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Top1. Introduction
Assignment problem (AP) is a special type of a well-known transportation problem (TP) in which the objective of the decision maker (DM) is to assign n number of machines to n number of tasks (jobs) at a minimum time / minimum cost (or maximum profit or maximum production).
An AP can be viewed as a TP in which all supplies and demands equal to the numerical value one (i.e., 1). AP is used worldwide in solving scientific/real-world problems. An AP plays a major role in assigning of the following:
The solid assignment problem (i.e., SAP or three dimensional assignment problem (3D-AP)) is a generalization of the classical or conventional AP in which three dimensional (3D) properties are taken into account in the objective function and constraint sets instead of machines and jobs.
In literature, the famous Hungarian method developed by Kuhn (Kuhn, 1955) is recognized to be the first practical method for solving the standard AP. (Pierskalla, 1967) introduced the 3D-AP as a straightforward extension of the classical two-dimensional AP (2D-AP). The multidimensional assignment problem (if the AP has more than three types of constraint sets then it is called a multidimensional assignment problem (i.e., MD-AP)) was proposed by (Pierskalla, 1968). Other than Pierskalla, a number of researchers (Burkard et al., 1996; Prajapati et al., 2021; Rezaul Karim, 2021; Amalia et al., 2021; Esakkiammal & Murugesan, 2021; Violina, 2021; Dinagar & Raj, 2021a; Brogiolo, 2021; Murugesan & Esakkiammal, 2021) have also studied the concept of different types of APs under crisp environment.
A genetic algorithm based heuristic to solve the generalized AP was presented by (Chu & Beasley, 1997). (Arora & Puri, 1998) introduced a variant of time minimizing AP (TMAP). (Tadei & Ricciardi, 1999) studied the multilevel AP under the assumption that the utility components for each pair-wise matching are stochastic. (Storms & Spieksma, 2003) showed geometric 3D-APs. (Bufardi, 2008) investigated the efficiency of feasible solutions/assignments of a multicriteria AP (MCAP). (Kuroki & Matsui, 2009) presented an approximation algorithm for MD-APs minimizing the sum of squared errors. (Odior et al., 2010) discussed a method for determining feasible solutions of a MCAP. (Goossens et al., 2010) developed the approximability of 3D-APs with bottleneck objective. (Sharify et al., 2011) presented solution of the optimal AP by diagonal scaling algorithms. (Perea, 2011) presented the applications and some thoughts of MD-APs. (Anuradha & Pandian, 2012) developed a new method (i.e., reduction method) for finding an optimal solution to 3D-APs. (Kavitha & Pandian, 2012) presented type-II sensitivity analysis in 3D-APs. (Ćustić et al., 2014) developed the planar 3D-APs with Monge-like cost arrays. (Frieze & Sorkin, 2015) have developed efficient algorithms for 3D axial and planar random APs. (Ćustić et al., 2015) gave geometric versions of the 3D-AP under general norms. (Mittelmann & Salvagnin, 2015) developed on solving a hard quadratic 3D-AP. Thus, many authors have proposed different approaches to solve the different types of APs when its parameter was in well known crisp numbers. That is, efficient algorithms have been developed for solving 3D-APs when the coefficient of the objective function (Z), jobs (Ji), machines (Mj) and factories (Fk), 
values are known precisely.