A HIERARCHICAL APPROACH BASED ON LINEAR

AND STOCHASTIC PROGRAMMING FOR THE EMPTY

CONTAINER ALLOCATION PROBLEM

Jorge Luis Morales B. and Ciro Alberto Amaya G.

Department of Industrial Engineering, Universidad de los Andes, Bogotá, Colombia

Keywords: Empty container repositioning, Empty container allocation, Stochastic programming, Sample average

approximation.

Abstract: This paper proposes a solution method to the problem of allocating an empty container fleet to a set of

stocking yards in order to minimize empty container stock and repositioning costs under uncertainties in

demand, supply, container damages and repairing times. We propose an approximate solution for the

problem based on a hierarchical approach. We used random data from different probability functions to

generate problem instances and evaluate robustness and performance. We find that the proposed model

solves the single location inventory problem in a very short time while obtaining high robustness and each

one can be solved independently. This approach allows liners to reduce the complexity of an aggregate

stochastic problem by solving multiple independent stochastic inventory problems. Additionally to other

similar works, the presented models consider random container damages and repairing times.

1 INTRODUCTION

In the new scenarios of increasingly globalized

commerce, repositioning and container allocation

problems are recurrent; however, commercial and

operational contingency decisions dominate tactical

policies and planning, leading to operational

inefficiencies.

Imbalances are frequently observed between

empty containers’ demand and supply.

Consequently, stock of empty containers may be not

enough if replenishment is not made. Liners need to

face the costs of these imbalances by allocating their

container fleets to different yards, and repositioning

empty containers in order to replenish yards with

stock-outs while unnecessary inventory from other

yards is evacuated. This problem has been

denominated as the Dynamic Empty Container

Allocation Problem.

This paper proposes a solution method to this

problem, and is organized as follows. In Section 2

we describe the problem and state the randomness

issues of the problem. In Section 3 we present and

describe the previous works that have tackled this

problem. In Section 4 we propose an approximate

solution for the Dynamic Empty Container

Allocation Problem based on a hierarchical approach

applying linear and stochastic programming,

presenting and describing in detail our proposed

models and the hierarchical structure of the solution.

In Section 5 we describe the performed experiments

for testing robustness and performance, and the data

structure used. Finally, Section 6 presents the

conclusions and perspectives.

2 PROBLEM STATEMENT

2.1 General Problem Statements

The dynamic empty container allocation problem

addressed in this paper considers a liner—decision

maker—that operates in a network of container

storage yards located in both ports and inland points.

The liner also offers shipping services on fixed and

periodic routes and owns a finite container fleet.

Each yard has an associated exogenous demand that

is met with empty container inventory available at

each location.

Shipping routes are weekly cyclic. The fixed and

periodic routes allow each yard to be reached from

any other.

476

Luis Morales B. J. and Alberto Amaya G. C..

A HIERARCHICAL APPROACH BASED ON LINEAR AND STOCHASTIC PROGRAMMING FOR THE EMPTY CONTAINER ALLOCATION PROBLEM.

DOI: 10.5220/0003758004760479

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 476-479

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Replenishment of empty containers at each yard

is done from imports, which must consider the time

elapsed after a container is discharged at a port and

returned empty by the client. The empty returned

containers can also present damages, and they are

not always immediately available to be assigned to

demand. Alternatively for replenishment, empty

containers can be repositioned from a storage point

to another when necessary, or the liner can sublease

a finite number of empty containers to meet

inventory requirements.

2.2 Present Uncertainties

The described process has a high level of uncertainty

over time. The major sources of randomness are

given by the demand, supply, damages and repair

times of empty containers. On the other hand, the

supply of containers in a yard is uncertain, when

containerized cargo is transported from one port or

yard to another, it loses time in waiting, from the

moment the client receives the cargo in the ending

port to the moment it is returned empty to storage

yards and equipment is available.

Additionally, there is the possibility of container

damages, in which case it must be repaired. Finally,

the repair of empty containers takes a random time,

after which the equipment will be available in

storage to be assigned to new demand.

3 RELATED LITERATURE

The literature related to this subject is relatively

short, and not until after the work of Crainic,

Gendreau and Dejax (1993) was the dynamic and

stochastic problem first really tackled. They

proposed both deterministic and stochastic dynamic

models for both single-product and multi-product

cases; for the deterministic model, a decomposition

strategy was proposed in subsequent works

(Abrache, Crainic and Gendreau, 1999).

Shen and Khoong (1995) proposed a hierarchical

decision support system for the multi-period

planning of the distribution of empty containers.

Cheung and Chen (1998) formulated the problem

through a two-stage stochastic network. Leung and

Wu (2004) presented a robust optimization model

for the stochastic problem. Li et al. (2007)

considered implementation of a (U,D) inventory

interval policy for the replenishment. Crainic, Di

Francesco and Zuddas (2007) presented a mixed-

integer programming model associated with a

stochastic program. Chang et al. (2008) use a

decomposition technique and branch and bound.

Belmecheri et al. (2009) used integer programming.

Yuanhui et al. (2009) used a hybrid integer code

hybrid genetic algorithm. Feng and Chang (2010)

used a revenue management model. Song and Zhang

(2010) modeled container transportation as a

continuous fluid flow. Wang et al. (2010) included

decision maker risk preference in a robust

optimization model. Yang (2011) considered an

integer programming allocation model. Finally, Shi

and Xu (2011) presented a Markov decision process

model for empty container repositioning.

4 MODEL DESCRIPTION

Parameters include random container demands,

transit times, expected return times for containers,

storing and transportation costs, leasing costs and

capacities, random supplies, damage probabilities,

and probability for repair times.

The proposed models is based on the following

assumptions: a) The whole observed demand is met

along the time horizon; b) Uncertainty is given by:

demand, supply, damages, repair times; c) The

random variables are independent and not self-

correlated along the time horizon; d) There is no

capacity limit for replenishment containers; e)

Storage capacity in yards is unlimited; f) A returning

yard or time period for subleased containers is are

not specified. Based on these assumptions and the

topology shown above, two optimization programs

will be presented.

4.1 Empty Container Aggregated

Replenishment (Agre) Model

The objective of the model will be to minimize the

costs of aggregate replenishment for empty

containers.

The decisions that must be made along the

planning horizon include the container stock to

having available the in each yard, transportation of

empty containers between yards, and containers

subleasing.

Following this, a cost function can be defined

conidering storage, transportation and

subcontracting costs. Contingency supplies must be

also defined to guarantee the problem feasibility.

Constraints of the Agre model include the

dynamic balance for the available stock of empty

containers, arrivals during a given time period,

subleasing capacities, contingency supplies, and

safety stock.

A HIERARCHICAL APPROACH BASED ON LINEAR AND STOCHASTIC PROGRAMMING FOR THE EMPTY

CONTAINER ALLOCATION PROBLEM

477

The Agre model does not take into account the

randomness of the parameters under uncertainty.

The impact of this randomness must be captured by

the safety stock that must be guaranteed at each sto-

rage point along the planning horizon. This

minimum stock should be sufficient to deal with

uncertainty. The model which estimates the

minimum stock level is described in the next section.

4.2 Minimum Stock (Mist) Model

The empty container minimum stock (Mist) model

should estimate a minimum inventory at each

storage point, to meet the randomnes of the problem.

Based on certain demand for initial stock at the

first time period, we must make a decision on the

inventory to stock. In each period, when making a

decision, the history of previous decisions and

realizations of random variables is known, ignoring

the future demand and supply.

The cost function of the Mist model considers

only the stock costs, specifically, the mean cost

along the planning horizon. Note that, since the

solution of depends on random parameters of

demand and supply, then it also is random.

Contrary to the aggregated replenishment model,

the demand and the supply are random variables

with known probability functions. Additionally, the

damage rate and the devolution times are considered

probabilistically.

Solution of the Mist model provides the

minimum container stock, which will be used as a

parameter by the Agre model in order to capture the

randomness of the system and face the uncertainty.

4.3 Solution to the Mist Model based

on the Sample Average

Approximation

In order to solve the Mist stochastic non-linear

model described previously, the sample average

approximation, or SAA, method is used. This

method generates an approximation of the

probability distribution of the random variables by

Montecarlo simulation. On each period of the time

horizon, realizations of the random variables are

generated, about which decisions should be made.

The motivation for using this method lies in the

following reasons: a) Multiple randomness sources

exist, with heterogeneous probability distributions;

b) The planning horizon considers multi-periods; c)

The probability distributions of the random variables

can be expressed through discrete functions; e)

Based on the previously established assumption of

independence and no self-correlation for the random

variables, random scenarios can be easily generated;

f) Since the Mist model does not include binary or

integer variables, the problem can be solved by a

polynomial size linear program.

5 IMPLEMENTATION AND

RESULTS

We code the models in GMPL to test the

performance robustness of the model. It is important

to note that the computational complexity of the

solution is determined by the complexity of the Mist

problem, given the large number of scenarios that

have to be generated in order to simulate the

randomness of the system. For an instance of one

yard, one container type, a fourweek planning

horizon, and a tree with 1,000 random scenarios, the

matrix of the associated program has a size of

62,000 columns and 5,000 constraints before pre-

processing.

We construct four data sets with four different

demand probability distributions, empirical, normal,

lognormal and uniform, for fourty experiments

corresponding to fourty probability trees. To test the

robustness of the model, the variation of the optimal

solution for expected cost and average minimum

stock over the four periods was recorded, using the

sample variability coefficient as the estimator. To

measure the performance of the model, solution

times and memory usage were recorded; note that it

is expected that the running frequency of the

application is not grater than one run per week.

5.1 Conclusions and Perspectives

Inspired by the work of Shen and Khoong (1995),

the model presented in this paper proposes a

hierarchical solution approach to the Dynamic

Empty Container Allocation Problem that divides

the problem into two models: Mist, the estimation of

the minimum stock necessary to deal with

uncertainty at every storage point over a planning

horizon, and Agre, the estimation of the flow of

container replenishment to ensure compliance with

those minimum stocks levels and business demand

budgets. These models are also inspired by several

of the works mentioned before and operating

conditions of the specific problem described.

However, unlike the works mentioned, the proposed

models also consider the possibilities of observing

damage to containers and repairs’ random times.

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

478

The computational complexity of the application

is given by the stochastic complexity of the Mist

model. For this program, we introduced the sample

average approximation method approach, which

offers advantages regarding uncertainty capture by

generating a large number of random scenarios

given by realizations of the random variables

considered. Through this method, the stochastic pro-

gram became a stochastic linear program and the

expected cost function was expressed as the sample

average cost observed over the generated scenarios.

The computational performance and robustness

of the models was tested, finding that the Mist

model, which governs the complexity of the

application, solved each of fourty runs and four data

set instances in a negligible time, considering that

the running frequency of the model should not be

greater than one run per week. Also, the solution, of

minimum stock values and optimal cost estimates,

was found to show good robustness. Only the

lognormal demand data set showed a high variation

coefficient for the solution, due the dispersion of the

lognormal distribution.

Further work may eliminate some assumptions,

such as the infinite capacity of empty container

shipping; include uncertainty in the capacity; and

include constraints on the container leasing process,

limiting the devolution time and location of leased

equipment to a time window and a specific

geographic location. Finally, is possible to perform a

statistical analysis to generate a measurement of effi-

ciency for the number of random scenarios that

should be generated in the tree, and to estimate

confidence intervals for the calculated solution.

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CONTAINER ALLOCATION PROBLEM

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