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cities is completed and that going from city i directly to i is not permitted, which means c = 0, The equivalent production problem is. n items are to be produced on a machine in continuation The set-up cost if item / is followed by j is er v,= 1 if item i is followed by ; directly and zero otherwise. Each item is to be produced only once, t.e. item i is not to be produced again after item Ite. C, = .0. Thus the travelling salesman problem or the multiple product production problem can be put in the form of an assignment problem as shown in table 4. 201 TABLE 4.201 To city (or item) f45 C2 C. 2C1 cy . From city (or item) 3 C1 C2 ! 2 3 Co C. c. Cica It may be noted that there must be only one x, for each value of and each value of j. The problem is to find the n elements, one in each row and in each column so as to minimize the sum total of the associated costs shown in the matrix above. The problem is similar to the assignment problem discussed earlier with the difference that there is the additional constraint that no city is to be visited again before the tour of all the cities is completed (or no item is to be produced twice till all the items have been produced) The above assignment problem can be solved and it can be hoped that the solution will satisfy the additional constraint. If it does not, it can be adjusted by inspection. The method, however, works well for small problems only. For large problems, a more systematic approach, developed by J.D.C. Little and his colleagues is used. EXAMPLE 4.10-1 A salesman wants to visit cities 1, 2, 3 and 4. He does not want to visit any city twice before completing the tour of all the cities and wishes to return to his home city, the starting station. Cost of going from one city to another in rupees is given in table 4.202. Find the least cost route. TABLE 4.202 To city 1 2 3 4 1 0 30 80 50 2 40 0 140 30 From city 3 40 50 0 20 4 70 80 1300