Background

This section provides a brief background to the TP with network efficiency. It introduces the DEA approach as a multicriteria tool which further helps us to measure the efficiency on each transportation arc and develop a new MOTP in Section 3.

6.2.1 DEA for multi-criteria decision analysis

DEA is a multi-criteria optimization tool that is widely used for evaluating the relative efficiency of a set of decision-making units (DMUs) with multiple max- and min-type factors. At the heart of DEA is an LP that measures the relative efficiency of DMUs as the ratio of weighted sum of outputs to the weighted sum of inputs. The first two DEAs model, known as the CCR and BCC models are due to Charnes, Cooper, & Rhodes (1978) and Banker et al. (1984), respectively, under constant and variable returns-to-scale assumptions. Suppose there are DMUs; DMU , ( = 1, … , ) that use inputs ( , = 1, … , ), to produce outputs ( , = 1, … , ). The IMCCR model for the relative efficiency of DMUs is given apriori follows: max = ∑ s. t. ∑ = 1 ∑ − ∑ ≤ 0 = 1, … , ≥ = 1, … , ≥ = 1, … , (6.1)

where the outputs weights and input weights are required to be greater than a small positive number (known as non-Archimedean infinitesimal) to forestall weights from being zero (see Toloo, 2014d). The DEA efficiency benchmarking in the CCR model is characterized by the identification of an efficient frontier determined by the non-dominated DMUs. In other words, DEA efficiency is a dominance-based concept which follows similar Pareto optimality conditions as the MOLP. A DMU is efficient (Pareto optimal) if and only if it is not possible to improve the performance of any input or output without worsening at least one other input or output. See Keshavarz & Toloo (2015) on further established efficiency status of the MOLP solutions as pertain to the DEA. The set of all efficient units is obtained by solving at least one optimization problem for each DMU in (6.1); whence = 1 indicates a Pareto efficient unit. Figure 6.1 shows a unitized space for one unitized input and two output case where P P indicates the piecewise DMU has efficient solution

Figure 6.1. Efficiency of DMUs in unitized space.

linear efficient frontier. DMUs A and F are weakly efficient, usually obtained if the lower bound is removed and = 1. Other DMUs on the frontier are Pareto efficient or strongly efficient. Notice that the distance measure CC indicate an efficient projection of the dominated DMU to the frontier.

6.2.2 The transportation problem with arc efficiency

First, consider a transportation network made of origions ( = 1, … , ) and distinations ( = 1, … , ). For each transportation made, a supply quantity from source is dispatched which is received as demand at destination . It is clear that the supply and demand indicate physical quantities and must be nonnegative, i.e. ≥ 0 ∀ , ≥ 0 ∀ . The process is

represented as a network with source nodes, sink nodes, and a set of × directed arcs (links). Let = _× denote the matrix of decision variables, in other words, the quantity of goods to be transported from all the sources to all the destinations. The conventional TP is the problem of minimizing the total transportation cost of the whole distribution and is mathematically expressed as:

( ) = min ∑ ∑ s. t. ∑ ≤ = 1, … , ∑ ≥ = 1, … , ≥ 0 = 1, … , , = 1, … , (6.2)

where represent the cost of transporting the products from source to destination . The condition ∑ ≥ ∑ is imposed for feasibility of the problem. The transportation problem considering arc efficiency developed under the generic name ‘extended transportation problem’ (ETP) is studied in Amirteimoori (2011). The ETP adopts DEA to measure efficiency of incommensurate multiple max-type and multiple min-type factors of transportation arcs. The idea was earlier introduced in Chen & Lu (2007) for the assignment problem37_{. The ETP considers a network system with single product made of multiple inputs} and multiple outputs. Each arc ( , ), as a DMU, has min-type factors (inputs) ( )_, ( )_{, … ,} ( ) _{and max-type factors (output)} ( )_, ( )_{, … ,} ( ) _where ( )_, ( )_{, … ,} ( ) _{≥ and} ( )_, ( )_{, … ,} ( ) _{≥ . Two scenarios are used to measure the} efficiency of each link; the origin – oriented scenario and destination – oriented scenario. The DEA model (6.3) measures the efficiency at the origin

( )_{= max ∑} ( ) s. t. ∑ ( ) = 1 ∑ ( )− ∑ ( )≤ 0 = 1, … , ≥ = 1, … , ≥ = 1, … , (6.3)

( )_{= max ∑} ( ) s. t. ∑ ( ) = 1 ∑ ( )− ∑ ( ) ≤ 0 = 1, … , ≥ = 1, … , ≥ = 1, … , (6.4)

While the DEA model (6.4) measures the efficiency at the destination. Here the vector and are the outputs weights and input weights respectively and the origin- and destination- efficiency are given by ( )and ( ). Figure 6.2 shows the transportation arcs of the two- scenario efficiency measurement.

( ) Origin – oriented scenario ( ) Destination – oriented scenario

Figure 6.2. Transportation problem with network efficiency

6.2.2.1 Amirteimoori’s Approach

Amirteimoori (2011) considers the maximum efficiency of the transportation arcs over profit and cost. However, unlike the extended assignment problem which suggests the use of composite efficiency index as a performance measure of each assignment (Chen & Lu, 2007), Amirteimoori (2011) considers the averages of the relative efficiency of the two scenarios of the transportation arc and compute the efficiency of the network with the following model:

min ∑ ∑ (1 − ) s. t. ∑ = = 1, … , ∑ = = 1, … , ≥ 0 = 1, … , , = 1, … , (6.5)

where 1 − is the inefficiency score of arc ( , ) and = 1/2 ( )+ ( ) . As a matter of fact, model (6.5) is a traditional transportation problem where the unit cost of shipping from factory to warehouse is 1 − and provides a single objective of maximum efficiency.

6.2.3 The drawback of the extended approaches

While the two scenarios of efficiency measurement are fascinating for operations research problems such as the transportation and assignment problems, the approaches of Chen & Lu (2007) – a composite efficiency defined as the product of ( ) and ( ) and Amirteimoori (2011) – average efficiency , induce a non-multiple criteria optimization framework characterizing the non-dominated solution of the problems. The main drawback of these extended approaches are pointed out in Keshavarz & Toloo (2015) and Shirdel & Mortezaee (2015). Shirdel & Mortezaee (2015) provides a counterexample to Amirteimoori (2011) ETP to show that solutions generated in the later are not necessarily a non-dominated solution, thus given that the transportation problem is usually a multiple choice or multiple objective problem requiring Pareto optimal to alternative feasible solutions (Roy, Maity, Weber, & Gök, 2017). On the other hand, Keshavarz & Toloo (2015) provides practical example to show that the efficient solution in the extended assignment problem of Chen & Lu (2007) is dominated. Keshavarz & Toloo (2014) further provide a multi-criteria framework to classify all the efficient solutions of the assignment problem.