B-1: Formulate and solve the transportation problems using both manual methods and the Excel Solver, and interpret the solutions.

In supply chain network design, the location of factories, warehouses, and distribution centers is a strategic decision that affects costs. Consequently, companies spend a significant amount of time and resources in considering and evaluating alternatives to choose the one that is most suitable for them. Specifically, the transportation model can aid companies in minimizing the overall costs of transportation and distribution from the various supply sources to the demand destinations by making decisions about the location of their plants or supply sources relative to the demand destinations and decisions about how many units of a particular product should be transported from each supply origin to each demand destination, to satisfy the existing demand for the company’s products.

The transportation model is a special case of linear programming problems in which the objective is to minimize the total cost of transporting goods from the various supply origins to the different demand destinations. The model is often classified as a linear programming problem because the relationship between the variables transportation costs and the number of units shipped is assumed to be linear. There are important decision rules that must be applied to the transportation problem. The transportation model requires the following assumptions:

a. Capacity at each supply location or origin is limited.

b. The demand requirements at each destination are known.

c. Regardless of their origin or destination, the items shipped are the same (homogeneous).

d. Regardless of the number of units shipped, the shipping cost on a per unit basis remains the same.

e. Between each origin and destination, the mode of transportation being used does not change, and there is only one route used.

Two methods that can be used to obtain the initial basic feasible solution are the northwest corner rule and the matrix minimum cost method. The advantage of the northwest corner rule is that it allows us to find an initial feasible solution to the transportation problem. It emphasizes finding an initial solution that satisfies all constraints without regard to the relative shipping costs of those orders. The northwest corner rule generates feasible solutions but not necessarily an optimum least cost solution. The objective of the matrix minimum cost method is to minimize total cost, making this method more intuitively appealing. This method also reduces the number of computations and the time required to determine the optimal solution.

The stepping stone method is intended to generate improved solutions for transportation problems. The model begins with an initial feasible solution, and it is then used to determine whether that initial solution is optimal. The term stepping stone refers to the occupied cells in the initial solution of the transportation matrix, which are used in arriving at an improved solution.

Excel Solver offers a means for finding optimal solutions to the transportation problem. The procedure for solving transportation problems using Excel Solver is almost identical to the steps used for solving linear programming problems (see Module A).

B-2: Apply transportation modeling to other situations.

If a transportation problem is not balanced, it has to be converted into a balanced situation by adding dummy supply sources or dummy demand destinations in the transportation matrix. A transportation problem is degenerate if one of the occupied routes (cell) fully exhausts the supply from a source and meets the demand requirements for a destination. If degeneracy exists, it is impossible to apply the stepping stone method and it is impossible to trace a closed path for one or more of the unoccupied cells or routes. To overcome the problem of degeneracy, we need to create an artificially occupied cell.

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