There are a number of typical models in the Operations Research field which can be applied to a wide range of supply chain problems. In other words, by learning a typical model various problems in...

Table of Contents
BACKGROUND:    1
MATHEMATICAL MODEL:    3
STEPS IN SOLVING A TRAVELLING SALES PROBLEM (TSP):    4
LITERATURE REVIEW:    8
Application of Travelling Sales Problem:    8
Metaheuristic solution procedure:    8
Artificial neural network:    8
Discrete version of the Cuckoo Search:    9
Different application of Traveling Salesman Problem:    9
Sequential Constructive crossover:    10
Analysis:    10
Suitability of using Traveling Salesman Problem model in supply chain analysis:    10
Limitations:    12
REFERENCES:    13
BACKGROUND:
Supply chain management is one of the rising concerns in the business houses. Maintaining the supply of the goods and services for the consumers is becoming extremely different in the dynamic business scenario. The business houses need to be more competitive to be in the industry and sustain in the long run. There are several problems associated with the maintenance of the supply chain like job shop scheduling, travelling thief problem, multiple traveling salesman problem (mtsp), flow shop scheduling, hamiltonian path problem, knapsack problem, parallel machine scheduling, etc. Solutions for these problems need to be found by the business houses to sustain in the long run. For this research paper, the discussions will be limited to the Travelling Sales Problem (TSP).
Travelling Sales Problem (TSP), is a problem faced by most of the companies. The problem include that with a given set of cities and the distance between the cities, the shortest possible route that can be taken to visit each of the city and then return to the starting point. It is one of the important problems in combinatorial optimization, operations research and theoretical computer science. The decision for this problem is to decide upon the correct route graph. Along with the increase in the number of cities, the worst case running time also increases. The problem date back to 1930 and is often used as the benchmark for a lot of optimization methods.
Reaching of the sales person to the largest possible number of consumers is important for the organization. The number of sales persons available with a company is also often limited and so it is very essential that those sales people save time and reach as many people as possible. Cost is also an issue. Companies have budget constraints and limited amount can be spent on the travelling of the sales persons. Often TSP is the sub problem in areas of planning, logistic, DNA sequencing, etc.
MATHEMATICAL MODEL:
TSP can be planned using an integer linear program. There are two notable formulation for the solving of TSP, which are Dantzig-Fulkerson-Johnson (DFJ) formulation and Miller-Tucker-Zemlin (MTZ) formulation. Here, the formulation by Dantzig-Fulkerson-Johnson has been used for the evaluation.
The main objective for the formulation of the function is to decide the shortest route possible between a set of cities and the origin city. The cities need to be leveled as 1, 2 up to n. It is assumed that Cij is the distance between city i and j.
Min
-
Where, 1 if i = s
−1 if i = t
0 else
xij ∈ {0, 1}
∀i ∈ V
∀i ∈ V
∀(i, j) ∈ A
STEPS IN SOLVING A TRAVELLING SALES PROBLEM (TSP):
There are several methods which can be adopted for the solving of the problem like exact algorithm, heuristic algorithms, etc. But for the purpose of this problem the method of exact algorithm has been used.
As per the method of exact algorithm all the possible permutations can be tried and then it can be seen as which one is the cheapest. A polynomial factor can be chosen like O(n!).
For example, there are four cities which a salesperson needs to travel. It can be assumed that there are seven cities.
The possible solution for this problem can be:
1. Any city can be considered to be the starting point but the same city will have to be the ending point.
2. All (n-1)! permutations of the cities need to be found.
3. Cost of every permutation need to be calculated and the permutation with minimum cost need to be checked.
4. Finally the permutation with the minimum cost is to be chosen.
The dynamic programming of the problem can be set as:
If size of S is 2, then S must be {1, i},
C(S, i) = dist(1, i)
Else if size of S is greater than 2.
C(S, i) = min { C(S-{i}, j) + dis(j, i)} where j belongs to S, j != i and j != 1
The difference between the seven different cities can be as...