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## Session 4a

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**Overview**More Network Models • Assignment Model (Contract Bidding) • “Big Cost” trick • Project Management (House Building) • More binary / integer tricks • Critical Path / Slack Time • Excel trick: Conditional Formatting • Cost Crashing • Changing an objective to a constraint • Issues with Integers • Location Analysis (Hospital Location) Decision Models -- Prof. Juran**Contract Bidding Example**A company is taking bids on four construction jobs. Three contractors have placed bids on the jobs. Their bids (in thousands of dollars) are given in the table below. (A dash indicates that the contractor did not bid on the given job.) Contractor 1 can do only one job, but contractors 2 and 3 can each do up to two jobs. Decision Models -- Prof. Juran**Formulation**Decision Variables Which contractor gets which job(s). Objective Minimize the total cost of the four jobs. Constraints Contractor 1 can do no more than one job. Contractors 2 and 3 can do no more than two jobs each. Contractor 2 can’t do job 4. Contractor 3 can’t do job 1. Every job needs one contractor. Decision Models -- Prof. Juran**Formulation**Decision Variables Define Xij to be a binary variable representing the assignment of contractor i to job j. If contractor i ends up doing job j, then Xij = 1. If contractor i does not end up with job j, then Xij = 0. Define Cij to be the cost; i.e. the amount bid by contractor i for job j. Objective Minimize Z = Decision Models -- Prof. Juran**Formulation**Constraints for all j. for i = 1. for i = 2, 3. Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Solution Methodology**Notice the very large values in cells B4 and E3. These specific values (10,000) aren’t important; the main thing is to assign these particular contractor-job combinations costs so large that they will never be in any optimal solution. Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Optimal Solution**Decision Models -- Prof. Juran**Conclusions**The optimal solution is to award Job 4 to Contractor 1, Jobs 1 and 3 to Contractor 2, and Job 2 to Contractor 3. The total cost is $182,000. Decision Models -- Prof. Juran**Sensitivity Analysis**• What is the “cost” of restricting Contractor 1 to only one job? • How much more can Contractor 1 bid for Job 4 and still get the job? Decision Models -- Prof. Juran**Conclusions**• The sensitivity report indicates a shadow price of –2 (cell E29). • (Allowing Contractor 1 to perform one additional job would reduce the total cost by 2,000.) • The allowable increase in the bid for Job 4 by Contractor 1 is 3. • (He could have bid any amount up to $43,000 and still have won that job.) Decision Models -- Prof. Juran**Network Representation**Con. 1 Con. 2 Con. 3 Job 1 Job 2 Job 3 Job 4 Decision Models -- Prof. Juran**Optimal Solution**Con. 1 Con. 2 Con. 3 44 51 40 47 Job 1 Job 2 Job 3 Job 4 Decision Models -- Prof. Juran**House-Building Example**Decision Models -- Prof. Juran**Managerial Problem Definition**Find the critical path and the minimum number of days needed to build the house. Decision Models -- Prof. Juran**Network Representation**F 6 1 3 4 G 3 A 5 E 4 0 Start 5 End C 10 B 8 D 5 2 Decision Models -- Prof. Juran**Formulation**Decision Variables We are trying to decide when to begin and end each of the activities. Objective Minimize the total time to complete the project. Constraints Each activity has a fixed duration. There are precedence relationships among the activities. We cannot go backwards in time. Decision Models -- Prof. Juran**Formulation**Decision Variables Define the nodes to be discrete events. In other words, they occur at one exact point in time. Our decision variables will be these points in time. Define ti to be the time at which node i occurs, and at which time all activities preceding node i have been completed. Define t0 to be zero. Objective Minimize t5. Decision Models -- Prof. Juran**Formulation**Constraints There is really one basic type of constraint. For each activity x, let the time of its starting node be represented by tjx and the time of its ending node be represented by tkx. Let the duration of activity x be represented as dx. For every activity x, For every node i, Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Solution Methodology**The matrix of zeros, ones, and negative ones (B12:G18) is a means for setting up the constraints. The sumproduct functions in H12:H18 calculate the elapsed time between relevant pairs of nodes, corresponding to the various activities. The duration times of the activities are in J12:J18. Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Optimal Solution**Decision Models -- Prof. Juran**Conclusions**The project will take 26 days to complete. The only activity that is not critical is the electrical wiring. Decision Models -- Prof. Juran**CPM Jargon**Any activity for which is said to have slack time, the amount of time by which that activity could be delayed without affecting the overall completion time of the whole project. In this example, only activity D has any slack time (13 – 5 = 8 units of slack time). Decision Models -- Prof. Juran**CPM Jargon**Any activity x for which is defined to be a “critical” activity, with zero slack time. Decision Models -- Prof. Juran**F**6 1 3 4 G 3 A 5 E 4 0 Start 5 End C 10 B 8 D 5 2 CPM Jargon Every network of this type has at least one critical path, consisting of a set of critical activities. In this example, there are two critical paths: A-B-C-G and A-B-E-F-G. Decision Models -- Prof. Juran**Excel Tricks: Conditional Formatting**Decision Models -- Prof. Juran**Critical Activities: Using the Solver Answer Report**Decision Models -- Prof. Juran**House-Building Example, Continued**Suppose that by hiring additional workers, the duration of each activity can be reduced. Use LP to find the strategy that minimizes the cost of completing the project within 20 days. Decision Models -- Prof. Juran**Crashing Parameters**Decision Models -- Prof. Juran**Managerial Problem Definition**Find a way to build the house in 20 days. Decision Models -- Prof. Juran**Formulation**Decision Variables Now the problem is not only when to schedule the activities, but also which activities to accelerate. (In CPM jargon, accelerating an activity at an additional cost is called “crashing”.) Objective Minimize the total cost of crashing. Decision Models -- Prof. Juran**Formulation**Constraints The project must be finished in 20 days. Each activity has a maximum amount of crash time. Each activity has a “basic” duration. (These durations were considered to have been fixed in Part a; now they can be reduced.) There are precedence relationships among the activities. We cannot go backwards in time. Decision Models -- Prof. Juran**Formulation**Decision Variables Define the number of days that activity x is crashed to be Rx. For each activity there is a maximum number of crash days Rmax, x Define the crash cost per day for activity x to be Cx Objective Minimize Z = Decision Models -- Prof. Juran**Formulation**Constraints For every activity x, For every activity x, For every node i, Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Solution Methodology**G3 now contains a formula to calculate the total crash cost. The new decision variables (how long to crash each activity x, represented by Rx) are in M12:M18. G8 contains the required completion time, and we will constrain the value in G6 to be less than or equal to G8. The range J12:J18 calculates the revised duration of each activity, taking into account how much time is saved by crashing. Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Solution Methodology**Decision Models -- Prof. Juran**Optimal Solution**Decision Models -- Prof. Juran**F**6 1 3 4 G 3 A 3 E 3 0 Start 5 End C 9 B 5 D 5 2 Conclusions It is feasible to complete the project in 20 days, at a cost of $145. Decision Models -- Prof. Juran**Conclusions**Decision Models -- Prof. Juran**An Alternative Solution**Decision Models -- Prof. Juran**Excel Tricks: VLOOKUP**Looks for a specific value in the left column of a table and finds the row where that value appears, then returns the value corresponding to another specified column in that row. Decision Models -- Prof. Juran