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McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Linear Programming Using the Excel Solver Appendix A

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A-2 Background The key to profitable operations is making the best use of available resources of people, material, plant and equipment, and money Today’s manager has a powerful mathematical modeling tool available for this purpose - linear programming In this presentation, we will show how the use of the Microsoft Excel Solver to solve LP problems

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A-3 Introduction Linear programming (LP) refers to several related mathematical techniques used to allocate limited resources among competing demands in an optimal way LP is the most popular of the approaches falling under the general heading of mathematical optimization techniques

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A-4 Linear Programming Applications Finding the minimum-cost production schedule Aggregate sales and operations planning Comparing how efficiently different service and manufacturing outlets are using their resources Service/manufacturing productivity analysis Finding the optimal product mix where several products have different costs and resource requirements Product planning Finding the optimal way to produce a product that must be processed sequentially through several machine centers Product routing Finding the optimal way to use resources such as aircraft, buses, or trucks Vehicle/crew scheduling

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A-5 Linear Programming Applications Minimizing the amount of scrap material generated Process control Finding the optimal combination of products to stock in a network of warehouses or storage locations Inventory control Finding the optimal shipping schedule for distributing products between factories and warehouses Distribution scheduling Finding the optimal location of a new plant by evaluating shipping costs between alternative locations Plant location studies Finding the minimum-cost routings of material handling devices between departments Material handling

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A-6 Linear Programming – Essential Conditions There must be limited resources There must be an explicit objective There must be linearity There must be homogeneity There must be divisibility

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A-7 Linear Programming Model

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A-8 Graphical Linear Programming Limited to solving problems with two variables Steps are: 1. Formulate the problem in mathematical terms 2. Plot constraint equations 3. Determine the area of feasibility 4. Plot the objective function 5. Find the optimal point

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A-9 Linear Programming Using Microsoft Excel Spreadsheets can be used to solve linear programming problems Microsoft Excel has an optimization tool called Solver for this purpose If the Solver option does not appear in your Data menu, you likely need to install the Solver Add-In Search Excel help for “Load the Solver Add-In” for detailed instructions specific to your version of Excel

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A-10 Linear Programming with Excel 1. Define changing cells 2. Calculate total profit 3. Set up resource usage 4. Set up Solver 5. Solve the problem

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A-11 Step 1 – Define Changing Cells A convenient starting point is to identify cells to be used for the decision variables in the problem These are H and C, the number of hockey sticks and number of chess sets to produce Excel refers to these cells as changing cells in Solver We have designated B4 as the location for the number of hockey sticks to produce and C4 for the number of chess sets Changing Cells – designate B2 as “H” and B3 as “C”

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A-12 Step 2 – Calculate Total Profit This is the objective function Calculated by multiplying profit associated with each product by the number of units produced We have placed the profits in cells B5 and D5 so the profit is calculated by the following equation: B2*B5 + B3*D5, which is calculated in cell F5 Objective Function – “=B2*B5+B3*D5”

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A-13 Step 3 – Resource Usage (Constraints) Our resources are machine centers A, B, and C For machine center A, 4 hours of processing time are used for each hockey stick produced and 6 hours for each chess set For a solution, the total amount of the machine center A resource used is calculated in F7 (=B7*B2 + D7*B3) We indicate in cell G7 that we want this to be less than the 120-hour capacity of machine center A, in H7 Resource usage for machine centers B and C are set up in the exact same manner in rows 10 and 11 Constraints – Row 7 – Machine Center A Row 8 – Machine Center B Row 9 – Machine Center C

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A-14 Step 4 – Set Up Solver 1. Set Target Cell: is the location where the value that we want to optimize is calculated This is the profit calculated in F5 2. Equal To: set to Max since maximize profit 3. By Changing Cells: are the cells that Solver can change to maximize profit Cells B2 through B3 are the changing cells 4. Subject to the Constraints: correspond to our machine center capacity Here click on Add and indicate that the total used for a resource is less than or equal to the capacity available

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A-15 Step 5 – Solve the Problem Solver has numerous options, but we will need to use only a few Use the Simplex LP solving method to tell Solver that we want to use the linear programming option for solving the problem We know our changing cells (decision variables) must be numbers that are greater than or equal to zero since it makes no sense to make a negative number of hockey sticks or chess sets We indicate this by selecting “Make Unconstrained Variables Non-Negative” as an option Click on “Solve” to find the answer

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