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Mathematical Theory and Modeling www.iiste.org

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Transshipment Problem and Its Variants: A Review

Gregory Okiemute AGADAGA* Nsikan Paul AKPAN

Department of Mathematics and Statistics, Faculty of Science, University of Port Harcourt, Rivers State, Nigeria.

*Corresponding author E-mail: gregagadaga@gmail.com, nsipaulakpan@gmail.com

ABSTRACT

The transshipment problem is a unique Linear Programming Problem (LLP) in that it considers the assumption

that all sources and sinks can both receive and distribute shipments at the same time (function in both directions).

Being an extension of the classical transportation problem, the transshipment problem covers a wide range of

scenarios for logistics and/or transportation inputs and products and offers optimum alternatives for same. In this

work the review of literatures from the origin and current trends on the transshipment problem were carried out.

This was done in view of the unique managerial needs and formulation of models/objective functions. It was

revealed that the LLP offers a wide range of decision alternative for the operations manager based on the

dynamic and challenging nature of logistics management.

Key words: Transshipment problem, Linear Programming Problems (LPP), model, objective functions, decision

alternative.

1. Introduction

The transportation of goods and/or services (from the plants/warehouse to the consumer/final destination) is a

major concern for every manufacturer. This concern forms an integral part of the production process.

Distribution managers are constantly being faced with this critical transportation need to which they must make

the best decisions for the benefit of the consumer at the same time considering the overall profit of the

organization. This decision is further complicated with the challenges arising from globalization, increasing

market competition, and accelerated technology development. Due to demand variability and market uncertainty,

achieving these goals requires ﬂexibility as much as possible, short response time and development of new

innovative solutions for reaching the customers (Noham and Tzur, 2014). However, this challenge faced by the

decision makers and/or distribution managers is easily surmountable by the decision alternatives provided for by

Operations Research.

One of the classic uses of Operations Research and, in particular of Linear Programming is to propose optimum

alternatives for the logistics or transport of inputs and products from a group of suppliers to a group of receivers

or petitioners (LPW, 2014). Among the Linear Programming Problems (LPP), the transportation problem is a

special category which has been widely studied due to its importance in the logistics and operations management

where distribution of goods and commodities from sources to destinations is an important issue (Chaudhuri and

De, 2013). In considering process with the inclusion of intermediaries between the sources and destinations, this

becomes an extension of the basic model of transport which is commonly known as a model of transportation

with transshipment (LPW, 2014).

2. History of Transshipment Problem

The Transshipment Problem has a long and rich history, dating back to the medieval times when trading started

becoming a mass phenomenon. It was first introduced by Orden (1956) in which he formulated an extension of

the original transportation problem to include the possibility of transshipment. This extension of the

transportation problem saw a modification in which each origin and destination can act as an intermediate point

through which goods can be temporarily received and then transshipped to other points or to the ﬁnal destination

(Gass, 1969: cited by Briš, 2010). From the original work done by Orden, the transshipment technique was and

is still used to find the shortest route from one point in a network. Both methods, that is the transportation

problem and transshipment problem are widely used for the planning of large scale distribution of goods

especially in areas where the distances travelled (via road or otherwise) are very far (NagoorGani et al., 2012).

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So many researchers in recent times have extensively studied this LPP and have developed different variants

based on the needed objective function.

This study therefore seeks to review literatures on the emergence of the transshipment problem as well as the

variants based on objective functions and approach.

3. Literature Review

The transshipment problem is a unique Linear Programming Problem. From the unset of its introduction by A.

Orden in 1956, its acceptance and/or use has greatly broadened over the years. The initial problem took into

consideration the use of intermediate points through which shipments were to be allowed to pass through to the

needed destination. From his work the transshipment problem was and still is the ideal for solving shortest route

problems.

Rhody (1963) considered the transshipment problem as a reduced matrix model while King and Logan (1964)

went further to formulate an alternative model that allows simultaneously the transportation of main goods

through processors to the market as end products as a transshipment problem. Hurt and Tramel (1965) adopted

the studies of King and Logan as an alternative formulations of the transshipment problem within the framework

of the transportation model but proposed that there was no need for the subtraction of artificial variables.

In 1965 Judge et al. proposed the use of the transshipment problem to cover a multiregional, multiproduct and

multi-plant problem. They formulated this transshipment problem in the form of general linear programming

model (Judge et al., 1965). The time minimizing transshipment problem was studied by Garg and Prakash in

1985. Their study proposed very clear algorithms as to how the optimal time can be achieved while

transshipping goods from different origins to different destinations. Osman and Ellaimony (1984) introduced an

algorithm for solving bi-criteria multistage transportation problems. Herer and Tzur (2001) studied the dynamic

transshipment problem. Their work was the first time transshipments are examined in a dynamic or deterministic

setting. They also considered a system of two locations which receives their stock from a single supply point,

and transshipments were possible between these locations.

The introduction of a multi-objective transportation problem (MOTP) with interval cost, source and destination

parameters and multi-objective transportation problem (MOTP) under fuzziness were studied by Das et al.

(1999) and Abd El-Wahed (2001) respectively. Saraj and Mashkoorzadeh (2010) studied the multi objective

transportation problem (MOTP) using interval numbers under fuzziness.

The transshipment problem considers that within a given time period each shipping source has a certain capacity

and each destination has certain requirements with a given cost of shipping from source to destination. The

Objective Function is to minimize total transportation costs and satisfy destination requirements within source

requirements (Gupta and Mohan, 2006).

Notably the main priority in the transshipment problem was first and foremost to obtain the minimum-cost

transportational route, however technological development slowly gave place to minimum-durational

transportation problems.

Hmiden et al. (2009) studied the transshipment problem that is characterized by the uncertainty relative to

customer demands and transfer lead time. They consider a distribution network of one supplier with several

locations selling a product. They used expert judgments to evaluate customer demands and the transfer lead time

which they represented by fuzzy sets. The purpose of their study was to identify a transshipment policy that takes

into account the fuzziness of customer demands and transfer lead times and to determine the approximate

replenishment quantities which minimize the total inventory cost. They proposed a new transshipment policy

where the transshipment decision is made within the period and the possible transshipment decision moments

belong to a fuzzy set to achieve their aim. For the expert judgments they also consider the decision maker

behavior types (pessimistic and optimistic) to determine the precise transshipment decision moment and the

transshipment quantity.

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Hoppe and Tardos (2000) concluded that the transshipment problem is defined by a dynamic network with

several sources and sinks and that there were no polynomial-time algorithms known for most of transshipment

problems. They gave the first polynomial-time algorithm for the quickest transshipment problem and this

algorithm provided an integral optimum flow. Herer and Tzur (2001) investigated the strategy of transshipments

in a dynamic deterministic demand environment over a finite planning horizon.

Tadei et al. (2009) noted in their research that in any transshipment problem the transportation process takes

place in two stages: from origins to transshipment facilities and from transshipments facilities to final

destinations. Economically put, the process could incure three kinds of cost: the fixed cost of locating a

transshipment facility, the transportation cost from an origin to a destination through a transshipment facility and

the throughput operation cost at each transshipment facility. Their study pointed out the overall cost implication

of the transshipment process which may not be incorporated in the transshipment LPP. Kestin and Uster (2007)

added that the throughput operations cost are generated by the freight treatment operations, such as

loading/unloading, but also to inventories and postponed processing, such as packaging, testing etc. While the

fixed cost and the transportation cost are usually well defined, easy to determine and measure, the throughput

operation cost is vague, stochastic and cannot be easily measurable, so that their probability distribution remains

in general unknown (Tadei et al., 2009). Malakooti (2013) further considered multi-objective transshipment

problem (MOTP).

Agarwal and Ergun (2008) considered transshipment in their study but did not consider its cost implication. They

explain this highlighting the tremendous increase it will cause to the size of the graph and concluded that “Our

results indicate high percentage utilization of ships’ capacities and a signiﬁcant number of transshipment in the

ﬁnal solution”. When the transshipment cost is not taken into account the transshipment in the optimal solution

will probably be inappropriately large (Ameln and Fuglum, 2015).

Most recent literatures in the study of transshipment covered areas like the Bi-criteria large-scale transshipment

problems. Alkhulaifi et al., (2014) presented a formulation of different structures of bi-criteria large-scale

transshipment problems and an algorithm for solving a class of them, which they proposed can be solved using

the decomposition technique of linear programming by utilizing the special nature of transshipment problems.

Al-Rajhi et al., (2013) studied the decomposition algorithm for solving a class of bi-criteria multistage

transportation problem. They noted that in transportation problems with one or multistages, single criterion of

minimizing the total cost is usually considered however in certain practical situation this may not be the case as

two or more objectives may be relevant. For example, the objectives may be minimizations of the total cost,

consumption of certain scarce resources such as energy, total deterioration of goods during transportation, etc.

Rajendran and Pandian (2012) introduced the splitting method which they proposed for obtaining an optimal

solution to a transshipment problem where all parameters are real intervals. For them, the total transportation

cost in the transshipment problem is always lesser than that of the classical total transportation cost. Somani

(2015) proposed an innovative method for finding optimal transportation cost which he called the Somani's

Approximation Method (SAM). Panda and Das (2014) presented a two-vehicle cost varying transportation

model. In the model they noted that the transportation cost varies based on the capacity of vehicles and the

amount of transport quantity.

Khurana et al., (2015) considered a transshipment problem with the objective to minimise the duration of

transportation. They proposed an algorithm that solves the original transshipment problem by transforming it

into an equivalent transportation problem and the optimal solution of the original problem is obtained from the

optimal solution of the transformed transportation problem. It involves a finite number of iterations that is easily

applicable. The algorithm finds out the optimal time for transportation from origins to destinations with

transshipment. With this transshipment variant managers are provided with another alternative solution to a

variety of production distribution problems.

In another 2015 study by Archana Khurana, she used the transshipment technique to find the shortest route from

one point in a network to another point and is very useful to reduce the cost of transportation. The study focused

on the shortest route as against the minimum-cost and minimum-duration from earlier studies. She formulated a

three-dimensional linear transshipment problem. This was due to constraints that may sometimes arise from the

production budget or politically induced, the total flow of transportation is specified by some external decision

maker which thereby results in restricted and enhanced flow in the market. The optimal solution of the specified

problem is obtained by transforming it into an equivalent transportation problem by adding an additional row

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and a column and avoiding the trouble of determining a least-cost route for every origin-destination pair. She

was also confident that arriving at the optimal solution of such a problem would be of practical interest to

decision makers with such peculiar challenge (Khurana, 2015b).

Akilbasha et al. (2016) proposed a new method for ﬁnding an optimal solution for integer transportation

problems where transportation cost, supply and demand are intervals. They called it the split and separation

method based on zero point method. This method can served as an important tool for the decision makers when

they are handling various types of logistic problems having rough variable parameters.

Bretschneider (2013) studied transshipment problem as an example of the dynamic flow problem which is

applicable for the model flow for evacuation planning. He, referring to the work done by Hoppe and Tardos,

(1994) noted that the dynamic network ﬂow models can be used to describe evacuation problems and that the

question of how to obtain the quickest ﬂow with multiple sources is often referred to as the evacuation problem.

For the case of evacuation planning, the quickest transshipment problem determines the evacuee ﬂow such that

all evacuees leave the danger zone in a minimum amount of time or, in other words, the clearance time of the

evacuation zone is minimized (Bretschneider, 2013).

Pascoal et al. (2006) studied the single path from a source to sink while minimizing the network clearance time.

Hoppe and Tardos (1994) considered the quickest transshipment problem with multiple sources and a single sink

called the evacuation problem. Fleischer (2001) investigated the quickest transshipment problem, where the

transit times of all arcs are zero, but the inﬂow is still assumed to be capacitated.

Ameln and Fuglum (2015) introduced the use of transshipment to the liner shipping network designs which was

less common a few decades ago. They noted that as at 1991 transshipment was not included in the studying route

planning for liner shipping network. Ameln and Fuglum citing Notteboom and Rodrigue (2008) stated that the

use and inﬂuence of transshipment on the liner shipping network design was described and whether or not

transshipment operations are allowed between the routes diﬀer based on each researcher while Meng et al.

(2014) noted that 8 out of 12 of the studies on ﬂeet deployment did not consider transshipment. Reinhardt and

Pisinger (2011) argue that transshipment of goods is frequently occurring in liner shipping and the associated

cost should not be ignored when designing the network. They claim that their paper presents the ﬁrst exact

solution methods to the LS-NDP (Liner Shipping Network Designs Problem) with transshipment cost (Ameln

and Fuglum, 2015).

More researchers have also studied the transshipment problem with mixed constraints i.e. equality and inequality

constraints, impaired/restricted and enhanced flow (Thirwani et al. 1997; Khurana and Arora, 2006, 2011;

Khurana, 2015).

4. Variants and Corresponding Models of Transshipment Problem

4.1 The General Model

The general linear programming model of a transshipment problem is

Subject to

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Where

Xij = Amount of unit shipped from node i to node j

Cij = Cost per unit of shipping from node i to node j

Si = Supply at origin node i

dj = Demand at sink node j

From the above model we understand that the supply point is the point(s) that can only send goods to another

point(s) but cannot receive goods. While the demand point(s) is/are also a point(s) that can only receive goods

from other points but cannot send any. Then the transshipment point(s) is that point(s) that can receive goods

from other points and also send goods to other point(s). In this model all goods within the network must be

transported/shipped without any reservations until all constraints are fully satisfied.

Another method that uses the above model considers the absence of an obvious transshipment point. The model

therefore assumes the possibility of transshipments. In this case all the source and sinks can serve as

transshipment points i.e. they can both receive and give shipments. This is unlike models with very clear points

for transshipments to take place.

The model is to minimize Z

subject to

:

4.2 The Unbalanced Model

In some cases the transshipment of the total supply (from the origin) does not equal the demand (in the

destination) thereby making it an unbalanced transshipment problem.

This model is converted to a balanced model by the addition of dummy supply or dummy demand depending on

the where the short fall exists.

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4.3 The Impaired flow Transshipment Model

This model arises in situations where a particular level of stock is reserved at the source for certain reasons like

emergencies. The total flow of the shipments is restricted. The level of restriction may be set at point Q which

gives the following mathematical model.

Or

′

Where Q′ = Q + (m+n)T

And T = max

The model is given

Min Z =

Subject to

′

′

(Impaired flow)

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It will be noted that the impaired flow constraint shows that a total

′

of the source

reserve will have to be kept at the various sources while at the destination the total

′

destination slacks will be retained at the various destinations.

The above model clearly differs from the general model as it takes into accounts an extra constraint known as the

impaired flow constraint. The flow constraint fulfils the desire of the operations manager without necessarily

altering the basics and standards of the original transshipment model. In this case an extra destination to receive

the source reserves and an extra source to fill up the destination slacks are introduced which gives the model the

needed balance.

4.4 The Enhanced flow Transshipment Model

Another model worthy of note is the enhanced flow model. Unlike the impaired flow where reservations are

made in the source, the model comes to play with extra quantities needed at the demand point(s). The enhanced

flow model therefore occurs when due of an extra demand in the market, the total flow has to be enhanced

(increased), forcing the factories (supply points) to increase their productions so as to meet the extra demand. In

this case the Q′ value is maximized. Mathematically put

′

The model is given

Min Z =

Subject to

′

′

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(Enhanced flow)

To deal with the enhanced flow in the above model, a related transportation problem has to be formulated by

adding an additional row with availability equal to′

and an additional column with

demand equal to′

. These were discussed extensively by Khurana et al., (2015).

The tilt from the impaired and enhanced flow models gives the decision maker more decision options for the

challenging and dynamic market needs.

4.5 Bi-Criteria Multistage Transshipment Models

Another model worthy of note is the bi-criteria large scale transshipment problems studied by Alkhulaifi et al.

(2014). They extended the use of the transshipment problem to cover cases of more than one objective functions

(bi-criteria) in large scales. We consider 2 cases and are given thus:

1) Bi-Criteria Multistage Transshipment Problem of the First Kind (BMTSP 1)

This case shows a multistage transshipment problem without any restrictions on intermediate stages. This can be

represented mathematically with certain assumptions.

Min Z1 =

Min Z2 =

Subject to

Where

aj= availabilities (Supply), j= 1, 2, 3, ..., n

bj = requirements (Demand), j= 1, 2, 3, ….., n.

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n = number of sources and destinations

cij and dij = The minimum transportation costs and deteriorations from i to j respectively, i and j= 1, 2, 3, …., n.

Xij denotes the quantity shipped from i to j;

xji is the neat amount transshipped through point j where xij ≥ 0.

cij = 0 for the quantity shipped from the source Si to itself and from destination dj to itself

2) Bi-Criteria Multistage Transshipment Problem of the Second Kind (BMTSP 2)

The second kind (BMTSP 2) deals a situation where the transportation at any stage is independent of the

transportation of the other stages. This is a multistage system characterized by independence in transportation. It

assumes a kth stage in the transportation and K = 1, 2, ..., N.

The model is given as:

Min

=

=

Subject to

Where

= 0 for the quantity shipped from the source (Si) to itself and from destination (dj) to itself

And the minimum transportation cost given as

Min

=

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It will be well noted that the above models covers transshipment at a very large scale. Operations managers who

deal on large scale would do better maximizing these models for their operations.

4.6 Other Transshipment Models

The transshipment problem is dynamic in the sense that it also offers the operations manager a model that covers

not just the cost and/or quantity to be transported that satisfies the supply and demand constraints but a variant

that can be used for locating the transshipment points.The capacitated transshipment location problem under

uncertainty was studied by Tadei et al. (2009). The model is given below.

Ế

θ

Subject to

Supply constraint

Demand constraint

capacity restriction at each transshipment facility k

Where

: set of origins

: set of destinations

: set of potential transshipment locations

: set of throughput operations scenarios at transshipment facility

: supply at origin

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: demand at destination

: throughput capacity of transshipment facility

: fixed cost of locating a transshipment facility

: binary variable which takes value 1 if transshipment facility

is located, 0 otherwise

: unit transportation cost from origin

to destination

through transshipment facility

: unit throughput cost of transshipment facility

in throughput operation scenario

: flow from origin

to destination

through transshipment facility

Ế

θ

denotes the expected value with respect to ;

The Capacitated Transshipment Location Problem under Uncertainty (CTLPU)is balanced if

5. Discussion and Conclusion

The literatures above clearly show the different views and approach of researchers on the transshipment

problem. Though there are more variants than those reviewed in this work. However, each model explains the

peculiar approach of the researcher to meet the dynamic needs of the operations manager in various industries

where the transshipment problem is applied.

The transshipment problem is unique in the sense that it offers a wide variety of decision alternatives to

accommodate the changing trends in dealing with shipments of goods to measure up to modern and global

practices. It also takes into accounts the scale (small and large) of transshipments for small, medium and large

firms.

Another advantage in the transshipment problem based on the objective function is its ability to cover varying

concerns. It is used for minimum costs problems, shortest route problems, minimum time problems and more

importantly the facility layout/location problems amongst others. Also models that cover more than one

objective function per time are available for the decision maker who is interested in solutions for multiple

concerns at the same time.

Further more in cases of emergencies (fire outbreak, natural disasters, military services, etc) and peak seasons

(festivities, marriages, etc) where the flow requires enhancing or from an alternate point of view for situations

like recession, budget/political constraints where the operations manager is forced to hold or restrict the flow of

goods, the transshipment problem with enhanced and impaired flow comes in handy.

Due to the dynamic nature of logistics, sometimes goods transportation may not move directly from suppliers to

customers. When such arises, the decision maker can optimize the transshipment problem into more than one

stage. He/she may wish to consider all efficient extreme points and their related distributions that will form basic

parts of the operations policies.

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