Rank-Based Solution Methods and their Applications in Determination of Non-Dominated Points Set For A Multi-Objective Integer Programming Model

For any single-objective mathematical programming model, rank-based optimal solutions are computationally difficult to find compared to an optimal solution to the same single-objective mathematical programming model. In this paper, several methods have been presented to find these rank-based optimal solutions and based on them a new rank-based solution method (RBSM) is outlined to identify non-dominated points set of a multi-objective integer programming model. Each method is illustrated by a numerical example, and for each approach, we have discussed its limitations, advantages and computational complexity. KeywordsExact and approximate methods for ranked-optimal solutions; K-ranked optimal solutions; Multi-objective integer programming model; Non-dominated point set; Rank-based solution method.


Introduction
Although, single-objective mathematical programming models and their solution approaches have made a lot of contributions in solving many industrial and real-life optimization problems in operations research and other related areas of study, yet many of these situations are better represented by a multi-objective model, and therefore a need for solution approaches for these multi-objective models is ever increasing. An optimal solution for single-objective models has played a significant role in complex decision-making situations. In the case of multi-objective integer programming (MOIP) problems, instead of finding a single optimal solution, one is required to find the non-dominated points set, which is a central problem of interest. The optimal solution for a single-objective mathematical programming model is relatively easy, as the optimality of the solution is defined by some known conditions. These conditions help us to identify the optimal solution. However, the same is not true for the ℎ ranked optimal solutions, ≥ 2.
The proposed rank-based solution method (RBSM) to identify the non-dominated point set of a given multi-objective integer programming model uses ℎ ranked optimal solutions, ≥ 2. Thus, justifying a need for developing methods that can determine K-ranked optimal solutions for a single-objective mathematical programming model. In this paper, several methods have been discussed that can find these K-ranked optimal solutions and later a rank-based solution method has been developed for identifying the non-dominated point set of the given multi-objective integer programming model. This paper has been organized into 9 Sections. Preliminaries are given in Section 2. Rank-based solution methods are discussed in Section 3. Literature concerning methods for determination of non-dominated point set has been reviewed in Section 4. The proposed rank-based solution method has been developed in Section 5. Some interesting observations about the ranked-optimal solutions have been stated in Section 6. The rank-based solution method for a multi-objective programming model is discussed in Section 7. Numerical illustrations, for the rank-based solution method discussed in this paper, are presented in Section 8. Finally, concluding remarks and future directions are presented in Section 9.

Preliminaries
The mathematical formulation of a single and multi-objective integer programming model is described as (1) and (2): ( ) = ( 1 ( ), 2 ( ), … , ( )) Here (1) and (2) are subject to x ∈ X, where X= {Ax ≤ b; x ≥ 0 and integer}, which is the set of all feasible solution of (1) and (2) and Z1(x), Z2(x),..., Zp(x) are the p linear objective functions for the multi-objective problem. The elements of matrix A are the coefficients of constraints, and b is a vector of non-negative integer values. X is called decision space and the image of X (Y = Z(x)) is called criterion or objective space. When x ∈ {0,1} n the problem is called a multi-objective combinatorial optimization problem. For this paper, the following notations are used.
The set of all efficient solutions x 0 ∈ X denoted XE is called the efficient solution set.
Definition 2.6. (Dominance) Let y 1 and y 2 be two points in the objective space, y 1 dominates y 2 , donated by 1 ≤ 2 , 1 ≤ 2 = 1, 2, … , . (NDP), and the set of all non-dominated points y 0 ∈ Y is denoted YND and is called the non-dominated points set.
For further details see, Ehrgott (2005) and Steuer (1986). Various methods of integer programming have been reviewed in Kumar et al. (2010)

Methods for Determination of Rank-Based Optimal Solutions
Rank-based optimal solutions are difficult to identify as there are no defined conditions to identify the optimality of the K th best solution, K ≥ 2. The K th best solution is defined as the best solution, excluding the (K-1) best solutions. This requirement makes identification and determination of the K th best solution to a given integer programming model more difficult. Some methods for determination of rank-based optimal solution are presented in this section, which is broadly classified into two categories. Determination of the K th best solution, K ≥ 2 for a linear integer program involving fractions are discussed in Section 3.1, and methods for unimodular models, which are free of fractional values are presented in Section 3.2. Kumar et al. (2007) obtained a linear relation from the objective row of the final simplex tableau of the LP relaxation of the given linear integer programming model as given in equation (3).

The Characteristic Equation Technique for Determination of Ranked-Optimal Solutions for A Single-Objective Linear Integer Program
In equation (3), 1 , 2 , … , are non-basic variables, is the objective function, +1, , = 1,2, … , are the coefficients of these non-basic variables in the objective row and P denotes the value of the objective function. Usually, in the LP relaxation of the given pure integer linear programming model, the constants +1, are fractional values. The value of the objective function is given by P under the condition that all non-basic variables at the optimal solution are zero. It may be noted that the values of +1, are not fractional values for the unimodal problems. Note that equation (3) can be re-written as equation (4) given below.
In equation (4), D is the common denominator, R is the integer residue of ( ′ ) and represents the integer coefficient of the non-basic variable = 1, 2, … , . Note that +1, = .
Equation (5) was later renamed by Kumar and Munapo (2012) as the characteristic equation for a pure integer programming model. Note that (5) is a necessary condition but it is not enough to ensure integer solutions. For integer solutions, the slack variables at the integer point need not be zero as in a linear programming model. However, when they are not zero, they also will be restricted to integer values. The LHS is a sum of these non-basic integer variables, which must be equal to + . The efficiency of the relation (5) in the identification of integer points is a function of these three parameters on the RHS i.e. the values , and . The largest value of is governed by the highest integer value of ⌈ ′ ⌉.
Equation (5) has interesting characteristics. For example:  A linear relation (5) will have positive R and D values ≥ 1, only when the objective function row of LP relaxation involves fractional values.
 In the case of unimodal functions = 0, = 1 and ⌈ ′ ⌉ = ′ , hence the CE approach is not suitable for a unimodular function. Other methods for unimodular problems are required and they are discussed in section 3.2.
 The solutions obtained with the help of relation (5) are ranked-optimal solutions in descending order. This is a unique feature of the CE, which is not seen in any other optimization method. Thus, the CE provides the optimal solution, as well as other ranked-optimal solutions in descending order.
Similarly, one can trace other solutions and verify them graphically.

Determination of Rank-Based Optimal Solutions for Unimodular Models
The CE approach is not applicable for the unimodular models, such as assignment and transportation models. For ease of presentation, we discuss the case of an assignment model and find the K th best solution by the Random Search Method developed by Kumar et al. (2018). It is presented in section 3.2.1. solution in the feasible domain. In an assignment problem of size n × n, if the optimal solution is unique and after determination of that optimal solution, one has (n(n−1)) unoccupied cells. These unoccupied cells offer possibilities of other ranked solutions. However, if there r1 number of alternative optimal solutions, the possibilities become r1(n(n − 1)). Therefore, the probability of a successful solution is given by: Here a successful solution means that a solution picked up randomly may be the required K th best solution with probability = 1 1 ( ( −1)) . In general, for the K th best solution, K ≥ 2, the probability will be given by:  The value of assurance factor, denoted by (PR), is a given probability assigned for assurance of the results we find by the computational experiments. Here we have assumed PR = 0.85.
 In the application of RSM one needs the rule to stop these random searches, which is given by relation (8), where (NS) denotes the number of random searches.
This random search approach has been executed as Algorithm 1.
We select the PR =0.80, ; K=2 and we find = Step 1. The optimal solution of the assignment problem is: Step 2.
The number of searches, NS is equal to 18. Choose randomly a non-basic cell and let that cell be 11 . Then the non-basic cells 11 and 23 replaced with basic cells 13 and 21 .
Step 3. These solutions are arranged as: 1,2, 1,3 , 1,4 , … , 1,18 in descending order. Many solutions are repeated, thus from the 18 randomly generated solutions, we obtain only 6 distinct solutions as given in Z value of Table 2. From Table2, we take 13 representing the 2nd-best solution with probability 0.80.
Step 4. We choose from Table 2, the solution 13 , then reapply step 2 and step 3 on the selected solution. This process is continued for K=5 ranked-best solutions, which is shown in  Murty (1968) suggested to solve the given assignment model by the Hungarian method and let the optimal solution be denoted by a string of n allocations satisfying the assignment constraints. Let the string representing the optimal solution be denoted by
Minimum total cost for these n-1 problems will be the required 2 nd best solution and the allocation will be represented by its corresponding string and the cost matrix will be denoted with appropriate link, ( , ) = ∞. The process will continue for the subsequent K th best solution, K ≥ 2. Thus, for every K, one has to solve at least − 1 more problems of size nXn,...,2X2, with one or more elements equal to ∞ in the given cost matrix.

Numerical Illustration 3
Murty's approach is illustrated, using the same numerical illustration 2, discussed above.
Step 1. The optimal solution of the assignment problem (9)  Step 2. Return back to problem (9) and set one basic cell equal to a large number, say 1000, which are the basic cells i.e., (1,3), (2,1), (3,4) and (4,2), therefore 4 new assignment problems will have to be solved. The first one will be as shown below.
The optimal solution for the assignment data with cell (1,3) =1000 is given by: Optimal cost =13, Optimal allocation is at cells (1,2), (2,1), (3,4) and (4,3). Similarly, results of the remaining three problems are shown in Table 4. Therefore, the second-best solution is 13, and optimal allocation is { 12 , 21 , 34 , 43}. The similar analysis will have to be carried out for the 3 rd best solution from the second-best solution. However, it may be noted that apparently the second-best solution is resulting in the same allocation from two different cases. Therefore, we must solve 8 problems, four for each case. Four problems will be solved with cell (1,3) =1000 and one at a time i.e. cell (1,2), (2,1), (3,4) and (4,3) =1000 and other four problems will be with the cell (4,2) =1000 and one at a time cell (1,2), (2,1), (3,4) and (4,3). Murthy does not consider this possibility.

Literature Review for Non-Dominated Point Set for a Multi-Objective Integer Programming Model
There are many approaches in the literature that have been developed to find the non-dominated points set, for a given multi-objective integer programming (MOIP) model, for example see, (Steuer, 1986;Ehrgott, 2005;Greco et al., 2016). Multi-objective combinatorial optimization (MOCO) problems are a special case of MOIP problems, where all variables are binary. For example, see the survey by Ehrgott and Gandibleux (2000), which deals with a variety of methods and models for MOCO problems. They have discussed some exact and approximate methods for the identification of non-dominated point set for MOIP models. Ulungu and Teghem (1995) have presented a two phases method (TPM) to find all non-dominated points for bi-objective assignment problems. In phase one, the supported non-dominated points are determined. In phase two, the nonsupported points are obtained. The TPM was improved by Przybylski et al. (2008) and they applied to solve the three-objective assignment problem (Przybylski et al., 2010b). TPM has also been applied on several other MOCO models Al-Hasani et al. (2019). Crema (2004, 2007) have provided algorithms for generating non-dominated points set with limited success, particularly for larger size instances. Özlen and Azizoğlu (2009) developed an approach based on the ∈ −constraint method, which was developed by Haimes (1971). Przybylski et al. (2010a) have presented an algorithm that generates all non-dominated points for MOIP problems; they tested their algorithm on three objective assignments and knapsack problems. The well-established εconstraint scalarization and partitioning mechanism were used by Kirlik and Sayın (2014) and compared with Özlen and Azizoğlu (2009). Özlen et al. (2014) modified their previous approach and presented a comparison with their previous work for MOA problems. Boland et al. (2015) have presented a new approach, called Balanced Box Method (BBM) for bi-objective integer programming models. The BBM solves IP's three times the number of non-dominated points and therefore, it takes a lot of time to generate the non-dominated set. The L-shape search method has been presented by Boland et al. (2016) and applied to three-objective integer programmes and compared with Özlen et al. (2014) and Kirlik and Sayın (2014). Al-Rabeeah et al. (2020) developed an improved recursive method that out preformed many existing methods, mainly due to adopting a different scalarization scheme. This method decreased the number of sub-problems solved and hence the related CPU time was improved accordingly.
The complexity of a multi-objective assignment problem has been described by many authors, for example, see Ehrgott (2005). Due to these complexities and difficulties, the researchers have justified a need for approximation approaches to find the non-dominated points set. One such approach is by Teghem et al. (2000), which generalizes the multi-objective simulated annealing (MOSA) method. An interactive approach to generate the set of non-dominated points was developed by Tuyttens et al. (2000). Laumanns et al. (2006) presented a method to generate an approximate set of non-dominated point set for the MOA problem.
In this paper, a new approach to generate the non-dominated points set for a multi-objective integer programming (MOIP) model has been developed. The method is based on identifying the ranked optimal solutions to a single-objective integer programming model. As seen earlier in three numerical illustrations, unlike the optimal solution of a single-objective model, the ranked optimal solutions are computationally more demanding, as they do not have defined conditions for optimality of the ℎ best solution. Several methods have been discussed in this paper and developing more methods is a future challenge for further investigation.
Although computationally these methods are demanding but are easy to understand, and implement with in terms of CPU time and number of IP's is much less compared to many previous methods such as Balanced Box method by Boland et al. (2015), MOIP-AIRA by Özlen et al. (2014) etc. International Journal of Mathematical, Engineering and Management Sciences Vol. 5, No. 6, 1249-1269, 2020 https://doi.org/10.33889/IJMEMS.2020.5.6.093

Rank Based Solution Method for Determination of the Non-Dominated Point Set for a Bi-Objective Linear Integer Programming Model
The RBSM method for Bi-objective linear integer programming (BOIP) model comprised of the following steps: Step 1. Find the upper bound = ( 1 1 , 2@ 1 1 ) and the lower bound (ZLB) = ( 1@ 2 1 , 2 1 ), which are the initial two non-dominated points for the given bi-objective problem. Here, 1 1 = 1 and, 2 1 = 2 respectively.The (ZUB) and (ZLB) determines the endpoint of the search for the nondominated points set.
Step 4. The non-dominated point set YND is a subset of potential non-dominated point set which is obtained by testing each potential non-dominated point for dominance.
The Algorithm 2 provide the stepwise description of RBSM for BOLIP model.

Algorithm 2. (RBSM Algorithm for BOIP Problem)
Let Z Bi-objective linear integer programming model

Some Interesting Observations about the Ranked Based Solutions
The Rank Based Solution Method finds the non-dominated points set YND for the given MOIP model. It first finds ordered integer optimal solutions with respect to an objective Zs, where s = 1,2,..p.
Observation 6.1. A unique optimal solution with respect to the objective Zs is a non-dominated International Journal of Mathematical, Engineering and Management Sciences Vol. 5, No. 6, 1249-1269, 2020 https://doi.org/10.33889/IJMEMS.2020.5.6.093 point for the given MOIP. In addition, the ranked optimal solutions with respect to each objective Zs are also potential non-dominated points for the given MOIP.
Observation 6.2. The relationship between the ranked best solutions for a single objective function Zs and non-dominated point set YND in the criterion space is very strong. All non-dominated points are ranked optimal solutions, but all ranked optimal solutions are not necessarily points of the non-dominated point set.
Observation 6.3. A unique optimal solution for a single objective integer programming model has a property that it always leads to an upper bound or a lower bound of the non-dominated point set for the bi-objective MOIP model. However, for the multi-optimal solutions, one and only one of them will result in an upper bound or a lower bound point of the non-dominated point set.
Observation 6.4. A single objective integer programming model may have r s K ≥ 1, number of multi ranked optimal solutions for a given objective s, s = 1, 2, ..., p. From these r s K number of alternative solutions, one and only one solution will become potential non-dominated point in the objective space.

Ranked Based Solution Method for Multi-Objective Integer Programming Model
The Rank Based Solution Method (RBSM) discussed above for the bi-objective integer model can be easily extended for determination of the non-dominated points set YND for a given multi-objective integer programming model. Once again, the approach is based on the determination of ordered optimal solutions with respect to an objective Zs, where s = 1, 2, ..., p. and each point is tested for dominance.

Computational Illustration of the Rank-Based Solution Method
In this section, two numerical examples are presented to illustrate the bi-objective and multiobjective problem for the application of the rank-based solution method. • Find the optimal solution in respect of 1 and 2 . Let the optimal be denoted by 1 1 , 2 1 , respectively. The value of 1 1 = 9 and 2 1 = 8.
• Using definition 6, find the non-dominated set from all potential non-dominated points as shown in Figure 1.
• All Non-dominated point are also shown in Table 7.
• Using definition 6, all potential non-dominated points in Tables 8, 9 and 10 are tested and the set of non-dominated point are shown in Table 11.