Abstract—The primary objective of the paper is to propose a new method for solving assignment problem under uncertain situation. In the classical assignment problem (AP), z pq denotes the cost for assigning the qth job to the pth person which is deterministic in nature. Here in some uncertain situation, we have assigned a cost in the form of composite relative degree F pq instead of pq z and this replaced cost is in the maximization form. In this paper, it has been solved and validated by the two proposed algorithms, a new mathematical formulation of IVIF assignment problem has been presented where the cost has been considered to be an IVIFN and the membership of elements in the set can be explained by positive and negative evidences. To determine the composite relative degree of similarity of IVIFS the concept of similarity measure and the score function is used for validating the solution which is obtained by Composite relative similarity degree method. Further, hypothetical numeric illusion is conducted to clarify the method’s effectiveness and feasibility developed in the study. Finally, conclusion and suggestion for future work are also proposed. Keywords—Assignment problem, Interval-valued Intuitionistic Fuzzy Sets, Similarity Measures, score function. I. INTRODUCTION SSIGNMENT problems deal with the question how to assign n objects to m other objects in an injective fashion in the best possible way. An assignment problem is completely specified by its two components: the assignments - which represent the underlying combinatorial structure and the objective function to be, optimized which models “the best possible way". To find solutions of assignment problems, various algorithms such as linear Programming [1]-[4], Hungarian algorithm [5], neural network [6] and genetic algorithm [7] have been developed. Over the past 50 years, many variations of the classical assignment problems are proposed e.g. bottleneck assignment problem, generalized assignment problem, quadratic assignment problem etc. But in real life situation, the parameters of AP are imprecise number instead of fixed real numbers. Zadeh [8] introduced the notion of fuzzy sets to deal with vague situation in real life. In recent years, fuzzy transportation and fuzzy assignment problems have received much concentration. Lin and Wen [9] proposed an efficient algorithm based on the labeling method for solving the linear fractional programming case. Atanassov Gaurav Kumar is with the Singhania University, Rajasthan, India as research scholar (e-mail: gaurav.joshi8403@gmail.com). Dr. Rakesh Kumar Bajaj is with the Department of Mathematics, Jaypee Institute of Technology, Waknaghat, H.P. India (rakesh.bajaj@juit.ac.in). [10]-[12] introduced a generalized concept of fuzzy sets i.e. intuitionistic fuzzy sets. Then Atanassov introduced prominent form by combining the concept of intuitionistic fuzzy sets and interval-valued fuzzy sets i.e., interval–valued intuitionistic fuzzy sets IVIFS [12] and this concept is widely used in multi criterion decision making problems, medical diagnosis etc. [13]–[17]. Bustince and Burillo [18] showed that vague sets are intuitionistic fuzzy sets. Further, they studied some distance measures, similarity measures, and correlation measures for IVIFS and applied the introduced concepts to many real-life problems in pattern recognition, decision making, etc. In addition to this, there are other applications which involve the information entropy of IVIFS [19]–[21]. Xu [22] defined the concept of the degree of similarity between interval-valued intuitionistic fuzzy sets and defines some distances measures between two IVIFS and proposed an approach for decision making with interval-valued intuitionistic fuzzy information [23], [24]. Some authors [25], [26] proposed different methods for decision making under intuitionistic or interval-valued intuitionistic fuzzy environment. Jahan [27] used the linear assignment method to rank the materials of a given engineering component in accordance with several criteria. Mukherjee and Basu [28] proposed an algorithm to solve Intuitionistic Fuzzy Assignment Problem by Similarity Measures and Score Functions. The paper is organized as follows: In Section II, we recall some basic notions related to Interval-valued intuitionistic fuzzy sets and Interval-valued Fuzzy Intuitionistic Fuzzy number. In Section III, we develop a new methodology to solve assignment problem in interval-valued intuitionistic fuzzy environment. In Section IV, by using the concept of positive and negative ideal for interval-valued intuitionistic fuzzy sets, we develop two algorithms to solve by IVIFAP . Section V provides an illustrative example and finally, in Section VI, we conclude the paper. II. PRELIMINARIES A. Interval-Valued Intuitionistic Fuzzy Sets and Interval- Valued Fuzzy Intuitionistic Fuzzy Number Let T be a finite non-empty set i.e. } ,..., , , { 3 2 1 n t t t t T = . Let ] 1 , 0 [ R be all subintervals of ] 1 , 0 [ . An IVIFS X in T is defined with the form, On Solution of Interval Valued Intuitionistic Fuzzy Assignment Problem Using Similarity Measure and Score Function Gaurav Kumar, Rakesh Kumar Bajaj A World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:8, No:4, 2014 715 International Scholarly and Scientific Research & Innovation 8(4) 2014 scholar.waset.org/1307-6892/9998707 International Science Index, Mathematical and Computational Sciences Vol:8, No:4, 2014 waset.org/Publication/9998707