Multi-Objective Optimization of Time-Cost-Quality Using Hungarian Algorithm

In this paper, we propose an algorithm for solving multi-objective assignment problem (MOAP) through Hungarian Algorithm, and this approach emphasizes on optimal solution of each objective function by minimizing the resource. To illustrate the algorithm a numerical example (Sec. 4; Table 1) is presented.


Introduction
General assignment problem includes "N" tasks that must assign to "N" workers where each worker has the competence to do all tasks.However, due to personal ability or other reasons, each worker may spend different amount of resource to finish different tasks.The objective is to assign each task to a proper worker so that the total resource that spends finishing all tasks can be minimized.
Many studies have been developed to solve the assignment problem [1]- [4], like time, cost, quality and risk in construction and development projects and investment has been taken into consideration and explanation [5], projects were completed by using the FDOT to establish a model to demonstrate the functional relationship between construction cost and time for the collected highway construction projects [6].A Multi-Objective Ant Colony Optimization is developed to analyze the advanced time cost-quality trade-off problem [7], relationship between time, cost and quality management and the attainment of client objectives [8].
Most of the developed methods for the assignment problem consider only one-objective situation, such as (1) the minimum cost assignment problem, (2) the minimum finishing time assignment problem.The minimum cost Table 1.Assigned cost mtrix (ACM).here cost unit: thousands, time unit: weeks, quality levels: 1, 3, 5, 7, 9.
assignment problem focuses on how to assign tasks to workers so that the total operation cost can be minimized.Such problems have been generally discussed and well developed in many operations researches.Geetha and Nair [9] provide a solution for an assignment problem that minimizes both time and cost.Tsai et al. [10] try to solve a multi-objective decision making problem associated with cost, time, and quality by fuzzy concept.Unfortunately, the provided approaches only deal with the 2-objective assignment problem.

Model Construction of Simple Assignment Problem
Assignment problem is one of the special cases of transportation problems.The goal of the assignment problem is to minimize the cost or time of completing a number of jobs by a number of persons.An important characteristic of the assignment problem is the number of sources is equal to the number of destinations.It is explained in the following way. 1) Only one job is assigned to person.
2) Each person is assigned with exactly one job.Management has faced with problems whose structures are identical with assignment problems.For example, a manager has five persons for five separate jobs and the cost of assigning each job to each person is given.His goal is to assign one and only job to each person in such a way that the total cost of assignment is minimized.
Balanced assignment problem: The number of persons is equal to the number of jobs.Minimize (Maximize): where 1, if the job is assigned to the machine.0, if the job is not assigned to the machine.
Problem definition: Consider a problem which consists a set of "n" machines , , , , m J J J J J =  which is to be considered to assign for execution on "n" available machines, with an execution of cost ( )

Scope Triangle
The triangle illustrates the relationship between three primary forces in an assignment.Time is available to deliver the assignment, cost represents the amount of money and quality represents fit for the purpose of assignment which should be a successful achievement.

Methodology
To determine the assignment of cost (C), time (T) and quality (Q) vs. machine (s) of an assignment problem for a set of "n" machines , which are to be considered as assigned for execution on "n" available machines with an execution are mentioned in the ACM of order, where m = n.First of all, we obtain the sum of cost, time, and quality in each job of the ACM (Figure 1).In this way we get single objective balanced assignment problem nature (Table 2).Now we apply the Hungarian algorithm approach [11]- [13] to obtain the exact optimum solution of balanced assignment problems.To solve this problem we follow the below algorithm.

Algorithm
Step 1: Consider "m" jobs on "n" machines costs given as a matrix (ACM), which is an balanced assignment problem, where m n = .
Step 2: Obtain the sum of cost, time, quality in each job of the ACM.
Step 3: If the total effectiveness of ACM is to be maximized, change the sign of each cost element in the effectiveness matrix and go to Step 4, otherwise go directly to Step 5 if ACM has the total value as minimum.
Step 4: If the minimum element in the th i row is not zero, then subtract this minimum element from each element in the row i ( 1, 2, 3, , i m =  ).
Step 5: If the minimum element in the column j is not zero, then subtract this minimum element from each element in the column j ( 1, 2, 3, , j m =  ).