lect11

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Rensselaer Polytechnic Institute *

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Industrial Engineering

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Oct 30, 2023

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196 The Transportation Problem Recall the following example of a Transportation Problem: CCC has 1000 computers at each of three plants this month. Three customers have requested 1100, 800, and 1100 computers. These data are summarized in the table below, along with the cost of shipping one computer from each plant to each customer. As Distribution Manager, you have been asked to develop a least-cost shipping plan for CCC. Customer 1 Customer 2 Customer 3 Supply Plant 1 5 3 2 1000 Plant 2 4 7 8 1000 Plant 3 6 7 4 1000 Demand 1100 800 1100

197 Formulating the Trans. Problem A FIA is proposed now for solving a general transportation problem. To do so, however, it is assumed that the problem is balanced , meaning that total supply equals total demand. If not, then... Using vertices to represent the plants and customers, and edges to connect each plant to each customer to indicate the possibility of shipping computers between the two associated locations, problem formulation resulted in the desire to find a least-cost shipping plan in the following complete bipartite graph: P3 P2 C2 C3 C1 P1 800 1100 1100 1000 1000 1000 4 5

198 Handling Too Much Demand (Supplies) Plants Customers (Demands) 2 1 2 3 3 1 4 3 7 6 8 7 4 2 5 1000 1000 1000 1100 800 1600 The algorithm for solving the transportation problem requires that total supply = total demand. Question: What do you do if total supply < total demand? Dum 3000 3500 0 0 0 500 3500 Note: Shipping one unit from the Dummy Plant to Customer j means that Customer j will not receive one unit of their demand. Means that some customers will not receive all of their demand.

199 Handling Too Much Supply (Supplies) Plants Customers (Demands) 2 1 2 3 3 1 4 3 7 6 8 7 4 2 5 1500 1000 1000 1100 800 1100 The algorithm for solving the transportation problem requires that total supply = total demand. Question: What do you do if total supply > total demand? Dum 3500 3000 0 0 0 500 3500 Note: Shipping one unit from Plant i to the Dummy Customer means that Plant i will have one unit of supply not shipped. Means that some plants will not ship all of their supplies.

200 Solving The Trans. Problem Based on the steps of the simplex algorithm , a FIA is now developed for solving a balanced Transportation Problem and works with the following transportation tableau : C1 C2 C3 Sup. P1 5 3 2 1000 P2 4 7 8 1000 P3 6 7 4 1000 Dem. 1100 800 1100 Step 0 ( Initialization ). Find an initial feasible solution f 0 ∈ F and set k = 0. Step 1 ( Move ). Use a movement mechanism in an attempt to find a new feasible solution f k +1 ∈ F with Z ( f k +1 ) strictly “better” than Z ( f k ). If successful, move to f k +1 , set k = k + 1, and return to Step 1. Otherwise, stop. Recall how a general Finite Improvement Algorithm works: Cells

201 Finding an Initial Shipping Plan C1 C2 C3 Sup. P1 5 3 2 1000 P2 4 7 8 1000 P3 6 7 4 1000 Dem. 1100 800 1100 = cost 1000 100 1000 0 100 0 900 0 100 100 0 800 0 800 0 12600 Step 0 ( Initialization ). Find an initial shipping plan (tableau). One approach for doing so is a greedy algorithm that works with the cells (building blocks) of the transportation tableau, one at a time.

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