A Modified Genetic Algorithm for solving uncertain Constrained Solid Travelling Salesman Problems q Samir Maity a,⇑ , Arindam Roy b , Manoranjan Maiti c a Dept. of Computer Science, Vidyasagar University, Medinipur, 721102 WB, India b Dept. of Computer Science, P.K. College, Contai, Purba Medinipur, 721401 WB, India c Dept. of Applied Mathematics, Vidyasagar University, Medinipur, 721102 WB, India article info Article history: Received 20 May 2014 Received in revised form 12 January 2015 Accepted 18 February 2015 Available online 9 March 2015 Keywords: STSP CSTSP Probabilistic selection Comparison crossover Modified Genetic Algorithm (MGA) abstract In this paper, a Modified Genetic Algorithm (MGA) is developed to solve Constrained Solid Travelling Salesman Problems (CSTSPs) in crisp, fuzzy, random, random-fuzzy, fuzzy-random and bi-random environments. In the proposed MGA, for the first time, a new ‘probabilistic selection’ technique and a ‘comparison crossover’ are used along with conventional random mutation. A Solid Travelling Salesman Problem (STSP) is a Travelling Salesman Problem (TSP) in which, at each station, there are a number of conveyances available to travel to another station. Thus STSP is a generalization of classical TSP and CSTSP is a STSP with constraints. In CSTSP, along each route, there may be some risk/discomfort in reaching the destination and the salesman desires to have the total risk/discomfort for the entire tour less than a desired value. Here we model the CSTSP with traveling costs and route risk/discomfort factors as crisp, fuzzy, random, random-fuzzy, fuzzy-random and bi-random in nature. A number of benchmark problems from standard data set, TSPLIB are tested against the existing Genetic Algorithm (with Roulette Wheel Selection (RWS), cyclic crossover and random mutation) and the proposed algorithm and hence the efficiency of the new algorithm is established. In this paper, CSTSPs are illustrated numerically by some empirical data using this algorithm. In each environment, some sensitivity studies due to different risk/discomfort factors and other system parameters are presented. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction The TSP was first formulated as a mathematical problem in 1930 and became increasingly popular after 1950. It is one of the most intensively studied problems in optimization even in recent years. A TSP is to find a possible tour along which a Travelling Salesman (TS) visits each city exactly once for a given list of cities and back to the starting city, so that total cost spent/distance cov- ered is minimal. TSP is a well-known NP-hard combinatorial optimization problem (Lawler, Lenstra, Rinnooy Kan, & Shmoys, 1985). Different types of TSPs have been solved by researchers dur- ing last two decades. These are TSPs with time windows (Focacci, Lodi, & Milano, 2002), stochastic TSP (Chang, Wan, & Tooi, 2009), double TSP (Petersen & Madsen, 2009), asymmetric TSP (Majumder & Bhunia, 2011; Mestria, Ochi, & Martins, 2013), TSP with precedence constraints (Moon, Ki, Choi, & Seo, 2002; Rakke, Christiansen, Fagerholt, & Laportei, 2012), etc. In TSP, it is assumed that a TS travels from one city to another using only one conveyance. But in real life, a set of conveyances may be available at each city. In that case, a TS has to design his/ her tour for minimum cost maintaining the TSP conditions and using the suitable conveyances at different cities. This problem is called Solid Travelling Salesman Problem (STSP). Traveling cost from one city to another city depends on the types of conveyances, condition of roads, geographical areas, weather condition at the time of the travel, etc., so there always prevail some uncertain- ties/vagueness. For this reason it is better to model the costs by uncertain parameters as fuzzy, random, random-fuzzy, bi-random and fuzzy random values. To analyses the large scale/amount of data throughout a long time interval, we observe that the data values are fluctuating over a period of time/year/session etc. So, for the decision making problem, twofold random phenomena is well suited/realistic approach. Also since TS may use different con- veyances to travel along different routes, there may be correspond- ing some risk/discomfort factors, which depend on the condition of roads, types and conditions of vehicles, law and order condition http://dx.doi.org/10.1016/j.cie.2015.02.023 0360-8352/Ó 2015 Elsevier Ltd. All rights reserved. q This manuscript was processed by Area Editor Mitsuo Gen. ⇑ Corresponding author. Mobile: +91 9874512894. E-mail addresses: maitysamir13@gmail.com (S. Maity), royarindamroy@yahoo. com (A. Roy), mmaiti2005@yahoo.com (M. Maiti). Computers & Industrial Engineering 83 (2015) 273–296 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie