# In Pentico (2007). Real world application of APs

In the real situation Decision Maker (DM) is more interested in representing discrete choices and an interval range of aspiration level in multi objective multi choice decision making problem. The present paper proposed Genetic Algorithm (GA) based hybrid approach to find the solution of fuzzy multi-objective multi-choice assignment problem (FMOMCAP) using a fuzzy exponential membership function with subject to some real constraints. In this proposed approach DM is mandatory to specify the different aspiration level as per his/her preference to obtain an efficient allocation plan with different shape parameter in the exponential membership function. A numerical illustration is provided to express the usefulness of the proposed approach with data set from realistic circumstances. This research concludes that developed hybrid approach in this work is provided analysis based effective output to take decision under uncertainty.

Keyword: Assignment problem; genetic algorithm; exponential membership function; multi-objective optimization

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1.      Introduction:

Assignment problem (AP) has been usually used to solve decision making problem in industrial organization, manufacturing system, developing service system etc,. It is to optimally resolve the problem of n- activities to n- devices such that total cost/time can be minimized or total profit/sales can be maximized. In AP, generally two types of objectives are considered: Minimize objective and Maximize objective. Minimization is used for minimizing activities cost or total duration etc while the maximization is used for maximizing overall profit or the overall quality etc. The general solution procedures of AP can be found in the literature survey of Pentico (2007).

Real world application of APs involves many objectives hence single objective assignment problem is rarely sufficient to hold all facets of the problem. In multi-objective assignment problem (MOAP), there is frequently a conflict between the different objectives to optimize simultaneously. Many researchers focused on the MOAP with the intention to provided better decision for real world applications. However, in many decision making situations, it is not easy to convene the preferences of the DM in view of single aspiration level for each objective. So, as an alternative of single choice, if there may be various choices involved associated with the decision parameters like cost, time, quality, profit etc, then the DM is confused to select the appropriate choice for these parameters. In these circumstances, the study of multi-objective assignment problem creates a new direction which is called multi-objective multi-choice assignment problem (MOMCAP). In the real situation, DM is more interested in representing discrete choices and an interval range of aspiration level in multi objective multi choice decision making problem of each objective function.

To find the solution of multi-objective multi-choice optimization problem, few methods are developed by the researchers. Maity G. and Roy S.K have proposed a utility function approach to solve multi-choice multi-objective transportation problem (MCMTP). Mukesh mehlawat has developed the multi-objective COTS products selection optimization model that involves multi-choice aspiration levels based on discrete choices. S. Acharya and Biswal have proposed a solution procedure to solve MCMTP. In particular MOMCAP, Mehlawat and S. kumar has studied MOMCAP with cost and time objectives under some realistic constraints including multi-job assignment. For both cost and time objectives, DM provides multiple aspiration levels using discrete choices as well as interval values is considered in this approach. Multi-choice goal programming technique is utilized to find efficient allocation plans of MOMCAP.

In an optimization model of MOAP, uncertain phenomena, such as random and fuzzy phenomena are often encountered due to many factors, including uncertainty and nondeterministic model parameters. Additionally, decision parameters of assignment problem are indistinct and imprecise according to linguistic terms given by DMs. In such conditions, MOAP turn into fuzzy multi-objective assignment problem (FMOAP). The fuzzy set theory concept was initiated first time by Zadeh 27, which provides a very useful technique for handling inaccurate data. In the decision making problem of the real world, FMOAP is more beneficial by fuzzy theory; the subjective preference of the DM.

Similar, In FMOAP, it is not easy to assemble the preferences of the DM by assuming single aspiration level for multi-objectives, i.e., the decisions are made to be using goal(s) that can be achieved from many choices of the aspiration levels. In such situation FMOAP creates new direction as fuzzy multi-objective multi-choice assignment problem (FMOMCAP).

In above mention cases, it is assumed that the decision parameters in a decision making problem are multi-choice in nature and presented with limited number of choices for a parameter of multi-objective optimization problem. This paper deals with the FMOMCAP and its solution.

2.      Fuzzy multi-objective multi-choice assignment problem:

Main characteristics and some assumption are used in FMOMCAP are (1) Each activity is finished by only one device, and a device can accept more than one activity, all the activities must be completed. (2) It is not compulsory to allow any activity to some devices. (3)  It is required to specify the number of devices who have been assigned to activities, to balance the quantity of work between the devices. (4) In the decision-making method, each device is considered by its working ability (). We suppose that each device should be assigned the number of activities in a definite range.To formulate the mathematical model of FMOMCAP, some parameters, the indices and variables are used (1) Parameters:  activities = devices = n; number devices assigned activities = s; maximum jobs assigned to each employee =; (2) Indices: i and j defined index of device and activity respectively (3) Decision variables:  is indicate that the whether the  devices is assigned for  activities or not. 2.1 Optimization models of fuzzy multi-objective multi-choice assignment problem:The mathematical model of FMOMCAP is defined as follows:Decision Problem: After finishing of all activities, the minimization of total cost and consumed time are given as follows:Model-1:Objective function:    Constraints:As per the above stated depiction of FMOMCAP, the constraints are formulated as follows:

(1)

(2)

(3)

(4)

(5)

where and  are the cost and time which associated with the  devices who has performed the  activity respectively.To define the different aspiration levels as discrete choice and interval range of aspiration level for FMOMCAP, first we have to convert the FMOAP into crisp MOAP. Some preliminaries are used to convert FMOAP into crisp MOAP which defined as follows:  1.      Some Preliminaries:Possibilistic programming approach: The incomplete information on real world problems is an important issue as it imposes a high level of uncertainty. Even though, past data are presented, the performance of the parameters does not require fulfilling with their past model in future. To deal with these issues with the concerned problem, the uncertain parameters are presented with fuzzy numbers. The possibility theory lies in such way that a significant part of the data deals with human choices and also depends on possibilistic in nature. The possibility distribution is estimated in the derived form of insufficient data and knowledge of DM. Possibilistic programming approach has used to solve fuzzy optimization model of many important applications. Possibilistic approach converts the fuzzy objective and/or constraints into crisp objective and/or constraints regarding their three scenarios as optimistic, most-likely and pessimistic scenarios. It is also utilized to sustain the uncertainty of the problem until the solution is obtained 22. Therefore, the possibilistic approach is used to convert the FMOAP model into a crisp multi-objective optimization model 22. Numerous studies in the literature are used possibility distribution to solve fuzzy optimization problem .Triangular Possibilistic Distribution: To represent the triangular uncertain parameters, the triangular possibility distribution is usually applied because of its simplicity and computational effectiveness 66.-level set is the main fundamental theory to set up the connection between fuzzy and traditional set theory which is initiated by Zadeh in 1965 16. -level set is used to reflect the confidence of DM in his/her fuzzy decision; it is also termed as a confidence level. The smallest -level yields a construction of interval judgment having large spared, which shows a high level of pessimism and uncertainty. The largest -level yields smaller but more optimistic judgment whose upper and lower bounds have a superior degree of membership in the initial fuzzy sets. Numerous researchers have used this -level sets concept for finding the solution of the fuzzy optimization related problem .Formulation of multi-objective optimization model:To convert the model-1 into auxiliary multi-objective optimization model, we used Triangular possibilistic distribution (TPD) strategy is used. Thus, the cost objective function is written as              (6)           where , which can be considered as follows.                                              (7)Eq. (6) and (7) are related with the optimistic, the most likely scenario and the pessimistic scenarios respectively. Using the -level sets concepts (), each can be stated as , where . Hence, Eq. (7) can be written as:                              (8)Similarly, the time objective function is as follows                              (9)Crisp multi-objective optimization model:To reflect the optimistic, the most-likely and the pessimistic scenarios with set concept, FMOAP is converted into crisp MOAP 22 is defined as follows: model-2:                                                                      (5.10)   Under the constraints (5.1) to (5.5)Fuzzy multi-objective multi-choice optimization models using different aspiration level:There are two cases are defined for FMOMCAP:1) discrete choice for various aspiration levels, 2) interval range for the various aspiration levels. The crisp multi-objective multi-choice optimization models of the FMOMCAP using discrete choices of aspiration levels for two objective functions are defined as follows .

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