . . . . . . . . . . . 106 5.3.2 Simulated annealing . . . . . . . . . . . . . . . . . . . 111 Numerical Experiments 115 6.1 Some issues regarding the threshold policy . . . . . . . . . . . iii 116 6.1.1 The average performance of the stationary threshold policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.1.2 The stationary and the non-stationary threshold policy 122 6.1.3 Some insights on implementing the stationary threshold policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2 How good is the perturbation approach? . . . . . . . . . . . . 125 6.3 Comparison on the average performance of the methods discussed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Conclusion and Future Direction 136 Bibliography 140 iv Acknowledgment I wish to express my heartfelt gratitude to my thesis supervisors, Associate Professor Lee Loo Hay; Associate Professor Chew Ek Peng and Dr. Peter Lendermann. I will like to thank Christie for her support these five years. In addition, I also like to show my appreciation to my family and friends who have helped me along the way. v Abstract The thesis is divided into two parts. In the first part, we introduce a revenue management model for the ocean carrier. The two classes of order, namely the ad hoc orders and the contractual orders, may arrive at each time instance and each class of order consists of a random amount of containers. A container from the ad hoc orders must be delivered by the first ship leaving the port. On the other hand, if a container from the contractual orders is accepted, the carrier can either deliver it by the first ship leaving the port or postpone the delivery to the next ship on the shipping schedule. Under this situation, we formulate the problem as a stochastic dynamic programming model and prove that a threshold policy exists which gives an optimal solution to the problem. We also show that the threshold policy is non-increasing with respect to the departure date of the ship. In the second part, we introduce a nonlinear optimization problem to determine the stationary threshold policy. We convert the nonlinear optimization problem into a mixed integer linear programming problem and propose a vi heuristic (known as the perturbation approach) to solve the resulted mixedinteger programming problem. In another approach, we apply two metaheuristics (genetic algorithms and simulated annealing) to solve the nonlinear optimization problem directly. From the numerical results, we demonstrate the effectiveness of the threshold policy based on the cases considered. It is also shown that the perturbation approach performs better than some of the methods used to solve the mixedinteger programming problem. vii Nomenclature Rtk (x, y) – The maximal total normalized revenue from decision period t of k th departure period onwards when the remaining capacities of ship and ship are x and y respectively pkt (A) – The probability of getting request in the tth decision period of k th departure period to ship A ad hoc containers pkt (C) – The probability of getting request in the tth decision period of k th departure period to ship C contractual containers At,k The number of ad hoc container arrived at tth decision period of k th departure period Ct,k The number of contractual container arrived at tth decision period of k th departure period t,k βAC – The number of ad hoc containers accepted viii the software and beyond the control of the programmer. 135 Chapter Conclusion and Future Direction In this thesis, we have introduced a stochastic dynamic programming formulation of the sea cargo revenue management model and shown mathematically that a threshold policy is optimal to allocate container on each ship. As the problem is large, it cannot be solved efficiently. Hence, a stationary threshold policy is proposed. To determine the stationary threshold, we next introduce an approximation technique, known as sample average approximation method. In summary, this method generates a sample path of demands which facilitates the determination of the stationary threshold policy that maximizes the average revenue. It turns out that the stationary threshold problem is not trivial due to existence of discrete variables. Several heuristics are used to solve this problem. We demonstrated how the genetic algorithm and the simulated annealing 136 can be applied here to obtain an approximately optimal stationary threshold policy. We have also introduced a novel approach (Perturbation and Shadow Price) to solve the mixed-integer formulation of the Stationary Threshold Problem. There are several approximations used throughout the thesis due to the size of the problem. We note that the original stochastic problem (i.e. dynamic programming formulation) can not be solved efficiently and currently only approximation methods (collectively known as neuro-dynamic programming approach) are available. It will be interesting to compare the result between neuro-dynamic programming approach and our approach. In the numerical experiments conducted, the following conclusions are made: • We demonstrated the improvement gained from the threshold policy under various situations. The stationary threshold policy will affect the average revenue significantly when the capacity of the ship is limited. • Demand variability does have an impact on the average revenue generated by the threshold policy. However, the impact is relatively small. Through standard deviation of demand for both ad hoc and contractual containers increases by times, the drop in average revenue for the zerothreshold policy was less than 3.2% and drop for the threshold policy 137 was lesser than 1.6%. We also observe that increase in the coefficient of variation of ad hoc demand has greater impact on the average revenue than increase in c.v. of contractual demand. • From the cases simulated, the difference in revenue between non-stationary and stationary threshold policies is at most three percent. However, the determination of the stationary threshold policy can be more efficient than that of non-stationary threshold policy. Furthermore, we have also shown that the stationary threshold policy may be determined from a rather small-sized problem. This further motivates the implementation of the stationary threshold policy. • From the numerical experiments, it is also shown that the stationary threshold policy can be obtained from a problem with reasonable number of departure periods. This will further reduce the computational effort to get the stationary threshold policy. • In determining the stationary threshold policy, the performance of the perturbation approach is comparable with other heuristics tested. The problem can be further extended in the following directions: • The carrier can postpone the delivery of contractual containers to multiple departure periods, so long that they reach their destinations on time. 138 Our mathematical formulation of the problem can be readily extended to such a case. However, the resulting problem formulation will be too complicated for mathematical analysis. In that case, the problem can be solved using some of the well-known approximate dynamic programming technique such as reinforcement learning approach. • Our formulation can be extended to consider the case where the maximum remaining capacity of the ship varies with the decision period. • It is noted that the number of scenarios tested in chapter is limited. 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Itinerary control • Allotments Due to these differences, the air cargo revenue management problem is considered far more complex than the airline passenger revenue management problem Saranathan et al (1999) presented an operational model for the air cargo revenue management at United Airlines However, no mathematical model for the air cargo revenue management problem is discussed so far Kleywegt (2002) also... control model is used for short-term allocation of capacity in the ship 13 2.3.4 Restaurant, internet service provider and manufacturing plant Bertsimas and Shioda (2003) developed a revenue management model for a sushi restaurant In a restaurant, the floor manager has to decide where and when to seat each group of arriving customers daily, in order to maximize the revenue Assuming that the total bill... applications and successful stories The airline industry in the United States started applying revenue management in the 1970s after the deregulation of air transportation With revenue management, the airline carriers ensure that there are enough seats reserved for the full-fare customers arriving at a later time and the remaining available seats are opened to the discounted-fare customers, hence maximizing... maximizing their revenue The impact of revenue management is illustrated in Belobaba (1987b): Delta Airlines estimated that selling just one seat per flight at a full fare rather than a discounted fare can add over $50 million to its annual revenue Davis (1994) also added in his article that American Airlines saved $1.4 billion in the period from 1989 to 1992 with the practice of revenue management Following... to act as buffer for demand uncertainty According to Barut and Sridharan (2004), the MTO manufacturing company needs to establish capacity management 16 policies in order to solve the short-run capacity allocation problem when the demand exceeds the capacity The chief capacity management issue is to ensure that the company utilizes the available capacity in the most effective and efficient manner to satisfy... do not naturally have the characteristics required for the application of revenue management However, we see some innovative approaches to modify the problem so that the concept of revenue management can be applied In the next section, we will focus on the airline industry, particularly in 17 the area of single leg seat inventory control This area is related to the Sea Cargo revenue management model, ... customer alienation • severe employee morale problems • a change in reward systems • a need for intensive employee training Kimes (1989) also pointed out that there is a lack of research in the managerial implication of revenue management In order to gain more from revenue management, we need to look at how the revenue management methodology can be integrated into an organization Following this article, Hansen... services and prices for each segment Although the largest air cargo companies have begun to practice this concept, the container shipping industry has yet to apply this Kasilingam (1996) compared the differences between the air cargo revenue management and the airline passenger revenue management He listed four main differences between these two applications: 12 • Uncertain capacity • Three-dimensional capacity... (Shukla and Pestian (1997)) • Internet Service Provider (Nair and Bapna (2001)) • Lodging (Ladany (1976)) • Manufacturing Sector (Barut and Sridharan (2004)) • Restaurant (Bertsimas and Shioda (2003)) It is noted that, due to the different nature, most industries do not take the same approach in applying revenue management to their areas However, these applications share some common characteristics already... covered are the airline industry, the hotel industry, the cargo industry, etc Following that, we will look at the classical single leg seat inventory control problem in detail • Chapter 3 will describe the Sea Cargo Revenue Management Model The mathematical formulation and its assumptions will be given in this 3 chapter • Chapter 4 will present some structural properties of the Sea Cargo Revenue Management . A REVENUE MANAGEMENT MODEL FOR SEA CARGO SIM MONG SOON NATIONAL UNIVERSITY OF SINGAPORE 2005 A REVENUE MANAGEMENT MODEL FOR SEA CARGO SIM MONG SOON (B.Eng.(Hons.),NUS) A THESIS SUBMITTED FOR. that, we will look at the classical single leg seat inventory control problem in detail. • Chapter 3 will describe the Sea Cargo Revenue Management Model. The mathematical formulation and its assumptions. the carriers use some tools to manage their allo cation of capacity. In this thesis, we propose using revenue management (also called yield management) to better manage their capacity. There are