Rough Genetic Algorithm for Constrained Solid TSP with Interval Valued Costs and Times

ABSTRACT This paper presents new rough set based genetic algorithms (RSGAs) to solve constrained solid travelling salesman problems (CSTSPs) with restricted conveyances (CSTSPwR) having uncertain costs and times as interval values. To grow the impreciseness in soft computing (SC), the proposed RSGAs, a rough set based age-dependent selection technique and an age-oriented min-point crossover are used along with three types of probability, p-dependent random mutations. A number of benchmark problems from standard data set, TSPLIB are tested against the proposed algorithms and existing simple GA (SGA). CSTSPwRs are formulated as constrained linear programming problems and solved by both proposed RSGAs and SGA. These are illustrated numerically by some empirical data and the results from the above methods are compared. Statistical significance of the proposed algorithms are demonstrated through statistical analysis using standard deviation. Moreover, the non-parametric test, Friedman test, is performed with the proposed algorithms. In addition, a post hoc paired comparison is applied and the out performance of the RSGAs.


Introduction
TSP (travelling salesman problem) is an NP-hard combinatorial optimisation problem [1]. During last two decades, several types of TSPs have been solved by the researchers. In TSP, a Travelling Salesman (TS) is sometimes asked to visit a given number of cities/nodes for which there are time windows for the earliest and the latest service start times. Focacci et al. [2] formulated a TSP with time window (TSPTW) and solved using a hybrid exact algorithm. Later, Chang et al. [3] extended the TSPTW with stochastic travel time and tight time-windows (TWs) called stochastic dynamic TSP with hard time windows (SDTSPHTD). To avoid congestion, the double TSP with multiple stacks (DTSPMS) was considered by Petersen and Madsen [4]. It finds the shortest routes performing pickups and deliveries in two separated networks/regions.
All the above-mentioned models assumed the edge costs to be symmetric. Majumder and Bhunia [5] formulated a TSP with asymmetric costs and imprecise travel times and solved using GA. Moon et al. [6] applied precedence constraints before visiting the nodes/cities in a TSP and solved using an improved GA. Xing et al. [7] presented a hybrid approach that combines an improved GA and optimisation strategies for solving the asymmetric TSP (ATSP). Bai et al. [8] proposed a max-min ant colony optimisation method for the solution of ATSPs bridging the gap between hybridisation and theoretical analysis. Jula et al. [9] considered a routing problem with stochastic travel times and time windows estimating means and variances of arrival times at nodes and removing routes that are dominated by others. Chang et al. [3] solved a stochastic dynamic TSP with hard time windows following more or less same procedures of Jula et al. [9]. Chang and Mao [10] developed a modified ant algorithm to solve TSPTWs for minimum cost tour. Dong et al. [11] proposed a new hybrid algorithm, cooperative genetic ant system to solve TSP. Yuan et al. [12] proposed a new crossover operator called two-part chromosome crossover for solving the multiple travelling salesman problem (MTSP). Recently, Miranda Bront et al. [13] formulated and solved a time-dependent travelling salesman problem (TTSP). Perturbed Decomposition Algorithm applied for multi-objective TSP by Cornu et al. [14].
In spite of the above different types of developments in the area of TSP, there are some lacunas in the realistic formulation of the problem and development of solution techniques. These are as follows.
In TSP, it is generally assumed that a salesman 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 salesman has to design his/her tour for minimum cost/time/distance maintaining the TSP conditions and using the suitable conveyances at different cities. This problem is called Solid Travelling Salesman Problem (STSP). In realistic situations, all conveyances may not be available from all places/ stations. According to the availability of the conveyances, the model is set up with some restricted conveyances at each station, i.e. in each node, equal number of conveyances may not be available to travel from one station to another. All the above-mentioned earlier TSP formulations considered only one conveyance for travel between different cities. Here we have considered several different conveyances at different stations.
Travelling cost and time from one city to another city depend on the several factors such as types of conveyances, condition of roads, geographical areas, weather condition such as rainy/foggy weather at the time of the travel, conditions of vehicles and so on. So there always prevail some uncertainties/vagueness. Under these circumstances, time and cost parameters may be considered as imprecise in nature in the form of uncertain interval values instead of a fixed (deterministic) values. Except Changdar et al. [15], earlier investigations did not consider TSPs with imprecise/uncertain costs and/or times. Here we have formulated STSPs with interval valued costs and times in the objective and constraint.
Moreover, a salesman may be asked to complete the entire trip within a specified time which acts as a constraint on the model. Such kind of constraint is called time constraint, which may also be taken as uncertain interval values. In realistic approach of TSP, as the conveyances are restricted, we formulate CSTSP with restricted conveyances say CSTSPwR. Till now, none has considered such CSTSPwR in crisp or imprecise environments.
Basically to grow the impreciseness in SC methods that prompted us to design such kind of inclusions of rough set in the proposed GA. The uncertainty is come by nature in realistic situation of the proposed model also. Some historical study is presented in Table 1. In our proposed RSGAs, we consider age of each chromosome in the form of a rough set/extended rough set as a selection operation which provides the creation of the probability of crossover (p c ). Thus p c is a function of the chromosome 'age'. For crossover operation, we determine the convex combination of the interval values, after modifying the parents, compare the costs between each node of the pair of chromosomes and the best one is concatenated in the new route/ offspring. For mutation operation, a generation dependent new probability of mutation (p m ) is created and with this p m , total number of nodes to be mutated are calculated. Then identified nodes/positions/locations are randomly selected and mutated within a chromosome/path.
In this paper, CSTSPwRs are formulated with asymmetric parametric crisp interval values for time and costs. A total time is imposed for the entire tour in the form of a constraint. CSTSPwR formulated in different environments are solved by both proposed RSGAs and SGA for some empirical data set. Alternative near-optimum paths along with optimum time and cost are presented for each CSTSPwR.
To establish the efficiency of the RSGAs, some standard test problems from TSPLIB [42] are solved by both RSGAs and SGA and their performance are compared. The transformed CSTSPwRs are also solved using both proposed RSGAs and SGA. These models are illustrated with some numerical data and the results from both methods are compared. In crisp environment, some sensitivity analyses are performed with different allowable times. Through standard deviation (SD) and error analyses, the non-parametric Friedman test is applied also post hoc paired analyses is done.
The present problem is more complicated due to the uncertainty/impreciseness of the costs and times through uncertain interval values. As optimisation of interval valued objective functions is not well defined, it is very difficult to formulate and optimise these. Tiwari et al. [43] proposed fuzzy closer and fuzzy-rough set. We use fuzzy possibility and necessity based approaches [44,45] to represent and to solve the CSTSPwR with uncertain interval values. Thus present investigation contains the following new features: • CSTSPwR are more realistic TSPs. None considered this type of uncertain problems with interval valued parameters. • For the first time, new algorithms of RSGAs with 'Rough set based selection', 'min-point crossover' and generation-dependent mutations are developed. • New RSGAs are tested with different data sets from TSPLIB [42] and the efficiency of the proposed algorithms is established. • Here objective function is formulated using geometric mean (this is a new concept) and constraint values are compared using different interval ranking procedures. • Non-parametric Friedman test is performed with the proposed algorithms to establish the differences amongst the algorithms. • A post hoc paired comparison is made to determine the degree of differences and thus the proposed algorithm outperforms the others.
This paper is organised as follows. In Section 2, we describe mathematical preliminaries with definitions. Section 3 gives different types of CSTSPwRs. In Section 4, RSGAs are presented. Finally, we illustrate the above problems using some empirical data in Section 5. In Section 6, some statistical tests using SD are performed. Last, in Section 7, we conclude the paper identifying the scope of future development.
In next section, some mathematical requisite are described.

Interval number
A (real) interval is a nonempty set of real numbers where a L and a R are the lower and upper bounds (end points) of the interval number A, respectively. A is a generic (arbitrary) element in the interval A. denotes the all real numbers. A is a more complex expression, the lower bound and the upper bound are also written as The set of all interval is denoted by IR. To represent an unknown number as an approximation plus/minus an error bound, the midpointǍ and with of an interval A are respectively introduced asǍ Hence A can be wid(A) represented aś The operations on intervals used in this paper may be explicitly calculated for two interval numbers, Let the uncertain costs from two alternatives be represented by two closed intervalsÁ = a c , a w ,B = b c , b w respectively. It is also assumed that the cost of each alternative lies in the corresponding interval. These two intervalsÁ andB any be of the following three types: Type I: Both the intervals are disjoint. Type II: Intervals are partially overlapping. Type III: One interval is contained in the other.
In optimistic decision making, the decision maker (DM) expects the lowest cost ignoring the uncertainty. According to Majumder et al. [5], the order relations of the interval numbers for minimisation problems in case of optimistic decision making are as follows:

Definition 2.2 (Majumder et al. [5]): Let us define the order relation ≤ omin between
Pessimistic decision making For pessimistic decision making, the DM expects the minimum cost for minimisation problems according to the principle ' Less uncertainty is better than more uncertainty'. According to Karmakar et al. [44] and Majumder et al. [5], the order relations of interval numbers for minimisation problems in case of pessimistic decision making are as follows: However, for Type III intervals with (a c ≤ b c )∧(a w < b w ), the pessimistic decision cannot be taken. Here, the optimistic decision is to be considered.

Remark 2.4:
Now as the interval valued objectives are not well defined, so we use common features of arithmetic mean (AM) and geometric mean (GM) as follows. Let A = [a L , a R ] be a common interval for a particular objective function. Since we know that for a minimisation problem, determine the minimum of the objective function (a m 1 L * a m 2 R ) 1/(m 1 +m 2 ) , here m 1 and m 2 are the given weights.

Optimisation problem with interval values
An optimisation problem can be stated as follows: whereĉ j ,â i j andb i are interval values with known lower and upper bounds. Then the above equation is transformed as where The interval valued c, c are the upper and lower bound values ofĉ, similar values are taken forâ andb. Now as the above objective functions are not well defined, so it is converted according to Equation (4) and interval valued constraints are represented with different approaches [46] as Following different approaches for transformation on an interval constraint. According to Moore's approaches [46], the above constraint in Equation (7) can be written as following Ishibuchi and Tanaka's approaches [46] subject to due to Chanas and Kuchta's approaches [46] for 0 ≤ s 0 < s 1 ≤ 1, by Hu and Wang's approaches [46] subject to and using Mahato and Bhunia's approaches [46] subject to (optimistic case) In every case of the constraint evaluation from the above equations (8)- (12) x j ≥ 0, for j = 1, 2, . . . , n.
Trust measure (cf. Mondal et al. [47]) Let ( , , κ, ) be a rough space. The trust measure of event A is denoted by Tr{A} and defined by Tr{A} Then we can get the trust measure of the rough eventξ ≥ r as Tr{ξ ≥ r} and its function curves ( Figure 1) are presented below where r is a crisp number,ξ is a rough variable given   In this paper, we introduce a new mathematical extension on the rough intervals. In the earlier research papers, rough intervals are taken as three subintervals ( Figure 1 and Figure 2) but in our present investigation, we consider a modification/ refinement of the rough intervals. Here we consider in Equations (15) and (16) five subintervals on the rough intervals ( Figure 3 and Figure 4). If the interval is divided into more regions, then the trust values ofξ are a rough variable In dividing the interval [c, d] into five subintervals instead of three intervals, we get more specific small intervals and define the age of the chromosomes by five linguistic words in the said intervals. Later identifying the parents by five words, we get several types of their combinations and five types of offsprings, where only three types of children were produced for three subdivisions of [c, d].
In the following section, mathematical formulation of the proposed problem is presented.

Classical TSP with time constraints (CTSP)
In a classical two-dimensional TSP, a salesman has to travel N cities at minimum cost. In this tour, salesman starts from a city, visit all the cities exactly once and comes to the starting city using minimum cost. Here time in travelling from one city to another is considered. The salesman should choose such a path in which the maximum allowable time limit is maintained. Now when the cost and times are interval numbers, then the classical TSP mathematically written as Then the above CTSP reduces to determine a complete tour (x 1 ,

STSP with time constraints (CSTSP)
In an STSP, a salesman has to travel N cities by choosing any one of the P different types of conveyances available using minimum cost maintaining maximum allowable time. Time taken to travel from one city to another using different conveyances are different. Let c(i, j, k) be the cost for travelling from ith city to jth city using kth type conveyance and t(i, j, k) be the time taken for this travel. Then the salesman has to determine a complete tour (x 1 , Determine a complete tour (x 1 , x 2 , . . . , x N , x 1 ) using any one of the available conveyance in each step from the vehicle types (v 1 , v 2 , . . . , v P ) so as

STSP using restricted conveyances with time constraints (CSTSPwR)
In real life, it is seen that in all stations, all types of conveyances may not be available due to the geographical position of the station, weather conditions, etc. So it is more realistic that restricted conveyances are available in different stations. Considering the availability of the conveyances, we design the STSP with restricted condition with time constraints as below.
Let c(i, j, k) be the cost for travelling from ith city to jth city using kth type conveyance and t(i, j, k) be the time taken in travelling from ith city to jth using kth type conveyance. Then the salesman has to determine a complete tour (x 1 , provides maximum available S( ≤ P) types of conveyances. Then the problem can be mathematically formulated.
Determine a complete tour (x 1 , x 2 , . . . , x N , x 1 ) using any one available corresponding conveyance in each step from the vehicle types (v 1 , v 2 , . . . , v S ) so as Thus the above model is written as and The interval valued objective and constraint are transformed as given in Section 2 following Karmakar et al. [44]. Thus the crisp version of the above model is , for m 1 , m 2 ∈ (0, 1) with one of the following constraints.
(pessimistic case) Now we go to the proposed algorithm.

Simple genetic algorithm (SGA)
GA provides a general-purpose search methodology, which uses principles inspired by natural genetics to evolve solutions to problems [18,21]. The SGA starts off with a population of randomly generated chromosomes, each representing a candidate solution to the problem and advances towards better chromosomes by applying genetic operators. An SGA for a particular problem must have the procedure such as follows.

Rough set based genetic algorithms (RSGAs)
Here RSGA with the rough set-based age-dependent selection, min-point crossover and p m -dependent random mutation is developed and used, among a set of potential solutions to get a new set of solutions. As usual, it is continued until terminating conditions are encountered.

Representation
Here a complete tour on N cities represents a solution. So an N-dimensional integer vec-

Rough set-based age-dependent selection.
In Last et al. [34], for the first time, an attempt was made to improve the performance of genetic algorithms by providing a new fuzzy-based extension of the Life Time feature. They used a Fuzzy Logic Controller (FLC) to adopt the crossover probability as a function of the chromosomes age. Also Fdez et al. [36], Roy et al. [35] used it in some refinement of the mechanism on inventory control system. They used the age of the chromosome in fuzzy environment. But, for the first time, we model the age in rough environment as rough set is more uncertain than fuzzy set in the uncertain paradigms. So rough set-based age is more effective. The general principle is that for both young and old individuals, the crossover probability is naturally low, while there is a certain age interval, where this probability is high. The concepts of young, old and middle-aged are modelled as linguistic variables. Again we extend these linguistic variables in rough environment.
Here, the concept of representation of age using linguistic variables in both fuzzy and rough environments are same, but the mathematical representations are different.
Moreover, the classification of linguistic variables by earlier authors [34][35][36] has been extended in the present analysis.
In our proposed RSGAs, the age of a chromosome is determined based on their fitness values and then a 'rough set-based age-dependent selection' (REA) technique is applied. Here the age of each chromosome lies in a region of the common age represented by a rough set. These regions are termed as young, middle and old for RSGA-I. So for the age of each chromosome, a linguistic value -young, middle or old -is created. Now according to the age distributions of the members (in pair) of the mating pool, similar linguistic variables such as low, medium and high are generated for the said chromosomes to fix p c 's. Using the trust measures of rough set, the probability of crossover, p c for each chromosome is assigned by the corresponding linguistic variables. Now M solutions in a generation with fitnesses represented by f (X i ) of the ith chromosomes. At the time of initialisation, each chromosome age is defined as null. Now in every generation the age is counted as using the following mechanism : where avgfit is the average fitness values, maxfit and minfit are maximum and minimum fitness values of the last generation and k = (maxfit + minfit)/2 . Also avg(age) means the average age of the set of chromosomes. Here the maximum and minimum ages depend on the requirement of the problems.
Outcomes of different types of ages are given in Equation 27. The above equation (27) shows that if common rough age region is ([a, b], [c, d]) , then the space c to a refers to as young age, a to b as middle age and b to d as old age. Also the pictorial representation is given in Figure 5. Then in Table 2, according to the linguistic values, rough trust-based probability of crossovers (p c ) is assigned which is shown in Figure 6.

Rough extended age-based selection.
To have more accurate classification, we make five classifications instead of above three and then, the region of common age is divided into very young, young, middle, old and very old for RSGA-II. As before, combining the eligible parents, the very low, low, medium, high and very high linguistic variables are assigned for p c 's of chromosomes. Now we consider the age in a different extended linguistic code, i.e. Young, Middle and Old, are replaced by Very Young, Young, Middle, Old and Very Old scales. So it is more realistic in the sense of classification and acceptable to design for the real-world problems. According to the requirement of the five linguistic values, we expanded the trust measure levels in five sections which are shown in Equations (15) and (16). Determined p c values of the extended linguistics are also given below in Figures 7 and 8.
According to the extended age of the chromosome in Table 3, it belongs to any one of the common rough age intervals such as Very Young, Young, Middle, Old and Very Old. The common rough age ([a, b], [c, d]) is extended to 0 ≤ c ≤ c 1 ≤ a ≤ b ≤ c 2 ≤ d and is

Min-point crossover 4.2.2.1. Crossover mechanism.
To select two individuals (parents) from the matting pool, generate the random number between [0,1]. If r < p c , then select that population for first parent (say P r 1 ). Similarly choose the another parents (say P r 2 ). Let these are P r 1 : a 1 , a 2 , . . . , a N and P r 2 : s 1 , s 2 , . . . , s N . Here (a 1 , a 2 , . . . , a N ) and (s 1 , s 2 , . . . , s N ) are nodes within (1, 2, 3, . . . , N ), and these are numbers of cities. Then choose a city randomly from 1 to N , say a i = s p (i = 1, 2, . . . , N ), p = (1, 2, . . . , N ). Parents are modified by placing a i or s p in the first place of P r 1 and P r 2 . Now modified parents are given by P r 1 : a i , a 1 , a 2 , . . . , a i < − < / 1 , a i < + < / 1 , . . . , a N , P r 2 : s p , s 1 , s 2 , . . . , s p < − < / 1 , s p < + < / 1 , . . . , s N . To get the first child (Ch 1 ), placing a i in the first place of Ch 1 , compare the next route a i to a 1 and a i to s 1 , minimum cost route be selected in Ch 1 . As considered the uncertain interval values, so we determine the values as convex combination of the upper and lower interval values, such as for any route cost between two nodes a i to a 2 becomes as c (a i , a 1 ) = λ * CL i1 + (1 − λ) * CR i1 , λ ∈ rand[0, 1], CL and CR, lower and upper values of the corresponding intervals between two node and then compare with minimum values which are defined as min-point values.
The procedure is as follows: Thus by the repetition of this process, find the first child Ch 1 : a i , s 1 , a 1 , . . . , a N (say). Now for the second child, we modifying the parents by placing a i or s p at the end of P r 1 and P r 2 . The modified parents are given by P r 1 : a 1 , a 2 , . . . , a i < − < / 1 , a i < + < / 1 , . . . , a N , a i and P r 2 : s 1 , s 2 , . . . , s p < − < / 1 , s p < + < / 1 , . . . , s N , s p . For the second child (Ch 2 ), placing a i in the first place of Ch 2 , compare the next route a i to a N and a i to s N , minimum cost route be selected in Ch 2 . According to first child it also form second child Ch 2 : a i , s N , a N , . . . , a 1 (say). Hence by the above mechanism, using two modified parents (P r 1 ) and (P r 2 ), find the two children (offspring) Ch 1 and Ch 2 .

Three different forms of p m dependent random mutations 4.2.3.1. Selection for mutation.
For each solution of P(t), generate a random number r from the range [0, 1]. If r < p m , then the solution is taken for mutation where p m be the probability of mutation.

Mutation process. At first determined the total number of mutated node (T ).
To mutate a solution X = (x 1 , x 2 , . . . , x N ), number of mutated node T = p m * N, N = total number of nodes. (iii) Generation oriented mutation (variable method): Here we model a new form of mutation mechanism where probability of mutation (p m ) is determined as follows: So, here proposed mutation mechanism follows the real-world demand and p m decreases smoothly as generation may increase. After calculating the p m , then mutation operation works both of the two methods, Type I and Type II.   Table 5.

Algorithm for RSGA
Some numerical experiments are done in the following section.

Testing with problems from TSPLIB [42]
To validate the feasibility and effectiveness of the proposed algorithms, we apply the proposed RSGAs on some standard TSP problems taken from TSPLIB [42]. The proposed algorithm was implemented in C++ with following parameters as 100 chromosomes and 2000 iterations (maximum). The best optimal results are presented. Table 4 gives the results of the test problems using RSGA-II and SGA, the results are compared in terms of optimal cost, iterations and computational time (CPU time in minutes) also individual run of each algorithm are presented. It is seen that the number of iterations and computational times are less in RSGA-II than SGA. In each instance, average result (Avg), best found results (Cost) and the SD are presented. Here BKS stands for as "best known solution" in the literature.

Comparison of results of RSGAs with respect to different types of operators
Moreover, for a particular test problem bayg29, both SGA and proposed RSGAs are used with different operators and parameters (p c 's, p m 's, p s 's). The obtained results are presented in Tables 5 and 6. Here in each case 25 individual run with best found result 1610 is considered.
In Table 5, we survey the importance of different types of selection, crossover and mutation parameters in proposed algorithms. It indicates that for the optimal solution of the standard TSP data bayg29, optimal result is found in the rough extended age-based selection mechanism with min point crossover and fixed mutation. These results are obtained quickly by 64 iterations only. Here also, RSGAs perform better than SGA. In this testing,   "probabilistic" selection takes less generations than that required for " Roulette Wheel" selection. Again keeping every thing same, with RW selection, higher value of p c requires more number of generations and hence it is undesirable. In Table 6, optimum results for the standard TSP, " bayg29" are obtained in different environments using different selection and mutation techniques. It is observed that though all approaches furnish the same optimum result, the RSGAs with Min-point crossover and

Experiments for CTSP with and without time constraint
Here we consider a deterministic TSP of 10 × 10 size given by Equation (20), whose cost and time matrices are given in Table 7.
For the above input data, the problem given by Equation 22 is solved by RSGAs and SGAs and the optimum results are presented in Table 8. Here we took maximum generation = 100, independent run of each algorithms and deterministic constraint obtained using only Moore approaches. In all cases, 20 individual run done.
From Table 8, it is observed that for the cases of CTSP having " without T max " and same " T max ", the better results are obtained with respect to the cost as well as number of generations by RSGA-II and variable mutation. Also when T max decreases, corresponding cost increases according to the realistic conditions of the present investigation.

Experiment for CSTSPwR with time constraint
Now for a CSTSPwR, we consider maximum available three types of conveyances. The cost and time matrices for the CSTSPwR of 10×10 size are presented below in Table 9.
Here we took maximum generation as 200 with 20 independent runs, and for proposed RSGA-II, REA selection with variable mutation is used. Only feasible constraints that are satisfied only in the corresponding interval are considered. The optimum results are presented in Table 10.
Comparing the corresponding results from the above table, we see that Table 10 supports the usual expectation, i.e. as the total travel times decrease, the corresponding costs increase for all approaches. Here, the evaluated costs are of dominated types expect one or two, i.e. lower and upper values of the cost intervals with higher T max are larger than the corresponding values with lower T max . Considering all the approaches, Moore's approach with RSGA-II furnishes the lowest travel cost. This cost is much less than the corresponding costs with SGA-I for both values of T max .

CSTSPwR with virtual data set (for large TSP)
For the large-scale data set, we randomly generate the costs within a range. Here, costs c Lij and c Rij (i = j) are taken for first, second and third conveyances respectively as follows:  Table 11.
From Table 11, it is evident that for all sizes of CSTSPwR, Moore's approach gives the best results, i.e. minimum cost is less than the other four approaches. Moreover, with the sizes of the cost matrices, the optimum cost increases which is obvious and expected. For comparison, here SGA-I with Moore's approach is used and it is seen that RSGA-II fetches much better results than SGA-I for all sizes of cost matrix. SGA-I gives the worst result than all other approaches for a particular size of cost matrix.
In next, some statistical tests are done.

Against different test problems only
Performance of the proposed method is statistically tested running it 25 times and calculating the average value, standard deviation and percentage relative error according to optimal solution against some standard test problems. The results obtained by the Here BKS stands for best known solution. Examining Table 12, it is concluded that the proposed method, RSGA-II has generated the closer results to the optimal solutions with minimal standard deviations for the problems fri26, bays29, bayg29, dantzig42, eil51, berlin52, st70, pr76, rat99, Lin105 and Eil101. It can be seen that except one problem kroa200, for all other 14 problems, best results by RSGA-II are the same as the corresponding best results in the literature.

Against different test problems and different algorithms
In Table 13, average value SDs and the corresponding errors have been calculated for 11 problems using 7 methods. In all cases, the average results given by RSGA-I and RSGA-II are less than the corresponding average results by SGA-I, II, III, IV and V. Moreover, as the SDs in RSGA-I and -II are quite small except three cases, it indicates that these methods are stable, results in different runs do not differ much from the mean. We also obtained the least percentage relative error in different cases. These errors are also very small indicating that derived average solutions are nearer to the best known solution in the literature. Thus the proposed methods have produced closer results to optimum.

The Friedman test
To compare the performance of the algorithms SGA-I, II, III, IV, V, RSGA-I and RSGA-II, we perform the Friedman test (cf. Derrac et al. [48]). Since it is a non-para metrical statistical procedure whose main aim is to detect significant difference between the behaviour of two or more algorithms. The assumptions of Friedman test are as follows: • The results over instances are mutually independent (i.e. the results within one instances do not influence the results within other instances) • Within each instance, the observations (average objective values) can be ranked.
Hypothesis: H 0 : Each ranking of the algorithms within each problem is equally likely (i.e. there is no difference between them) H 1 : At least one of the algorithms tends to yield larger objective functions than at least one of the other algorithms.
Here number of algorithms (k) = 7 , number of instances (b) = 11. The Friedman ranking table is given below which is prepared according to the average results of Table 13.
Since T 2 > F 0.99,6,60 , we reject the null hypothesis. Hence there exist some algorithms whose performances are significantly different from others.

(Post hoc) paired comparisons
Here the algorithms i and j are considered different after the rejection of the null hypothesis from the Friedman test. Following the post hoc paired comparison technique (cf. Derrac et al. [48]), calculate the absolute differences of the summation of the ranks of algorithms i and j and declare i and j different if : where t 1−(α/2) is the 1 − (α/2) quantile of the t distribution with (b − 1)(k − 1) degrees of freedom. Here t 1−(α/2) for α = 0.01 and 60 degrees of freedom is 2.660 and the critical value for the difference is 2.66[(2 * 11(1540 − 1508.36))/((11 − 1)(7 − 1))] = 9.06. The following table summarises the paired comparisons, underline values indicated that the algorithms are different.
From Table 15, we conclude that RSGA-I and RSGA-II have outperformed than all other algorithms and RSGA-II is the best out performer amongst the other algorithms.
In the last section, overall conclusion and future plan is described.

Conclusion
Here, for the first time, new GA (RSGAs) has been proposed with rough set-based selection, min-point crossover and generation-dependent mutation processes. Here rough set-based age-dependent selection with three and five (extended) classifications, min-point crossover and three different p m -dependent mutations are developed. For STSPs, p m -oriented random mutation accelerates to get wide variety of node combinations. If p m is high, then the mutation rate is also much high. So it is in fine tuning to the optimisation problem, particularly this type of node oriented problems such as TSP, vehicle routing problem, network optimisation, etc. Again in accounting the complexity in mutation mechanism, type I is much high against the type II because in type I, random exchange occurs with searching the node in each step of the mutation whereas type II does no searching in location exchange. But type I is more affective to find the global optimum. For type III, its complexity is better against other two and efficiency is much high. With these new features, RSGAs are used for test problems from TSPLIB and its efficiency is proved. The supremacy of RSGA is established through the Friedman test and post hoc paired comparison. Later, two TSP problem-constrained TSP and constrained solid TSP are solved and the optimum results along with near optimum results are presented. The developed RSGAs are quite general, these can be used for the decision making problems in other areas such as inventory control system, supply chain and portfolio management. Moreover, RSGAs will be very useful for the large problems with large-scale data. The proposed RSGAs can be extended/modified to be applied for the optimisation of multi-objective problems.