Optimal price schedules for storage of inbound containers

Abstract

This paper discusses a method of determining the optimal price schedule for storing inbound containers in a container yard. The price schedule in this study is characterized by the free-time-limit during which a container can be stored without any charge, and by the storage price per unit time for the storage beyond the free-time-limit. The profit or cost models for optimal price schedule are developed from the viewpoint of a public terminal operator as well as a private terminal operator. The probability distribution of delivery times is expressed by a continuous probability function. Various characteristics of the optimal solution are analyzed by numerical experiments.

Introduction

Storage space is an important resource in container terminals. Efficient utilization of the storage space significantly improves the productivity and the profitability of the container terminal. Fig. 1 illustrates a container yard with two blocks with nine and six bays, respectively. Each bay consists of six stacks each of which has four slots. The container yard in container terminals plays the role of a temporary storage space for inbound and outbound containers and thus it is not usually used for long-term storage. Long-term storage of containers would create high congestion in container yards and thus low productivity in handling operations. To encourage customers not to store containers for a long time, terminal operators usually charge a storage fee – which is higher than the storage fee in an outside container depot – for storing a container beyond a pre-specified free-of-charge time-limit (free-time-limit). Based on the price structure of the storage fee, customers determine how long to store their containers in the container yard in order to minimize their own cost.

This paper addresses the pricing problem for storage of only inbound containers. Fig. 2 illustrates a theoretical distribution of delivery times of inbound containers after unloading. There is a tendency that most containers are delivered within several days after unloading and the number decreases rapidly afterward. Watanabe (2001) suggested the exponential distribution as the distribution for delivery times.

There are two possible storage and delivery schedules for unloaded containers. Unloaded containers may be stacked at container yard (CY) in the port container terminal (PCT) temporarily and then transported directly to its consignee. Otherwise, unloaded containers may be stored at CY in the PCT for a free-time-limit, and then moved to an outside container depot, which is operated by a private company and called an off-dock container yard (ODCY), and stacked for another period of time before being delivered to their consignees. When the terminal operator provides a free-time-limit, it is always better to utilize the benefit of the free-time-limit than to move containers directly to an ODCY. The delivery and storage schedule is selected based on the delivery date requested by its customer and the costs for storing and transporting it from the port to its final consignee.

Until recently, little attention has been paid by academic researchers to the operational aspects of storage spaces in container terminals. Related to space-allocation problems in container terminals, Kozan (2000) addressed the problem of allocating storage spaces to various different flows of containers. Taleb-Ibrahimi et al. (1993) analyzed the space-allocation problem of a constant or cyclic space requirement in container terminals. Castilho and Daganzo (1993) addressed the stacking problem of inbound containers in port container terminals. Kim (1997) proposed a formula for estimating the number of rehandles in inbound container yards. Kim and Kim (2002) addressed the space-allocation problem and a method for determining the optimal size of storage space and the number of yard cranes, respectively, for inbound container yards.

The following three papers are directly related to the pricing issue in this paper. Castilho and Daganzo (1991) addressed a pricing problem for temporary storage facilities at ports. They analyzed customer behaviors for different price structures when they maximize their cost saving – resulting from using temporary storage space instead of warehouses at a far distance – minus the sum of moving expenses and holding costs. However, they considered the capacity of the temporary storage facility as a fixed quantity and did not analyze the effect of the storage amount (congestion) on the handling cost of the temporary storage facility.

Holguin-Veras and Jara-Diaz (1999) proposed a method for determining space allocation and prices for priority systems in CYs. They analyzed three different pricing rules: welfare maximization, welfare maximization subject to a breakeven constraint, and profit maximization. For expressing the effect of the congestion in a CY on the handling cost, they assumed a continuous function whose variables are the storage amount and the capacity of a CY. They extended this study to the case with elastic arrivals and capacity constraint (Holguin-Veras and Jara-Diaz, 2006).

This paper also proposes methods for determining prices for the storage of containers in CYs. However, unlike the work of Holguin-Veras and Jara-Diaz, 1999, Holguin-Veras and Jara-Diaz, 2006, this study is based on detailed cost models of handling activities in CYs for customers and terminal operators. Three different objectives are assumed: the maximizing profit of the terminal operator, maximizing the profit of the terminal operator subject to a customer service constraint, and minimizing the total public cost.

This paper assumes a storage charge that is proportional to the length of storage time beyond the free-time-limit. Thus, the price structure can be expressed by the free-time-limit (F) and the storage price per unit time (S) for the storage beyond the free-time-limit. This is a typical price schedule which is used in most container terminals in Korea. Fig. 3 illustrates two real examples of the price schedule being used in terminals in Busan, Korea. The free-time-limit was 4 days in both the cases. The curve for the price schedule is almost linear with respect to the storage duration beyond the free-time-limit. In Terminal A and B, F was 4 days, S was approximately US$14.7 and US$ 18.9, respectively.

When a terminal operator proposes a price schedule of (F, S) to customers, customers will select a storage schedule for their containers based on costs and delivery requirements. To store a container in an ODCY, an additional transportation cost from the PCT to an ODCY and the handling cost at the ODCY are incurred. The cost of a container as a function of the delivery time can be represented by Fig. 4a. Note that the fixed cost for the ODCY results from the additional handling cost at the ODCY and the transportation cost from the PCT to the ODCY. Because the storage cost at the ODCY is lower than that at the PCT, containers whose delivery time is longer than t_S will be moved to an ODCY after staying at the CY of the PCT until the free-time-limit. Thus, from the viewpoint of the terminal operators, the curve for the revenue per container can be drawn as shown in Fig. 4b.

A longer free-time-limit and a lower storage price at the PCT will induce customers to leave their containers at the PCT for a longer period, which results in higher stacks in container yards and higher congestion during delivery and receiving operations. A shorter free-time-limit and a higher storage price result in more customers to move their containers to an ODCY after the free-time-limit despite additional transportation and handling before deliveries to their final consignees.

When a container terminal is owned and operated by a private company, it is natural to determine the storage price and the free-time-limit in a way of maximizing its own profit, which is a possible case when the demand on the port capacity exceeds the supply of the capacity. This is the first administration regime assumed in this paper. However, when the total demand of the port capacity becomes similar to or smaller than the total supply of the capacity, the competition among terminals for securing customers becomes severe. In this case, it is important to consider not only the profit of the terminals but also the service level for customers for deciding the price schedule of storage. In this case, the terminal operator will attempt to guarantee the minimum service level, while maximizing its own profit. The storage price schedule affects the density of container storage in the yard and the density in turn influences the turnaround time of customers’ road trucks which is a key measure of the service level for the customers. This is the second administration regime assumed in this paper. Because a large amount of investment is usually required to construct a container terminal, the container terminal is often constructed, owned, and operated by a government agency as in some developing countries. Because the terminal was constructed by the public investment, it is natural for the terminal operator to pursue the objective of the minimization of the total public cost instead of the maximization of the profit of the terminal. This is the third administration regime that we propose.

Thus, this paper proposes three mathematical models for optimally determining the pricing structure from different viewpoints on the objective function: maximizing the profit of the PCT; maximizing the profit of the PCT with a constraint on the service level for customers; and minimizing the total cost for both the PCT and its customers.

The following section introduces the relationship between the price schedule and the duration of stay of containers. Next, we suggest a mathematical formulation and properties of the optimal solution for the case where a terminal operator tries to maximize his/her profit. The following two sections analyze the cases where the terminal operator tries to maximize his/her own profit subject to a customer service constraint and where a cost model for public terminal operators is used, respectively. The next section compares two price schedules: one without a free-time-limit and the other with a positive free-time-limit. The final section provides concluding remarks.