Complexity

Volume 2017 (2017), Article ID 2813816, 12 pages

https://doi.org/10.1155/2017/2813816

## Optimal Investment Timing and Size of a Logistics Park: A Real Options Perspective

^{1}School of Traffic & Transportation Engineering, Central South University, Changsha, Hunan 410075, China^{2}Key Laboratory of Traffic Safety on Track of Ministry of Education, Central South University, Changsha, Hunan 410075, China^{3}College of Transportation and Logistics, Central South University of Forestry and Technology, Changsha, Hunan 410004, China

Correspondence should be addressed to Shuangyan Li

Received 2 July 2017; Accepted 7 November 2017; Published 5 December 2017

Academic Editor: Eulalia Martínez

Copyright © 2017 Dezhi Zhang 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

This paper uses a real options approach to address optimal timing and size of a logistics park investment with logistics demand volatility. Two important problems are examined: when should an investment be introduced, and what size should it be? A real option model is proposed to explicitly incorporate the effect of government subsidies on logistics park investment. Logistic demand that triggers the threshold for investment in a logistics park project is explored analytically. Comparative static analyses of logistics park investment are also carried out. Our analytical results show that investors will select smaller sized logistics parks and prepone the investment if government subsidies are considered; the real option will postpone the optimal investment timing of logistics parks compared with net present value approach; and logistic demands can significantly affect the optimal investment size and timing of logistics park investment.

#### 1. Introduction

Various logistics centers have recently been established for quickly distributing freight [1]. However, this raises many important issues, such as traffic congestion, air pollution, and high energy consumption. It has been shown that freight transportation contributes to about 5.5% of global greenhouse gas emissions [2]. To respond to these issues efficiently, it has been proposed to combine multiple distribution centers and logistic operators into a logistics park. A logistics park, also known as a “logistics village” in Germany, “distribution park” in Japan, and “logistics platform” in Spain, is a particularly important component of a city logistics network [3]. In general, a city logistics network is composed of different types of logistics nodes (e.g., distribution centres and logistics parks) and logistics links. A distribution center is a logistics node mainly for end customers to provide distribution services, which remains as the following several characteristics, that is, small service radiation range and multispecies, small batch, multibatch, and short cycle, while a logistics park is a comprehensive logistics node with large size, which is commonly located in a strategic area that can easily be accessed from main highways, railways, and airports. Moreover, a logistics park typically has a large space for ample trucks, mass warehousing, office parking, and logistics services such as information transaction, distribution processing, multimodal function, and support service functions. It is also a hub for different transportation modes and local and long-distance traffic [4, 5].

Logistics parks have led to significant environmental effects (e.g., reducing CO_{2} emissions and air pollution) in Germany [6]. Owing to successful logistics park operations in Germany and Japan, there is a growing trend to introduce logistics parks in other developing countries. In China, for example, the number of logistics park projects has continued to increase, from 207 in 2006 to 457 in 2008, 754 in 2012, and 1210 in 2015 according to the fifth survey report conducted by the China Federation of Logistics & Purchasing [7]. However, there are some important problems in the planning and operation of logistics park projects in China. China’s logistics park planning and construction has great blindness due to the lack of systematic theoretical and scientific planning means and assessment. There are many factors that affect the development of logistics parks. An important factor is how to determine the size of logistics park [8]. Moreover, the design of logistics parks has seldom considered the uncertainty of logistic demands and logistic users’ behavioral responses to project investment, leading to a lower usage rate of logistics parks. Investment in logistics parks also involves a high level of risk because of the length of the construction period and the volatility of logistic demands. Hence, the timing and size of logistics parks should be carefully considered.

There are a number of related studies on logistics parks design and investment.

A logistics park is an important logistics facility and can have a significant impact on the corresponding city logistics network. Therefore, the location of logistics parks should be integrated into the entire city logistics network design. Taniguchi et al. [11] proposed a bilevel model to determine the optimal size and location of public logistics terminals and solved the model using queuing theory and nonlinear programming techniques. Nguyen et al. [12] noted that logistics efficiency and cost were related not only to the structure of supply chains but also to the logistics network design and logistics infrastructure. Soysal et al. [13] presented a network model of wagonload traffic that aimed to determine hub location and size considering the total cost and efficiency of the network system. Vieira et al. [14] investigated a hub location problem from the perspective of network design while also considering the transportation cost and travel time and proposed a mixed integer programming formulation. Tang et al. [15] presented an optimization model for the location planning problem of logistics parks with variable capacity. Their model sought to determine optimal locations and allocate customers to the logistics parks using a hybrid heuristic algorithm. Chen et al. [16] addressed the facility layout problem in nonrectangular logistics parks with split lines, proposing corresponding mathematical programming models to obtain competitive solutions for the facility layout problem in a logistics park, solving the optimal model with an adaptive genetic algorithm with scatter search.

A logistics park is an important provider of logistic activities, linking long-haul transportation in different regions of a distributed city [17]. This facility plays an important role in promoting regional economic development, enhancing the level of logistics services, improving intensive land use, relieving pressure from roads and the environment, and increasing employment opportunities. Zacharias and Zhang [18] addressed a two-tiered freight distribution system in a big city, presenting a location-routing model to determine the optimal locations of logistics facilities (primary facilities and secondary facilities) and optimal sizes and routes for different vehicle fleets. Rivera et al. [8] considered the logistics agglomeration arising from logistics parks based on data from a survey conducted in the Zaragoza (Spain) Logistics Cluster and using structural equation modeling, demonstrating that further agglomeration into a logistics park positively impacts collaboration, specifically transportation capacity sharing.

In recent years, green logistics network design has attracted the attention of more practitioners and researchers, given that freight transportation is a major contributor to climate change and various pollution emissions. Green logistics network design focuses on improving logistics service efficiency, decreasing corresponding logistics costs, and reducing externalities while achieving a sustainable balance between economic, environmental, and social objectives (McKinnon et al. [19]. Sadjadi et al. [20] addressed a multicommodity, capacitated intermodal freight transportation network planning problem, which considered greenhouse gas emissions as the primary objective. Rudi et al. [21] assessed the effect of the traditional cost optimization approach to strategic modeling on overall logistics costs and CO_{2} emissions by considering the supply chain structure and different freight vehicle utilization ratios. Rao et al. [22] addressed the selected sustainable location of city logistic centers and proposed a fuzzy, multiattribute group decision-making technique based on a linguistic 2-tuple. Zhang et al. [3] proposed a model to address the design problem of a city logistics network; they considered the interaction between logistics authority and users, as well as the effect of economy size and taxes for CO_{2} emissions. Their results showed that the optimal location and size of logistics parks depend on the realized logistic demand and the size of the economy.

The aforementioned studies on design issues of logistics parks mainly focused on a static and deterministic problem. However, the regional economy, industrial structure, population size, and regional trading pattern will keep changing in the future, leading to logistics demand uncertainty that dynamically fluctuates over time. This is particularly true for some of the fastest-growing cities and the corresponding city logistics service demand pattern. It is therefore necessary to incorporate the dynamics and uncertainty of logistics demand over time into city logistics network design models.

Previous studies tended to use the net present value (NPV) approach. However, the traditional NPV approach does not consider the change of project value in investment; this approach also ignores the impact of postponement, abandonment, or expansion of an investment opportunity on project value in an irreversible and uncertain investment environment [23]. The real option (RO) valuation approach considers the opportunity during an investment period that goes unrecognized in NPV analysis [24]. There is a compelling need to account for time-dependent uncertainty within network designs and project investment due to their irreversible characteristics [25, 26]. Li et al. [10] proposed a RO model to address investment and selection in transit technology given a volatile city population by considering the spatial use equilibrium of a city. Chow et al. [27] investigated the management of a transportation network in an uncertain market, applying the RO approach and dynamic programming to obtain the value of flexibility and to defer and redesign a network. Chow and Regan [28] proposed two models that incorporate RO into network modeling: the first is the network option design problem, which maximizes the expanded NPV of a network investment; the second model decomposes the deferral option of a network investment into individual, interacting links, or project investments. Xiao et al. [29] studied the airport capacity choice problem using a real option model. They point out that if demand uncertainty is low and capacity and reserve costs are relatively high, an airport will not acquire a real option for expansion. Gao and Driouchi [30] examined rail transit infrastructure investment by treating population scale and the attitudes of decision-makers or social planners as sources of risk and ambiguity. They developed an alpha-max-min multiple-priors expected utility framework to solve for the option value of rail transit investment under knighting uncertainty. Li et al. [31] proposed a real options approach for valuing the investment of a new technology for producing cellulosic biofuels based on construction lead times and uncertain fuel prices. Chow and Sayarshad [32] investigated the reference policies for nonmyopic sequential network design and timing problems, proposing a scalable reference policy value defined from theoretically consistent real option values based on sampled sequences that are estimated using extreme value distributions. Cortazar et al. [33] presented a model to determine the optimal timing for a firm to invest in environmental technologies and analyzed the key parameters affecting the optimal decisions. Bockman et al. [34] considered investment timing and optimal capacity choice for small hydropower projects using a real options-based method with continuous scaling; they found a unique price limit for initiating the project. Boomsma et al. [35] adopted a real options approach to analyze investment timing and capacity choice for renewable energy projects under different support schemes, aiming to examine investment behavior under the most extensively employed support schemes, namely, feed-in tariffs and renewable energy certificate trading. Welling [36] investigated the size and timing of a renewable electricity investment, analyzing the effects of governmental support on the optimal capacity of a renewable electricity generating system. Li and Cai [37] applied a real option model to address the impacts of government incentives on the private investment behaviors with uncertain demand, including the choices of investment timing, capacity, and price.

To the best of our knowledge, existing studies that integrate the economic scales of logistics parks construction and operator and government subsidies based on a real option method are still scarce. This study aims to address the above knowledge gap by attaining two objectives. The first objective is to prove conjectures on determining the optimal investment timing and size of a logistics park project simultaneously under logistics demand uncertainty. The second objective is to address the impacts of government subsidies on the logistics parks investment timing and size.

The main contributions of this paper are as follows.

First, a real option model is proposed that incorporates economics of scale on logistics park projects and government investment subsidies to determine the optimal investment timing and size simultaneously. The model captures the effect of logistic demand volatility and economies of scale of a logistics park on investment timing and size decisions. Second, the thresholds of logistics demand that trigger investments in a logistics park with different sizes are explored. Third, we conduct comparative statistical analyses of investments in a logistics park. Sensitivity analyses are also conducted to assess the effect of key model parameters, such as logistics volatility, duration of construction, and discount rate. We also estimate loss in project value as a result of the adoption of the NPV method instead of the RO.

The rest of the paper is organized as follows. Section 2 presents the formulation of the model and the solution algorithm. Section 3 presents two numerical examples of the application of the model. Section 4 concludes the paper.

#### 2. Model Formulation

##### 2.1. Cost of a Logistics Park Project

Economies of scale in the construction of logistics parks refers to the phenomenon wherein the average construction cost per unit area of a logistics park decreases as the size of the logistics park increases [38–41]. Economies of scale in the operation of logistics parks refer to the phenomenon wherein the average operating cost per unit of shipment decreases as the size of the logistics park increases because of clustering and synergetic effects among logistics service providers [42, 43]. These effects should be considered in the design of logistics parks, particularly in an era of (capital and land) resource shortages and climate change.

Two kinds of costs are involved in the construction and operation of a logistics park (i.e., annual construction and annual operation costs). Both factors include economies of scale. We denote the average annual construction cost and operator cost as and , respectively. They are described in detail as follows:where is the unit construction cost and captures the effects of the economies of scale for a logistics park.

The annual average operation cost of a logistics park is expressed in the following: where is the variable operator cost relevant to the corresponding size of logistics park .

##### 2.2. Joint Optimal Timing and Size Problem of a Logistics Park

Regarding the logistics park as a firm, the logistic demand is equivalent to the product and the service charge is the product price; therefore, the service charge for unit transfer should be proportional to logistics demand in service and inversely proportional to the size of the logistics park. According to logistics service demand function, we can use the following logistics service supply function to capture the above relationships. where is the external logistics service demand of the market, is the size of the logistics park that the firm intends to build, and is a nonnegative constant that indicates the slope of the liner demand function and is used for the sensitivity of the service charge to the size of logistics park. As seen in (3), the size of the logistics park is inversely proportional to the service charge for unit transfer, and the function form of this assumption implies that the change of the logistic demand will directly react to the reverse demand curve.

The revenue of a logistics park is directly affected by the potential logistics demand. The derived value from a logistics park is not defined because of the uncertainty of the annual potential logistic demand. To describe the change in potential logistic demand over time, we denote as logistic demand of market at time . Given that the logistics demand follows the geometric Brownian motion, which can be captured by [23, 24]where is the growth rate of logistics demand, is the volatility rate of logistic demand, is infinitesimal time increment, and is an increment of a standard Wiener process, for any given period , satisfies the equation , where is a random variable that follows the standard normal distribution of mean 0 and standard deviation 1.

The value of investment is the duration of a project operation, which is equal to the corresponding expected discounted producer surplus. We let denote the net present value of the project, which can be expressed aswhere is the duration of the entire construction project, which is assumed to be a constant in this study, and is the coefficient of unit government subsidy.

By calculating (5), we can obtain the following equation (the proof procedure is shown in Appendix A).

When project value exceeds investment cost, investment through NPV is the prudent course of action. Investment should be accomplished immediately when . Based on (5), the logistics demand in NPV approach is obtained as

However, the logistics project faces many uncertainties caused by the external logistics demand, price of logistics service land price, operator cost, and duration of the investment process. In this study, we focus on the volatility in logistics demand and use RO method to analyze the optimal investment timing and size of a logistics park project. Unlike the NPV approach, the options value accounts for the uncertainty of investment, which leads to the uncertainty of the project value and loss of opportunity for future repeat investment. This method considers the opportunity cost in the project, namely, the value of investment opportunities or the options value. This study compares the logistics demand obtained by RO and NPV.

According to RO theory [24], the expected value of an investment opportunity is equal to its expected rate of capital appreciation over the short time . This relationship is represented by the Bellman equation for option value and can be expressed as

Equation (8) actually represents the Bellman equation for the option value . Applying Ito’s lemma, we can determine the logistic demand threshold and the option value function , which are shown as in(9) and (10), respectively. The proof procedure is shown in Appendix B.

When , the best strategy is to defer the project investment due to smaller logistics demand. Waiting for the future optimal investment time is a sensible strategy, and the value of investment is the option value at that time.

When , investing immediately is the best action, and the value of making the investment is NPV.

There are several methods for determining the optimal size of logistics park. The first method maximizes the intrinsic value of the project. The second method which selects the optimal size of the logistics park by maximizing the option value determines the capital density of investment. The third method is to determine the optimal size and the optimal output based on the corresponding maximal net present value (NPV). The first method may be better because the option value is equal to the intrinsic value at the moment of investment, while the value of the second choice is the option value before investment. The net present value is more suitable for the deterministic environment, but it ignores the opportunity cost in the process of investment in an uncertain environment. Therefore, this paper adopted the first method to determine the optimal size of a logistics park.

Let be the intrinsic value at the state of implementing the option. That is,

The optimal size of a logistics park must satisfy the first-order condition of (12) in order to maximize the intrinsic value:

Calculating (9) and (12), we can obtain the optimal investment size of a logistics park and the investment demand threshold as the follows:

As we know, the optimal investment size should be positive. In other words, the parameters in (13) and (14) need to satisfy the following conditions:

Solving these two inequalities, we know that the parameters need to meet the conditions and or meet the conditions and .

Given the model parameters, we can obtain the relationship between logistics demand under NPV and RO approaches from (7) and (9), as follows:

Equation (16) implies that the logistic demand under the RO approach is always larger than that under the NPV approach because the RO approach incorporates the value of flexibility through the option to wait and defer investment.

##### 2.3. Static Analysis of Threshold of Logistics Demand

The effects of key model parameters, such as interest rate , construction period , logistic demand , government subsidy , sensitive coefficient , and the change rate of demand are investigated. In accordance with (13), the following inequalities will hold, , , , and . Similarly, we can find that and ; however, the symbols of and are difficult to determine. The proof procedure is shown in Appendix C.

The signs of and with regard to the growth rate of logistics demand and discount rate (i.e., , , and ) are also difficult to determine analytically. Therefore, a simulation method is used to ascertain their relations in the later numerical example.

#### 3. Case Study

##### 3.1. Data and Parameter Settings

In this section, two test examples are used to illustrate the application model and the contribution of this study. The first example is designed to reveal the effect on the investment of a logistics park with and without consideration of government subsidies. The effect of the key model parameters (e.g., the growth rate of the demand, discount rate, the construction of the project, and the demand volatility) on the trigger logistics demand threshold and the optimal investment size are also investigated. The second example illustrates the real application of the proposed model for Jinxia logistics park project located in Changsha, Hunan Province. We can obtain the investment timing and the optimal size of the logistics park to confirm the validity of the model. Table 1 provides statistics data of logistics service demand of Changsha City from 2000 to 2014. Table 2 shows the other input parameters.