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Class_Notes_Part_4 - Class Notes for DMOR Set 4...

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Class Notes for DMOR: Set 4 Introduction to Network Optimization Algorithms 1
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Key Idea: What’s Hard for You is Hard for the Computer Consider six project leaders: Amanda, Bert, Elvira, Grover, Layla, and Sara. There are five projects that must be completed by the company. Each leader can take on one project, so one leader must be left out. Skills differ among the leaders, and some leaders will perform more efficiently on projects than others. The company wishes to pick which five leaders will be assigned to which project. The goal is to minimize the total amount of time necessary to complete all projects. Below is a matrix of how fast each leader can complete each project (in terms of days). Project 1 Project 2 Project 3 Project 4 Project 5 Amanda 27 20 31 31 20 Bert 14 21 19 41 19 Elvira 18 25 15 24 16 Grover 21 19 14 27 22 Layla 33 16 29 33 15 Sara 17 19 25 39 10 Formulation: 2
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Is this a Network Problem? Remember, we need nodes and arcs, capacities and costs, supplies and demands. There is an easier way of dealing with network flow problems than by using the simplex algorithm directly. But this one is even easier! 3
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Assignment Problem This is a special type of problem called the “Assignment Problem.” The AP assigns a set of n people to a set of n tasks in order to minimize total cost. This is often described as assigning jobs to machines, and so on. There is a specialized algorithm for the Assignment Problem called the Hungarian Algorithm. First, formulate the cost matrix. It needs to be n × n . We will have to assign one person to a “dummy project.” That person will not take on any project, and hence won’t contribute to the objective function. Project 1 Project 2 Project 3 Project 4 Project 5 Dummy Project Amanda 27 20 31 31 20 0 Bert 14 21 19 41 19 0 Elvira 18 25 15 24 16 0 Grover 21 19 14 27 22 0 Layla 33 16 29 33 15 0 Sara 17 19 25 39 10 0 Step 1: Scan each column, and identify the smallest cost, Δ. Subtract Δ from each entry in the column. For instance, in column 1, we will subtract 14 from every entry in that column. The reason why is that we have to incur at least 14 days for this project, regardless of who takes it. What’s remaining in that column is the time beyond the shortest possible completion time. Step 2: Scan each row, and identify the smallest cost, Δ. Subtract Δ from each entry in the row. Steps 1 and 2 can of course be done in any order. In the above example, we won’t be able to subtract anything from the rows, because Δ = 0 in each row. 4
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Hungarian Algorithm for the Assignment Problem, continued. Step 3: Look for an assignment of n people to n tasks, such that all costs are zeros. Does one exist? If yes, then that’s your optimal solution. If not, go to Step 4. Project 1 Project 2 Project 3 Project 4 Project 5 Dummy Project Amanda Bert Elvira Grover Layla Sara Step 4: Assign as many people to tasks as you can, using only zero costs. Say you get q of these assignments. Based on your assignment, draw q lines (some horizontal, some vertical) through the cost matrix so that each zero is covered by at least one zero. Now find the smallest element that is not covered by any lines, and call that element Δ. Subtract Δ from
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