# McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.1 Integer Programming.

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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.1 Integer Programming

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.2 Integer Programming When are “non-integer” solutions okay? –Solution is naturally divisible e.g., \$, pounds, hours –Solution represents a rate e.g., units per week –Solution only for planning purposes When is rounding okay? –When numbers are large e.g., rounding 114.286 to 114 is probably okay. When is rounding not okay? –When numbers are small e.g., rounding 2.6 to 2 or 3 may be a problem. –Binary variables yes-or-no decisions

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.3 The Challenges of Rounding Rounded Solution may not be feasible. Rounded solution may not be close to optimal. There can be many rounded solutions. –Example: Consider a problem with 30 variables that are non- integer in the LP-solution. How many possible rounded solutions are there?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.4 How Integer Programs are Solved

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.5 How Integer Programs are Solved

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.6 Applications of Binary Variables Making “yes-or-no” type decisions –Build a factory? –Manufacture a product? –Do a project? –Assign a person to a task? Set-covering problems –Make a set of assignments that “cover” a set of requirements. Fixed costs –If a product is produced, must incur a fixed setup cost. –If a warehouse is operated, must incur a fixed cost.

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.7 Example #1 (Capital Budgeting) Norwood Development is considering the potential of four different development projects. Each project would be completed in at most three years. The required cash outflow for each project is given in the table below, along with the net present value of each project to Norwood, and the cash that is available each year. Cash Outflow Required (\$million) Cash Available (\$million) Project 1Project 2Project 3Project 4 Year 19761128 Year 2643013 Year 3604010 NPV30162214 Question: Which projects should be undertaken?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.8 Algebraic Formulation Let y i = 1 if project i is undertaken; 0 otherwise (i = 1, 2, 3, 4). Maximize NPV = 30y 1 + 16y 2 + 22y 3 + 14y 4 subject to Year 1:9y 1 + 7y 2 + 6y 3 + 11y 4 ≤ 28 (\$million) Year 2 (cumulative):15y 1 + 11y 2 + 9y 3 + 11y 4 ≤ 41 (\$million) Year 3 (cumulative):21y 1 + 11y 2 + 13y 3 + 11y 4 ≤ 51 (\$million) and y i are binary (i = 1, 2, 3, 4).

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.10 Additional Considerations (Logic and Dependency Constraints) At least one of projects 1, 2, or 3 Project 2 can’t be done unless project 3 is done Either project 3 or project 4, but not both No more than two projects total Question: What constraints would need to be added for each of these additional considerations?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.11 Example #2 (Set Covering Problem) The Washington State legislature is trying to decide on locations at which to base search-and-rescue teams. The teams are expensive, so they would like as few as possible. Response time is critical, so they would like every county to either have a team located in that county or in an adjacent county. Question: Where should search-and-rescue teams be located?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.12 The Counties of Washington State

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.13 Algebraic Formulation Let y i = 1 if a team is located in county i; 0 otherwise (i = 1, 2, …, 37). Minimize Number of Teams = y 1 + y 2 + … + y 37 subject to County 1 covered:y 1 + y 2 ≥ 1 County 2 covered:y 1 + y 2 + y 3 + y 6 + y 7 ≥ 1 County 3 covered:y 2 + y 3 + y 4 + y 7 + y 8 + y 14 ≥ 1 : County 37 covered:y 32 + y 36 + y 37 ≥ 1 and y i are binary (i = 1, 2, …, 37).

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.15 Example #3 (Fixed Costs) Woodridge Pewter Company is a manufacturer of three pewter products: platters, bowls, and pitchers. The manufacture of each product requires Woodridge to have the appropriate machinery and molds available. The machinery and molds for each product can be rented at the following rates: for the platters, \$400/week; for the bowls, \$250/week; for the pitcher, \$300/week. Each product requires the amounts of labor and pewter given in the table below. The sales price and variable cost are also given in the table. Labor Hours Pewter (pounds) Sales Price Variable Cost Platter35\$100\$60 Bowl148550 Pitcher437540 Available130240 Question: Which products should be produced, and in what quantity?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.16 Algebraic Formulation Let x 1 = Number of platters produced, x 2 = Number of bowls produced, x 3 = Number of pitchers produced, y i = 1 if lease machine and mold for product i; 0 otherwise (i = 1, 2, 3). Maximize Profit = (\$100–\$60)x 1 + (\$85–\$50)x 2 + (\$75–\$40)x 3 – \$400y 1 – \$250y 2 – \$300y 3 subject to Labor:3x 1 + x 2 + 4x 3 ≤ 130 hours Pewter:5x 1 + 4x 2 + 3x 3 ≤ 240 pounds Allow production only if machines and molds are purchased: x 1 ≤ 99y 1 x 2 ≤ 99y 2 x 3 ≤ 99y 3 and x i ≥ 0, and y i are binary (i = 1, 2, 3).

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.18 Capital Budgeting with Contingency Constraints (Yes-or-No Decisions) A company is planning their capital budget over the next several years. There are 10 potential projects they are considering pursuing. They have calculated the expected net present value of each project, along with the cash outflow that would be required over the next five years. Also, suppose there are the following contingency constraints: –at least one of project 1, 2 or 3 must be done, –project 4 and project 5 cannot both be done, –project 7 can only be done if project 6 is done. Question: Which projects should they pursue?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.19 Data for Capital Budgeting Problem Cash Outflow Required (\$million) Cash Available (\$million) Project 12345678910 Year 1140443282625 Year 2222224233625 Year 3325242348225 Year 4445453121125 Year 5110655511225 NPV20252230422518352833(\$million)

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.21 Electrical Generator Startup Planning (Fixed Costs) An electrical utility company owns five generators. To generate electricity, a generator must be started up, and associated with this is a fixed startup cost. All of the generators are shut off at the end of each day. Generator ABCDE Fixed Startup Cost\$2,450\$1,600\$1,000\$1,250\$2,200 Variable Cost (per MW)\$3\$4\$6\$5\$4 Capacity (MW)2,0002,8004,3002,1002,000 Question: Which generators should be started up to meet the total capacity needed for the day (6000 MW)?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.23 Quality Furniture (Either-Or Constraints) Reconsider the Quality Furniture Problem: –The Quality Furniture Corporation produces benches and picnic tables. The firm has a limited supply of two resources: labor and wood. 1,600 labor hours are available during the next production period. The firm also has a stock of 9,000 pounds of wood available. Each bench requires 3 labor hours and 12 pounds of wood. Each table requires 6 labor hours and 38 pounds of wood. The profit margin on each bench is \$8 and on each table is \$18. Now suppose that they would not produce any fewer than 200 units of either product (i.e., either produce 0 or at least 200). Question: What product mix will maximize their total profit?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.25 Meeting a Subset of Constraints Consider a linear programming model with the following constraints, and suppose that meeting 3 out of 4 of these is good enough –12x 1 + 24x 2 + 18x 3 ≥ 2,400 –15x 1 + 32x 2 + 12x 3 ≥ 1,800 –20x 1 + 15x 2 + 20x 3 ≤ 2,000 –18x 1 + 21x 2 + 15x 3 ≤ 1,600

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.26 Meeting a Subset of Constraints Let y i = 1 if constraint i is enforced; 0 otherwise. Constraints: y 1 + y 2 + y 3 + y 4 ≥ 3 12x 1 + 24x 2 + 18x 3 ≥ 2,400y 1 15x 1 + 32x 2 + 12x 3 ≥ 1,800y 2 20x 1 + 15x 2 + 20x 3 ≤ 2,000 + M (1 – y 3 ) 18x 1 + 21x 2 + 15x 3 ≤ 1,600 + M (1 – y 4 ) where M is a large number.

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.27 Facility Location Consider a company that operates 5 plants and 3 warehouses that serve customers in 4 different regions. To lower costs, they are considering streamlining by closing one or more plants and warehouses. Associated with each plant are fixed costs, shipping costs, and production costs. Each plant has a limited capacity. Associated with each warehouse are fixed costs and shipping costs. Each warehouse has a limited capacity. Questions: Which plants should they keep open? Which warehouses should they keep open? How should they divide production among the open plants? How much should be shipped from each plant to each warehouse, and from each warehouse to each customer?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.28 Data for Facility Location Problem Fixed Cost (per month) (Shipping + Production) Cost (per unit) Capacity (units per month) WH #1WH #2WH #3 Plant 1\$42,000\$650\$750\$850400 Plant 250,000500350550300 Plant 345,000450 350300 Plant 450,000400500600350 Plant 547,000550450350375 Fixed Cost (per month) Shipping Cost (per unit) Capacity (per month) Cust. 1Cust. 2Cust. 3Cust. 4 WH #1\$45,000\$25\$65\$70\$35600 WH #225,00050254060400 WH #365,00060204045900 Demand:250225200275