Transportation Research Part E 47 (2011) 204–215

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Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

Economically-efficient port expansion strategies: An optimalcontrol approach

Sander Dekker a,⇑, Robert Verhaeghe b, Bart Wiegmans c

a Transportation & Mobility Division, Grontmij Nederland BV, PO Box 203, 3730 AE De Bilt, The Netherlandsb Department of Civil Engineering and Geosciences, Delft University of Technology, The Netherlandsc OTB Research Institute for the Built Environment, Delft University of Technology, The Netherlands

a r t i c l e i n f o

Article history:Received 23 March 2010Received in revised form 15 June 2010Accepted 1 September 2010

Keywords:Port expansionOptimal controlEconomic efficiencyCost-benefit analysis

1366-5545/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.tre.2010.10.003

⇑ Corresponding author. Tel.: +31 30 2207729; faE-mail address: sander.dekker@grontmij.nl (S. D

a b s t r a c t

This paper proposes an analytical model with a control approach to obtain an optimal portexpansion strategy by balancing investment costs for the port and congestion costs for itsusers. Starting point is the optimality condition that marginal investment costs should bal-ance marginal benefits. Particularly the scale effect in investment costs is considered; theconsequence that the investment will be made in different stages is included in the solu-tion. By relaxing some assumptions in the model, a numerical optimization algorithm isproposed which is applied to show how the approach can be used to deal with a practicalexpansion problem.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Investment in port expansion aims at an improved/expanded cargo-handling process to stimulate economic activity andgrowth. This causes, first of all, lower service costs and times for port users (freight carriers). When these benefits are passedon from the port to society, it leads to the ultimate (public) goal of port investment: namely, ‘‘to increase producers’ surplusof those who originate the exports passing through it, and to increase the consumers’ surplus of those who ultimately con-sume the imports passing through it’’ (Goss, 1990, p. 211). When the highest total benefits are obtained for the lowest invest-ment costs, then the port expansion is economically efficient or optimal.

Developing an optimal port expansion strategy essentially constitutes the establishment of an optimal expansion size atthe appropriate time and place by looking into the future on the basis of forecasts (e.g. Dekker, 2005). Another important andrelated aspect concerns the optimal utilization of the infrastructure being added. For example, the demand for a port’s ser-vices highly depends on the level of traffic congestion, which depends, in turn, on the utilization rate of the port’sinfrastructure.

Social cost-benefit analysis is the standard approach for evaluating the economic efficiency of port expansion strategieswhich are financed with public funds (e.g. Haralambides, 2002). Assessing the welfare effects requires considering a numberof interrelationships including the cost of investment for the government, and lower service costs and times (lower conges-tion costs) for port users. In practice, this involves attempting to trace the optimal port expansion strategy with a limitednumber of expansion alternatives. It is the premise of this paper that an analytic approach systematically covering thoseinterrelationships has the potential to contribute to identifying the expansion alternative with the highest net contributionto economic efficiency. Starting point for the approach is the optimality condition that the marginal investment cost of port

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x: +31 30 2205084.ekker).

S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215 205

expansion should balance its marginal benefit – a standard approach in the literature on economics (e.g. Zhang and Zhang,2003). An optimization approach that supports finding the port expansion strategy that meets this economic-efficiency con-dition is called for.

Two essential questions in deriving a port expansion strategy are: (1) When should the port be expanded? and (2) What isthe size of the expansion? These questions are strongly related and should be addressed jointly. An optimal control approachis developed in the present paper to solve these questions simultaneously. An analytic solution resulting in a numerical solu-tion algorithm is developed and checked against a recent analysis (CPB/NEI/RIVM, 2001a,b) of a major expansion for the Portof Rotterdam. Of particular interest is then how the economic-efficiency condition can be incorporated in the decisions onexpansion. Zhang and Zhang (2003, 2006) did so in their analyses of decisions on airport charges and capacity expansionmade by airports but they did not account for scale effects in investment costs. Furthermore, they used an analytical modelas tool for their analyses instead of a numerical model that can be useful for practical expansion problems.

This paper adds two improvements to the set-up proposed by Zhang and Zhang (2003, 2006). The first improvement isthat it explicitly takes into account scale effects in the investment costs for the port as well as in the congestion costs for itsusers. The second improvement is that this paper proposes – in addition to an analytical model – a numerical optimizationalgorithm that is applied to a practical capacity expansion problem.

The remainder of this paper is divided into five sections. In Section 2, the port expansion problem is formulated. In Section3, we present a review of the literature on the various approaches to derive expansion strategies. In Section 4, we solve theport expansion problem analytically with an optimal control approach, and propose a numerical optimization algorithm. InSection 5, we present and discuss the results of an application of the proposed optimization algorithm to the plan for Rot-terdam port expansion. Section 6 concludes.

2. Formulation of the port expansion problem

In principle, the government should base its transport investment decisions on maximizing the net benefits to society.Balancing investment costs from public sources against reduced congestion costs for the users form major components inthis process. The expansion problem, when facing growing demand, is characterized by relatively large capacity adjustmentsat certain points in time. Extra capacity is called for if demand approaches existing capacity and the associated service timestarts to build up (here: port congestion). Potential economies of scale in investment costs tend to increase the time betweenexpansions (relief interval) and the size of the expansion (e.g. Haralambides, 2002). This evolution of demand and capacityresults in a cyclical pattern of rising and falling utilization rates and associated levels of congestion.

A port’s throughput is determined by many factors including market demand, port capacity, inter-port competition, land-side accessibility, handling equipment utilization and speed, as well as the logistical dynamics of shipping lines (e.g. Luo andGrigalunas, 2003; Tongzon and Heng, 2005). The present paper uses the example of a port that is expanding its area to meetsteadily growing demand. This system, comprising the port and its users, is subject to change. Each year, demand is assumedto increase due to autonomous market growth. As a result, the congestion costs for port users also increase, and these arepassed onto the rest of society. From a social welfare point of view, there is a need for a port expansion strategy that main-tains an economically-efficient capacity in terms of an optimal balance between investment costs and reduced congestioncosts.

As mentioned in Section 1, two major questions are involved in deriving such an expansion strategy: (1) When should theport be expanded? And (2) What is the size of each expansion? In particular the first question – representing the timing-is-sue in capacity planning – is most relevant. If capacity expansion is delayed, users will incur higher congestion costs owing toshortages in port capacity. In a competitive market, other – less congested – ports may then become more attractive, whichleads to less demand, and, in turn, to fewer revenues for the port considered. On the other hand, premature expansion that ispublicly funded may lead to a large opportunity cost for society because some resources have been unnecessarily tied up(excess capacity). An optimal balance between shortages in capacity and excess capacity forms part of the optimal expansionstrategy.

Another relevant issue in the expansion problem concerns the scope of the welfare effects, which in the present set-up arelimited to congestion cost savings for the users of the port. These cost savings are, however, passed onto others in societyrepresenting external and indirect economic effects. Such effects increase social welfare insofar as they create more effi-ciency by network effects and other effects such as the reduction of labour market imperfections (e.g. Zhu et al., 2009).The expected increase of social welfare will not be realized if the associated markets, including those for hinterland trans-portation, are not sufficiently efficient to allow for passing on such effects to consumers through prices. Incorporating theseissues in the decision problem is beyond the scope of our optimization approach but is obviously part of the development ofeconomically-efficient expansion strategies.

3. Review of literature on deriving expansion strategies

The literature on deriving expansion strategies offers particularly partial approaches to determine the values of the deci-sion variables ‘when to expand’ and ‘expansion size’. We distinguish here three types of approaches, namely: (1) cost-benefitanalysis; (2) optimization of an investment cost function; and (3) optimization of a system cost function.

206 S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215

Cost-benefit analysis is often based on standardized methods (e.g. Bristow and Nellthorp, 2000) for computing the ex-pected total net benefits for sets (‘strategies’) of selected values for the decision variables (Zhou, 1995). It has been appliedto many port investment plans: for instance, by the Netherlands Bureau for Economic Policy Analysis (CPB/NEI/RIVM,2001a,b) to assess the economic feasibility of the second seaward expansion of Rotterdam Port (see Appendix A).

In most cost-benefit analyses, only a limited number of potential (expansion) strategies were evaluated. They were usu-ally identified separately in previous studies. There is a need to assess a larger (continuous) set of options which allow thebehaviour of the system to be observed and the true maximum net benefit to be found; this means essentially optimizationof a function.

Manne (1967) wrote a classical introduction to capacity expansion problems based on the optimization of investmentcost functions characterized by economies of scale. He established an approach for steadily growing demand in which therelief interval and expansion size are solved simultaneously by minimizing the present value of the investment costs. Contin-uing with this, Freidenfelds (1981) provided a functional format based on dynamic programming that could be used for anydemand pattern. Haynes and Krmenec (1989), for instance, included this format in an application to infrastructure expan-sion. To take into account the utilization rate and associated congestion costs in the decision to expand, an overall approachbased on the optimization of a system cost function should be introduced.

Shneerson (1981), Jansson and Shneerson (1982) and Jansson (1984) optimized the size of investments in transportationsystems by optimization of a system cost function. They used queuing theory as a framework for the optimization. Paelinckand Paelinck (1998) used this concept to determine the optimal design of container ports. Shneerson (1981), Koh (2001) andAversa et al. (2005) developed optimization models for identifying optimal port development plans. Dekker (2005) deter-mined the optimum expansion size for a node in a transportation network with competition based on traffic assignmentmodelling. As a functional format, he used an optimization model that maximizes user benefits (i.e. transport-efficiencygains) subject to, among other things, a cost recovery constraint. Zhang and Zhang (2003, 2006) established a similar func-tional format for their analyses of airport charges and capacity expansion.

The studies discussed above do not present an analytic solution for the simultaneous determination of the ‘when to ex-pand’ and ‘expansion size’ decisions based on the economic-efficiency condition that the marginal cost of port expansionshould balance its marginal benefit, and/or do not take into account scale effects in investment costs. Dekker (2005) didso by including scale effects and addressing the ‘when’ decision in a scenario but did not internalize this decision in his over-all economic optimization. Internalizing this decision is highly relevant in view of reducing premature public investmentswhich lead to an opportunity cost for society, as well as preventing delayed investments which cause – for growing demand– increasing congestion costs for the users and, consequently, a loss of market share for the port.

For the integrated solution of the capacity expansion decision based on minimization of an overall system cost function,optimal control theory provides an interesting concept. Dorfman (1969) and Seierstad and Sydsaeter (1987) discussed theapplication of the conceptual framework of this theory to investment decision making. Eijgenraam (2006) applied this tothe decision problem of increasing the heights of river dikes by finding a balance between the marginal investment costsand the economic value of the protected polder.

Developing on the concept of Dorfman (1969) and Seierstad and Sydsaeter (1987), and the application of Eijgenraam(2006), an optimal control approach for the port expansion problem is proposed in the present paper. An elaboration is madeof the theoretical concept followed by an application to the expansion of Rotterdam Port and then scoping/testing the resultsby comparing the outcome of a traditional cost-benefit analysis made by the Netherlands Bureau for Economic PolicyAnalysis.

4. Solution for the expansion problem

4.1. Basic equations

The port expansion problem involves a dynamic time-varying system, triggered by changing demand, and steered bydecisions to expand. As a consequence, capacity and the utilization rate, affecting congestion costs for port users, change overtime.

The analytic solution for this expansion problem requires setting up a series of basic equations. The first basic equationdescribes the pattern of demand for a port’s services. A steady linear growth is assumed (but can be relaxed to include non-linear growth later), which has, for instance, been applied by Dekker (2005) for planning a port’s capacity.

Demand at time t, Qt, assumed to be linearly growing in the plan horizon, can be described as follows:

Qt ¼ Q 0 þ c � t if ðt < hÞ ð1aÞ¼ Qh if t ¼ h ð1bÞ

in which: h is the plan horizon; Q0 the demand for a port’s services at time t = 0; Qh the demand for a port’s services at timet = h; and c is the annual demand growth (c > 0).

In port planning practice, demand projections are highly uncertain due to market uncertainty, potential advances in logis-tics and transport technology, and the associated volatility of transport routes.

S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215 207

The second basic equation comprises an approximation of the congestion costs at user level as a function of the utilizationrate (Qt/Kt). The average congestion costs at user level at time t, ACCt, can be typically expressed by the following functionalformat (e.g. Dekker, 2005):

ACCt ¼ ACC0 þ VOT � tff � 1þ a � Q t

Kt

� �b !

ð2Þ

in which: ACC0 is the out-of-pocket cost at user level; VOT the Value-Of-Time; tff the free flow service time (without conges-tion); a the parameter (a > 0); b the scale factor (b > 1); and Kt is the capacity at time t.

Here, ACC0 is assumed to be constant and thus independent from demand and capacity. ACCt can then be interpreted asthe average user cost.

For the VOT, different values need to be applied for different transportation modes. In the application case in the presentstudy, the VOT for container handling activities in the port is set at €156 per TEU/hour (CPB, 2004).

Different values for the parameters a and b can be used for different circumstances: for example, these parameters can bemodified to include the approximate effect of intersection delay in a transport link. If a > 0 and b > 1, an increasing delay costfunction is considered. In the present paper, a and b are set to 0.15 and 4, respectively, representing what is called the ‘Bu-reau of Public Roads’ (BPR) formula that is often used in the transportation sciences (e.g. Ortúzar and Willumsen, 2000). Jour-quin and Limbourg (2006) applied it in a study on multimodal freight transport. It is assumed here that such a functionalformat can also be used to simulate port congestion. It should be realized that port capacity (a port’s maximum cargo han-dling capability) and port congestion (higher service time than can ideally be achieved by the port (free flow service time)due to demand approaching port capacity) are rather complex concepts. This is because a port consists of different stages orlinks, and as a whole is sensitive to disturbances in one or more of these links.

The overall (or total) congestion costs at time t, TCCt, which represents the sum of the congestion costs over all users, is bydefinition equal to the product of demand and congestion costs at user level. The combination of Eqs. (1) and (2) then leads to:

TCCt ¼ Q t � ACCt ¼ Q t � ACC0 þ VOT � tff � 1þ a � Q t

Kt

� �b ! !

ð3Þ

In the course of time, capacity is expanded with Ke,i at the times Ti. The development of capacity over time can then bedescribed as:

K�T1¼ K0 ð4Þ

_Kt ¼ 0 if t–Ti ð5aÞ¼ KþTi

� K�Ti¼ Ke;i > 0 if t ¼ Ti ð5bÞ

Kh not fixed ð6Þ

in which: K0 is the capacity at time t = 0; K� the capacity just before expansion; K+ the capacity directly after expansion; Ti

the time of investment Ii; Ke,i the capacity expansion i associated with investment Ii; and Kh is the capacity at the end of theplan horizon.

Eq. (5a) implies that capacity does not change between two successive investments. Because capacity K is not continuousin time, its time-derivative is also not continuous (see Eqs. (5a) and (5b)). The end-capacity Kh at time h is not fixed; in otherwords: Kh is not set in advance to a certain value.

The investment costs of infrastructure expansion are often characterized by economies of scale (e.g. De Neufville, 1990).The investment costs Ii, associated with investment i, are then expressed by:

IiðKe;iÞ ¼ I� � Kke;i if Ke;i > 0 ð7Þ

¼ 0 if Ke;i ¼ 0 ð8Þ

in which: I� is the parameter and k is the scale factor (0 < k < 1).

Here, the investment costs function Ii is continuous assuming that expansion is available at any size. In other words: thereare no indivisibilities. In practice, it should however be realized that standardization of components, site irregularities, andoperational reasons – to name all but a few – may preclude continuous capacity expansion but the assumption helps to im-prove analytic tractability.

The value for the scale factor k determines the scale effect of the investment costs. In the present paper, based on analysisof the existing plan for Rotterdam port expansion, k is estimated at 0.6 representing a positive scale effect. If k > 1, the scaleeffect is negative; an expansion strategy in which the investment is made phased with relatively large capacity adjustmentsis less attractive then due to increasing unit investment costs.

From Eq. (3), what is called a connection equation can be derived. This comprises the difference between overall conges-tion costs just before and directly after an expansion:

TCCðKþTiÞ � TCCðK�Ti

Þ ¼ A � ðQTiÞbþ1 � ðKþTi

Þ�b � ðK�TiÞ�b

n oð9Þ

with: A ¼ VOT � tff � a.

208 S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215

4.2. Objective function

Using Eqs. (3) and (7), the following objective function can be derived representing the present value PV of all future sys-tem costs, comprising the sum of overall congestion costs (TCC) and investment costs (I):

minKe;i

PV ¼Z h

0TCCt � expð�dtÞdt þ

Xi

Ii � expð�dTiÞ ð10Þ

in which: PV is the present value of all future system costs; and d is the discount factor.The first term represents the present value of the overall congestion costs (for all present and future users) over plan hori-

zon h. The stream of discounted investment costs is represented by the second term.

4.3. Analytic solution based on optimal control theory

For the derivation of the analytic solution for the expansion problem based on optimal control theory, Pontryagin’s Min-imum Principle is used (for an application to economic problems, see, e.g., Dorfman, 1969; Seierstad and Sydsaeter, 1987).The objective function in Eq. (10) and its basic equations are interpreted in an optimal control theory format. From a math-ematical point of view, this consists of finding that Ke,i which satisfies the boundary conditions. These boundary conditionsare derived from Hamiltonians, which represent a formal statement of the necessary conditions for Pontryagin’s MinimumPrinciple. The solution comprises approaches for ‘expansion size’ and ‘when’ should the port be expanded?

4.3.1. Necessary conditionsThe objective function in Eq. (10) is transformed from a stream of congestion costs and (interdependent) investments into

a more approximate representation: the aforementioned Hamiltonians, which enable the objective function (Eq. (10)) to bedecoupled over time into a series of: (1) periods after re-investment (the first term of Eq. (10)); and (2) times of re-invest-ment (the second term of Eq. (10)). The underlying assumption is therefore that the principle of linear superposition can beused, which means that the successive investments and their effect on overall congestion costs are treated as mutually inde-pendent, and can be summed linearly.

Below, the time t and a re-investment i are indicated with subscripts. For a period after re-investment, it holds that:

Ham ¼ TCCt � expð�dtÞ ð11Þ

This equation expresses that, for a period after re-investment, system costs only change as a result of due to decreasing over-all congestion costs.

For the times of re-investment Ti, and the period of time after these times, indicated by a ‘+’, it holds that:

IHam ¼ Ii � expð�dTiÞ þWþi � Ke;i ð12Þ

in which: W is the added variable.This equation expresses that, at the time of re-investment, the system costs are affected by the occurrence of investment

costs (first term), as well as by a change of overall congestion costs after the investment due to adding a capacity incrementKe,i (second term). W can be interpreted as the additional (marginal) overall congestion costs due to adding a capacityincrement.

The Hamiltonians are used to derive the necessary conditions for optimal control. Following Dorfman (1969) and Seiers-tad and Sydsaeter (1987), and Eijgenraam (2006), for a period after re-investment, the following necessary condition applies:

@

@KHam ¼ _Wt for Ti < t < h ð13Þ

This necessary condition expresses that a marginal decrease of the overall congestion costs due to adding a capacity incre-ment should balance the annual growth of the additional (marginal) overall congestion costs (due to demand growth) at anytime after re-investment, since Kh is not fixed (see, in particular, Eijgenraam, 2006).

Further the following connection condition holds between the period of time directly after a time of re-investment andthe period following that time:

limt#Ti

Wt ¼ Wþi ð14Þ

Using the First Year Rate of Return (FYRR) criterion for the optimal timing of investment projects, the following expres-sion for a re-investment i should apply:

<¼ 0 if Ti ¼ 0

TCC KþTi

� �� TCC K�Ti

� �� �� expð�dTiÞ þ d � Ii � exp �dTið Þ ¼ 0 if 0 < Ti < h

>¼ 0 if Ti ¼ h

ð15Þ

S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215 209

Eq. (16) represents the first order linearization for the optimal solution of Eq. (12):

@

@Ke;iIhamð Þ

� �� ðKe � Ke;iÞ 6 0 for Ke;i >¼ 0 ð16Þ

The remainder of this section derives approaches for ‘when’, ‘relief interval’ and ‘expansion size’, which are based on the nec-essary conditions discussed above.

4.3.2. When should the port be expanded?Elaborating Eq. (15) gives:

<¼ 0 if Ti ¼ 0

TCC KþTi

� �� TCC K�Ti

� �þ d � Ii ¼ 0 if 0 < Ti < h

>¼ 0 if Ti ¼ h

ð17Þ

This equation determines the time of re-investment. The social benefits at the time of re-investment (here: reduction ofoverall congestion costs) are at that point equal to the social rate of return in the first year of investment (the FYRR-criterion).

Substituting Eq. (9) in Eq. (17) gives for 0 < Ti < h:

Q Ti¼ �d � Ii

A � ðKþTiÞ�b � ðK�Ti

Þ�bn o

24

35

1bþ1

ð18Þ

The timing Ti for investment i can be determined by substituting Eq. (18) in Eq. (1a).By using Qi = Qi–1 + c�Di, and substituting this expression in Eq. (18), it follows that:

Di ¼1c� �d � Ii

A � KþTi

� ��b� K�Ti

� ��b� �

2664

3775

1bþ1

� 1c� �d � Ii�1

A � fðKþTi�1Þ�b � ðK�Ti�1

Þ�bg

" # 1bþ1

ð19Þ

Di is then the relief interval between the investments i � 1 and i.

4.3.3. Expansion sizeFrom substituting Ke = 0 and Ke = 2Ke,i in Eq. (16), it follows that:

I0i � expð�dTÞ þWþi ¼ 0 ð20Þ

in which I0i represents the first derivative of the investment function Ii in Eq. (7).Integration of Eq. (13) and using Eq. (11) gives:

Wt ¼Z h

t

@

@KTCCs � expð�dsÞ@s ð21Þ

With the use of Eq. (3), it follows that:

@

@KTCCs ¼ �b � A � Qs

Ks

� �bþ1

: ð22Þ

Substituting Eqs. (22) in (21) gives:

Wt ¼ �b � A � Q t

Kt

� �bþ1

�Z h

texp �dsð Þ@s ¼ b � A

d� Q t

Kt

� �bþ1

� expð�dhÞ � expð�dtÞð Þ ð23Þ

Note that Qt and Kt have been placed outside the integration operator; Qt = QTi and Kt ¼ KþTi are introduced below.From Eq. (14) it follows that Wt is continuous. From this it follows that Eq. (23) should also apply to t = Ti for which Eq.

(20) holds:

I0i � expð�dTiÞ ¼ �b � A

d�

Q Ti

KþTi

!bþ1

� expð�dhÞ � expð�dTiÞð Þ ð24Þ

This can be rewritten as:

I0i ¼ �b � A

d�

Q Ti

KþTi

!bþ1

� expð�dðh� TiÞÞ � 1ð Þ ð25Þ

Eq. (26) represents the condition for economically-efficient expansion: the marginal investment costs should be equal to themarginal decrease of the overall congestion costs.

210 S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215

Rewriting Eq. (25) by substituting the first derivative of Eq. (7) leads to the function that gives the optimal expansion size:

k � I� � Kk�1e;i ¼ �

b � Ad�

Q Ti

KþTi

!bþ1

� expð�dðh� TiÞÞ � 1ð Þ ð26Þ

4.4. Numerical optimization algorithm

To determine the simultaneous solution for the two questions/decision variables ‘when to expand’ and ‘expansion size’for a specific situation, the system of Eqs. (18), (19), and (26) can be used with KþTi ¼ K�Ti þ Ke;i.

The above derivation provided the analytic solution which an optimal solution should satisfy. An iterative process canthen be used to derive the solution which satisfies these conditions.

The proposed optimization algorithm comprises two components. With the first component, the optimal first expansioncan be determined iteratively. Building upon a given K0, this component consists of the following six steps:

1. take a capacity expansion Ke,1;2. calculate investment cost I1 with Eq. (7);3. calculate demand QTi with Eq. (18);4. calculate the timing of the first investment, T1, by using (QT1–Q0)/c;5. calculate the right term of Eq. (26);6. calculate the difference between I01 and the result of Step 5.

This should be repeated until Step 6 is equal to 0. Note that Eq. (19) is not used here; there is no preceding investment Ii–1

that can be used for Eq. (19).With the second component of the optimization algorithm, the second and further expansion phases can be determined

iteratively. Building upon the result of the first optimization component, the second component consists of the followingseven steps:

1. take a capacity expansion Ke,i;2. calculate investment cost Ii with Eq. (7);3. calculate demand QTi with Eq. (18);4. calculate relief interval Di with Eq. (19);5. calculate the timing of the investment, Ti, by using Ti = Ti–1 + Di;6. calculate the right term of Eq. (26);7. calculate the difference between I0i and the result of Step 6.

This should be repeated until Step 7 is equal to 0. Note that Eq. (19), in contrast to the first component of the optimizationalgorithm, is used here; for each iteration, there is a preceding investment Ii–1 that can be used for Eq. (19).

The two components of the optimization algorithm have been implemented in a spreadsheet; the optimization makes useof the built-in solver.

Step 6 of the first component of the algorithm and Step 7 of the second component represent an objective function. Step 1in both components therefore represents the control (or decision) variable that determines the outcome of the objectivefunction.

5. Results and discussion

5.1. Results of the optimization algorithm

The proposed numerical optimization algorithm has been demonstrated with an application to the plan for RotterdamPort expansion. The existing expansion plan, as assessed by the Netherlands Bureau for Economic Policy Analysis (see Appen-dix A), is used to situate our optimization approach. Information on demand and supply in Rotterdam Port provided input toset the situation in the base year (2000). The expansion strategy as proposed in the existing plan is used for setting the valuesof the major input variables for our optimization algorithm. The algorithm has been implemented in a spreadsheet. The val-ues of the input variables, which are in line with literature sources, are presented in Appendix B.

A summary of the results of the optimization algorithm applied to the Rotterdam Port expansion plans is presented inTable 1. The second and third columns represent, for each expansion stage, answers to the questions: (1) When shouldthe port be expanded? And (2) What is the size of the expansion? Furthermore, the utilization just before expansion andthe investment costs for each expansion stage are indicated in the fourth and fifth column, respectively.

It can be observed from the results for the first five expansion stages that – for the given input variables – the expansionsize, and thus also the investment costs, increase in time. From analysis of a set of container ports, De Neufville and Tsunok-awa (1981) conclude that there are strong gains in efficiency when the total size of the port increases. This means that the

Table 1Summary of the results of the optimization algorithm. Source: by authors.

Stage When (year) Size (ha) Utilization rate justbefore expansion (%)

Investment cost(in bln. euros, in 2000 prices)

1 2002 78 69 0.52 2006 95 67 0.63 2015 115 66 0.64 2029 140 65 0.75 2050 170 64 0.8

Total 598 3.2

S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215 211

unit cost per container decreases for an increasing port area. This can be interpreted as a cost function which leads todecreasing unit costs for increasing capacity. In the present analysis, this has been approximated with an expansion decisionincorporating decreasing congestion cost per unit (TEU) for expanding total capacity. If everything else stays the same (e.g.annual demand growth), such decision making obviously results in increasing expansion sizes in order to keep the marginalbenefit (here: marginal decrease of the overall congestion costs) in balance with its marginal investment cost.

Based on the above-presented results of the optimization approach, we also estimated the welfare effects which are herelimited to congestion cost savings for the users of the port. Using the reference development as discussed in Appendix A, thewelfare effects of the expansion strategy resulting from the optimization algorithm are estimated at 5.4 billion euros (pres-ent value in 2000).

5.2. Sensitivity analysis

We conducted a sensitivity analysis of the expansion strategy resulting from our optimization algorithm in terms of tim-ing (‘when’) and size, to changes in the values of major input variables (a, b, c, k, I�, tff and VOT). The detailed results of thesensitivity analysis are presented in Appendix C. It appears that the decision on expansion size is not affected by the (hereassumed to be linear and positive) annual demand growth (c). To include also non-linear demand patterns, the optimizationalgorithm is recommended to be modified. The sensitivity analysis further pinpoints the relevance of (1) incorporating scaleeffects (in congestion as well as in investment costs) in optimization approaches for deciding upon expansion plans and (2)selecting proper values for the scale factors, since timing and size appear to be highly sensitive for changing values of thescale factors (b and k). Relaxing the assumption that the investment cost function is continuous will affect the results ifthe minimum required expansion size exceeds the smallest expansion size resulting from our optimization algorithm; anadditional ‘non-continuity’ condition should then be included in the proposed approach.

5.3. Comparison with the existing strategy for Rotterdam port expansion

In order to situate the optimization approach, the approach and results of the application case have been compared withthose of the existing plan for Rotterdam Port expansion. The details of this comparison can be found in Appendix D. It ap-pears that the timings resulting from the optimization approach are earlier – for the first two stages even 11 years – and theexpansion sizes are smaller. But there are also similarities: in both approaches only the timing depends on the demand sce-nario used, and the total expansion size appears to be equal for both approaches (600 ha). Despite differences in terms ofmethodology, scope and estimated values of welfare effects, and optimal expansion decisions, it appears there are also majorsimilarities between the approach followed for the existing plan for Rotterdam Port expansion and our optimization ap-proach, in particular similar total expansion sizes after five expansion stages.

6. Conclusions

The port expansion problem involves a dynamic time-varying system, triggered by changing demand, and steered bydecisions to expand. As a consequence, capacity and the utilization rate, which affect congestion costs for port users, changeover time. Two essential questions in deriving an optimal port expansion strategy are: (1) When should the port be ex-panded? and (2) What is the size of the expansion?

Starting point in our analysis of this expansion problem has been the optimality condition that the marginal investmentcost of port expansion should balance its marginal benefit – a standard approach in the literature on economics. Zhang andZhang (2003, 2006) did so but did not account for scale effects in investment costs. Furthermore, they used an analyticalmodel as tool for their analyses instead of a numerical model that can be useful for practical expansion problems. Our papertakes explicitly into account scale effects in the investment costs for the port as well as in the congestion costs for its users,and proposes – in addition to an analytical model – a numerical optimization algorithm that has been applied to a practicalcapacity expansion problem. Further review of approaches for deriving expansion strategies showed that for the simulta-neous solution of the questions ‘when’ and ‘size’, optimal control theory provides an interesting approach.

212 S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215

Our optimal control approach consists of minimizing the sum of investment costs for the port and the congestion costs forits users (system costs), while taking into account the utilization rate and scale effects. This has been employed to develop aneconomically-efficient port expansion strategy for container handling activities in Rotterdam port. It is notable that the re-sults show that the expansion size, and thus also the associated investment costs, increases in time. This is in accordancewith a similar observation in the literature; apparently, such decision making leads to increasing expansion sizes in orderto keep the marginal benefit in balance with its marginal cost.

We conducted a sensitivity analysis of the expansion strategy resulting from our optimization algorithm in terms of tim-ing (‘when’) and size, with respect to the values of major input variables. It appears that the decision on expansion size is notaffected by the annual demand growth (here assumed to be linear and positive). To be able to include also non-linear de-mand patterns, the optimization algorithm is recommended to be modified. The sensitivity analysis further pinpoints therelevance of (1) incorporating scale effects (in both congestion and investment costs) in optimization approaches for decid-ing on expansion plans and (2) selecting proper values for the scale factors in the congestion and investment functions, sincetiming and size appear to be highly sensitive for changing values of the scale factors. Relaxing the assumption that theinvestment cost function is continuous requires including an additional ‘non-continuity’ condition in the proposed approach.

In order to situate the optimization approach, the approach and results of the case study have been compared with thoseof the existing plan for Rotterdam Port expansion. Despite differences in terms of methodology, scope and estimated valuesof welfare effects, and optimal expansion decisions - including whether or not accounting for scale effects -, it appears thereare also major similarities in particular similar total expansion sizes after five expansion stages.

Extension of the optimization algorithm, such as adding an inter-port competition model (relevant in view of the optimaltiming-issue of the port expansion problem), and widening the scope of the welfare effects (in addition to congestion effectsfor the users, also profit for the port and welfare effects that are passed onto others in society) makes a contribution to largerstudies such as the existing analysis for Rotterdam Port expansion. Furthermore, the optimization algorithm could also beapplied to other transport/capacity problems to test the balancing of investment and congestion costs, while taking into ac-count the utilization rate and scale effects.

Acknowledgements

We are grateful to an anonymous referee and the Editor-in-Chief (Wayne Talley) for their constructive comments, whichhave substantially improved the paper. Sander Dekker also thanks Grontmij Nederland for encouraging and funding furtherstudy on port development issues.

Appendix A. Existing plan for Rotterdam Port expansion

Somewhere in the future, the use of the existing Rotterdam Port area will reach its capacity boundary. The precise mo-ment will depend on the future growth of demand. For the preparation of the existing plan for Rotterdam Port expansion, theNetherlands Bureau for Economic Policy Analysis used demand projections for different commercial sectors in the port.These projections were based on three scenarios for Dutch and global economic development until 2035 (here referred toas Scenarios I, II and III). The resulting demand projections for the container sector are presented in Table A1.

The actual need for extra port area depends strongly on the efficiency in the use of space that can be realized in the future.The Netherlands Bureau for Economic Policy Analysis considered an efficiency figure of about 29,000 TEUs/hectare/yearlikely for the container sector on the reclaimed land of Maasvlakte 2 (CPB/NEI/RIVM, 2001a).

In the analysis of supply, capacity at t = 0 (i.e. K0 in Eq. (4)) is the starting point for the decision to improve port capacity.For the container sector in Rotterdam, The Netherlands Bureau for Economic Policy Analysis estimated K0 in the existing planat 10.5 million TEU/year.

The selected supply option to meet projected future demand comprises seaward expansion of the port by land reclama-tion: the Maasvlakte 2-project. The plan for this reclamation, which is presently being constructed, comprises a total area of1000 ha. Sixty percentage of the total area (600 ha) is reserved for container handling activities and 40% for other activitiesincluding storage and chemical industries. The expansion strategy as recommended by The Netherlands Bureau for EconomicPolicy Analysis is presented in Table A2 with the year 2000 as the base year.

The Netherlands Bureau for Economic Policy Analysis determined this strategy on the basis of a study on the match be-tween demand for space and its supply over time, and by considering economically-efficient construction strategies. The

Table A1Container demand projections for Rotterdam port. Source: CPB/NEI/RIVM, 2001a.

Scenarios 2000 (mln. TEUs) 2020 (mln. TEUs) 2035 (mln. TEUs) Average growth(mln. TEUs/year)

I 6.3 11.0 13.4 0.197II 6.3 14.1 20.8 0.403III 6.3 16.4 27.6 0.592

Table A2Expansion strategy for Rotterdam Port as recommended in the cost-benefit analysis. Source: CPB/NEI/RIVM, 2001a.

Stage When (year) Size (ha) Of which for thecontainer sector (ha)

Investment cost(in bln. euros, in 2000 prices)

I II III

1 – 2013 2010 150 60 0.62 – 2017 2012 185 120 0.23 – 2023 2017 185 120 0.34 – 2031 2022 150 120 0.25 – – 2027 330 180 0.5

Total 1000 600 1.8

S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215 213

resulting expansion strategy was evaluated for each of the three demand Scenarios (I, II and III) via an assessment of welfareeffects, which included direct, as well as indirect and external, effects.

It can be observed from Table A2 that only the timing (‘when’) depends on the scenario used. For Scenario I and the laststage of Scenario II, it was concluded that the construction should start after 2035; more precise indications of ‘when’ forthese combinations of scenario and stage are not provided in the existing plan.

Because of the absence of alternative data concerning site-specific expansion costs, the investment cost function in ouroptimization algorithm (i.e. Eq. (7)) has been estimated on the basis of information in the cost-benefit analysis for the Maasv-lakte 2-project (CPB/NEI/RIVM, 2001a). Analysis of the information on the investment costs for the subsequent expansionstages leads to an investment cost function for land reclamation, I, that varies exponentially with capacity expansion Ke

by land reclamation. This is represented here by the function IðKeÞ ¼ I�Kke (in euros; see Eq. (7)) with I� = 77,430, and scale

factor k = 0.6.For the existing strategy for Rotterdam port expansion also the welfare effects have been estimated by The Netherlands

Bureau for Economic Policy Analysis. For the reference development in the cost-benefit analysis, demand growth was as-sumed to continue up to 2035. Furthermore, investments were assumed to be made to increase the stock of land withinthe existing port area (CPB/NEI/RIVM, 2001a); we estimated the resulting capacity (to be implemented in 2018) at 14.9 mil-lion TEU/year. For Scenario II, the total welfare effect of the existing strategy for Rotterdam port expansion was estimated at0.5 billion euros (present value in 2003). The effects for the users of the port (according to The Netherlands Bureau for Eco-nomic Policy Analysis mainly related to the container sector) were estimated at 0.2 billion euros (also present value in 2003).

Appendix B. Input variables for the optimization algorithm

See Table B1.

Appendix C. Sensitivity analysis

To test the results of our optimization algorithm, we conducted a sensitivity analysis of the resulting expansion strategyfor Rotterdam port (in terms of timing (‘when’) and size of the fifth expansion stage; see Table 1) to changes in the valuesselected for the input variables a, b, c, k, I�, tff and VOT. This is established here by a 10% increase of the values as presented inTable B1. The outcome of this sensitivity analysis is given in Table C1.

We report the sensitivity of the expansion strategy to changes in the values selected for the input variables for the: (1)demand pattern, (2) congestion cost function, and (3) investment cost function.

Table B1Input variables for the proposed optimization algorithm.

Input variable Meaning Value Unit Source

a Parameter in BPR-formula 0.15 – Ortúzar and Willumsen, 2000b Scale factor in BPR-formula 4 – Ortúzar and Willumsen, 2000c Annual demand growth 402,778 TEU/year Authors’ analysisd Discount factor 0.04 – CPB/NEI/RIVM, 2001a,bk Scale factor investment cost function 0.6 – Authors’ analysish Plan horizon 1000 Years AssumptionI� Parameter in investment cost function 77,430 €/TEU/year Authors’ analysisK0 Capacity in base year (2000) 10,500,000 TEU/year CPB/NEI/RIVM, 2001a,bQ0 Demand in base year (2000) 6,300,000 TEU/year CPB/NEI/RIVM, 2001a,btff Free flow service time 1 Days AssumptionVOT Value-Of-Time 156 €/TEU/day CPB, 2004

Table C1Outcome of the sensitivity analysis of the results for the fifth expansion stage. Source: byauthors.

10% Increase of Leads to a variation of the results for the fifth expansion of

Timing (years) Size (%)

a �1 0b �3 �14c �5 0k +2 �30I� +1 0tff �1 0VOT �1 0

214 S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215

C.1. Demand pattern (c)

The value of c reflects the annual demand growth. It appears that the selected value affects the timing (‘when’) of theexpansion strategy. An earlier timing of the expansion stage corresponds to a higher demand growth. The expansion sizeis, however, not affected.

A lower demand growth, in contrast, leads to a later timing of the expansion stage. An infinitely later timing would cor-respond then with a zero growth (see Eq. (19); the relief interval becomes infinitely large for c = 0 and, consequently, thetiming becomes infinitely later). In other words: expansion is not necessary then. For a negative demand growth (c < 0), alogical outcome of the optimization algorithm is not defined. If demand growth is not a constant, for instance, if periods withpositive growth are varied by periods with zero or even negative growth, the optimization algorithm should be modified tobe able to trace also non-linear demand patterns.

C.2. Congestion cost function (a, b, tff and VOT)

Increasing the input variables of the congestion cost function leads to the same direction of the sensitivity of the resultsfor timing and size: it leads to an earlier timing. The explanation for this is by that the (marginal) congestion costs increasefaster, which leads to an earlier balance with the (marginal) investment costs.

Of the four input variables for the congestion cost function, the selected value for b, the scale factor in the BPR-formula,has the biggest influence on the results of the optimization algorithm. It is the only congestion cost-variable that also affectsthe expansion size. This leads to the conclusion that the (assumed) scale effect of the congestion costs contributes substan-tially to the outcome of our optimal control approach.

C.3. Investment cost function (k and I�)

The values of k and I� reflect the characteristics of the investment costs. Particularly the selection of k, the scale factor,affects the resulting expansion strategy substantially. It appears that a 10%-increase of the value for the scale factor leadsto a substantial decrease of the expansion size. Combined with the fact that for k > 1 a phased expansion strategy is lessattractive for the port due to increasing unit investment costs (see Section 4.1) and assuming a positive demand growth,it is likely that when k exceeds the value of 1, the port should decide to make the investment continuously. This pinpointsthe relevance of (1) incorporating the (assumed) scale effect of investment costs in optimization approaches for decidingupon expansion strategies and (2) selecting a proper value for the scale factor in the investment cost function.

Regarding the investment cost function used in the optimization algorithm, it should be noted that it is continuousassuming that expansion is available at any size (see Section 4.1). Relaxing this assumption - taking into account that portcapacity is non-continuous such as building a new container terminal - affects analytic tractability of the optimization algo-rithm since a minimum required capacity expansion size should be introduced as additional condition. If this minimum re-quired size exceeds the smallest expansion size resulting from applying the optimization approach (here: 78 hectares), theadditional ‘non-continuity’ condition should be included which, as a consequence, leads to a larger expansion size. The im-pact on the timing depends on the (new) balance between marginal investment cost and marginal benefit.

Appendix D. Comparison with the existing strategy for Rotterdam Port expansion.

The existing strategy for Rotterdam Port expansion as recommended by The Netherlands Bureau for Economic PolicyAnalysis (see Appendix A) is determined on the basis of two components: (1) supply–demand analysis; and (2) cost-benefitanalysis based on assessing all welfare effects. The expansion strategy that results from our optimal control approach isbased on the optimization of a system cost function incorporating the congestion-investment balance in the decisions onexpansion and taking into account scale effects. Both approaches show major methodological differences. Another relevant

S. Dekker et al. / Transportation Research Part E 47 (2011) 204–215 215

difference concerns the scope of the welfare effects, which in our approach is limited to congestion cost savings for the usersof the port.

For our optimal control approach, the average annual demand of Scenario II in the existing plan served as input for thedemand pattern. The resulting expansion strategy of the optimal control approach (Table 1) differs, however, from the strat-egy in the existing plan (Table A2). The timings are earlier – for the first two stages even by 11 years – and the expansionsizes are smaller (although the associated investment costs are higher). An optimization approach taking into account theutilization rate and scale effects in the decisions on expansion apparently leads to a different strategy. Referring to the out-come of the sensitivity analysis (see Appendix C), it is obvious to conclude that the scale effects substantially contribute tothis.

Focussing on the welfare effects for users of the port, the difference between the existing plan and our approach is sig-nificant (in our approach about 4.9 billion euros higher). This is caused by methodological differences (e.g. in the existingplan a market share model has been used) and the fact that in our approach the timings are earlier leading to a less fasterbuild up of congestion costs compared to the reference development.

The results of the two approaches also show major similarities. It appears that in the existing plan for Rotterdam Portexpansion only the timing depends on the demand scenario used. On the basis of the results of the sensitivity analysis ofthe demand pattern in our optimal control approach (see Appendix C), a similar observation is made for this approach. Inboth approaches the fifth expansion stage is timed after 2035. Furthermore, the total expansion size resulting from the opti-mal control approach (598 hectares) is almost equal to that in the existing plan (600 ha).

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