QUANTITATIVE ANALYSIS FOR MANAGERS TRANSPORTATION MODEL LECTURE 3

Learning Objectives When you complete this chapter, you should be able to identify or define: Transportation modelling to explain or use: Northwest-corner method Least Cost method Vogel’s Approximation method Stepping-stone method LECTURE 3

Outline Transportation Modelling Developing an Initial Solution The Northwest-Corner Method The Least-Cost Method The Vogel’s Approximation Method The Stepping-Stone Method Special issues in modelling - Demand Not Equal to Supply LECTURE 3

Transportation Problem How much should be shipped from several sources to several destinations Sources: Factories, warehouses, etc. Destinations: Warehouses, stores, etc. Transportation models Find lowest cost shipping arrangement Used primarily for existing distribution systems LECTURE 3

A Transportation Model Requires The origin points, and the capacity or supply per period at each The destination points and the demand per period at each The cost of shipping one unit from each origin to each destination LECTURE 3

Transportation Problem Solutions steps Define problem Set up transportation table (matrix) Summarizes all data Keeps track of computations Develop initial solution Northwest corner Method Vogel’s Approximation Method Find optimal solution Stepping stone method LECTURE 3

Methods for finding Initial Solution 1. North-West Corner Method (NWCM) Begin in the upper-left-hand corner of the transportation table for a shipment and allocate as many units as possible equal to minimum between available capacity and requirement; i.e. min (a1, b1). Allocate the maximum that is possible, min (100, 90) = 90. Now move horizontally to the second column in the first row. Repeat the above steps

Example 1: A company has three factories F1, F2 and F3 with production capacity 100, 250 and 150 units per week respectively. These units are to be shipped to four warehouses W1, W2, W3 and W4 with requirement of 90, 160, 200 and 50 units per week respectively. The transportation costs (in Rs) per unit between factories and warehouses are given as follows: Warehouse W1 W2 W3 W4 Capacity F1 30 25 40 20 100 F2 29 26 35 250 F3 31 33 37 150 Requirement 90 160 200 50 Factory

The total cost of transportation is obtained as follows: Route Units per unit From To shipped X cost (Rs) = Total Cost F1 W1 F1 W2 F2 W2 F2 W3 F3 W3 F3 F4 Total =

2. The Least Cost Method Identify the cell with the lowest cost. Arbitrarily break any ties for the lowest cost. Allocate as many units as possible to that cell without exceeding the supply or demand. Then cross out that row or column (or both) that is exhausted by this assignment. Find the cell with the lowest cost from the remaining cells. Repeat steps 2 & 3 until all units have been allocated.

Total Transportation cost = Rs 15, 020 Condition: LCM Total Transportation cost = Rs 15, 020 Condition: Occupied shipping routes = (no. of rows + no. of columns) – 1 = 3 + 4 –1 = 6 This cost is less than the cost determined by NWCM. Therefore, this method is preferred over the NWCM.

3. Vogel’s Approximation Method (VAM) This method is preferred over the other two methods because the initial basic feasible solution obtained is either optimal or very close to the optimal solution. For each row and column, find the difference between the two lowest unit shipping costs. Identify the row or column with the greatest opportunity cost or difference. Assign as many units as possible to the lowest-cost square in the row or in the column selected.

W1 W2 W3 W4 Capacity F1 30 25 40 20 100 F2 29 26 35 250 F3 31 33 37 150 Requirement 90 160 200 50 500

Stepping-Stone Method Optimality Test Stepping-Stone Method This method starts with an evaluation of each of the unoccupied cells to decide whether it would be economical to introduce any of these cells into the current solution.

The Stepping Stone Method Apply any of the three methods to obtain the initial basic feasible solution Select any unused cell to be evaluated Begin at this cell. Trace a closed path back to the original cell via cells that are currently being used (only horizontal or vertical moves allowed) Place + in unused cell; alternate - and + on each corner cell of the closed path Calculate improvement index: add together the unit cost figures found in each cell containing a +; subtract the unit cost figure in each cell containing a -. Repeat steps 1-4 for each unused square LECTURE 3

Stepping Stone Method contd. Check the sign of each of the net change in the unit transportation costs. If all net changes are plus (+) or zero, then an optimal solution has been achieved, otherwise go to next step Select the unoccupied cell with most negative net change among all unoccupied cells. If two minus values are equal, select that one which will result in moving as many units as possible into the selected unoccupied cell with the minimum cost Assign the maximum unit that can be shipped on the new route. This is done by looking at the closed paths (-) sign and we select the smallest number found in the cells with (-) signs and make the transfer

Example 2: A company is spending Rs1000 on transportation of its units from three plants to four distribution centres. The availabilities and requirements of units with units cost of transportations are given as: Distribution Centres D1 D2 D3 D4 Availabilities P1 19 30 50 12 7 P2 70 40 60 10 P3 20 18 Requirements 5 8 15 What can be the maximum saving for the company by optimum distribution?

Special Issues in the Transportation Model Demand not equal to supply Called ‘unbalanced’ problem Add dummy source if demand > supply Add dummy destination if supply > demand Degeneracy in Stepping Stone Method Too few shipping routes (cells) used Number of occupied cells should be: m + n - 1 Create artificially occupied cell (0 value) Represents fake shipment LECTURE 3