Formulating LP Models 1 Given a problem first

Formulating LP Models 1. Given a problem, first, determine the objective or goal. Maximize (or minimize) what? 2. Identify & define the decision variables (unknowns). • What should they represent and how many do we need? 3. State the objective as a linear function of the decision variables. 4. Translate the requirements, restrictions, or wishes, that are in narrative form to linear functions. 5. Identify any lower or upper bounds on the decision variables (non-negativity constraints are v. common) 1

Solver Videos and more from Frontline. Solvers u http: //solver. com/videodemos. htm u http: //www. solver. com/getting-your-modelbuilt. htm 2

Formulate & solve u Electro-Poly Corp. Model 1 Model 2 Model 3 3, 000 2, 000 900 Hours of wiring/unit 2 1. 5 3 Hours of harnessing/unit 1 2 1 Cost to Make $50 $83 $130 Cost to Buy $61 $97 $145 Number ordered The firm has 10, 000 hours of wiring capacity and 5, 000 hours of harnessing capacity. 3

Electro-Poly Formulation Mi = Number of model i slip rings to make in-house, i=1, 3 Bi = Number of model i slip rings to buy from competitor, i=1, 3 MIN: 50 M 1 + 83 M 2 + 130 M 3 + 61 B 1 + 97 B 2 +145 B 3 S. T. M 1 + B 1 = 3000 M 2 + B 2 = 2000 M 3 + B 3 = 900 2 M 1 +1. 5 M 2 + 3 M 3 <= 10000 1 M 1 +2. 0 M 2 + 1 M 3 <= 5000 M 1, M 2, M 3, B 1, B 2, B 3 >= 0 A standard (efficient) Excel setup using SUMPRODUCT 4

Agri-Pro – a blending problem u Agri-Pro has received an order for 8, 000 pounds of chicken feed to be mixed from the following feeds Percent of Nutrient in Nutrient Feed 1 Feed 2 Feed 3 Feed 4 Corn 30% 5% 20% 10% Grain 10% 30% 15% 10% Minerals 20% 20% 30% Cost per pound $0. 25 $0. 30 $0. 32 $0. 15 u The order must contain at least 20% corn, 15% grain, and 15% minerals. 5

Formulating the Agri-Pro u Define the decision variables: u Xi = Pounds of feed i to use in the mix, i=1, 4 u Formulate the objective function: u MIN: 0. 25 X 1 + 0. 30 X 2 + 0. 32 X 3 + 0. 15 X 4 u Formulate the constraints: Produce 8, 000 pounds of feed X 1 + X 2 + X 3 + X 4 = 8, 000 u Mix consists of at least 20% corn 0. 3 X 1 + 0. 05 X 2 + 0. 2 X 3 + 0. 1 X 4 >= 1600 (20% of 8000) u Mix consists of at least 15% grain 0. 1 X 1 + 0. 3 X 2 + 0. 15 X 3 + 0. 1 X 4 >= 1200 u Mix consists of at least 15% minerals 0. 2 X 1 + 0. 2 X 2 + 0. 2 X 3 + 0. 3 X 4 >= 1200 u Nonnegativity conditions X 1, X 2, X 3, X 4 >= 0 or Xi >= 0 i=1, 4 u 6

More Formulations u PROTRAC – another product mix problem Go to course web page click on “LP formulation exercises” u How to document and format for deliverables. u Another “solver” LINDO u u Bagging Nuts problem - formulate, setup & solve u Direct Marketing problem u #28 Maintenance at a… - formulate, setup & solve u #27 Riverside oil… u Taco-Viva example in Chap. 3 7

u PROTRAC produces the largest member of its earth-moving lines (the E-9 and E-10) and the largest member of its forestry equipment line (the F-9) with essentially the same equipment. Demand for these items is very high at this time, and management believes it is possible to sell as many as the firm can produce. The profit is $5800 for an E-9, $5000 for an E-10 and $4000 for an F-9. In planning production for the next 2 weeks management knows it is necessary to produce a combined total of at least five E-9 s and F-9 s in order to meet previous commitments. All of the products requires processing in department A and department B. Over the next two weeks these departments have 200 and 220 hours of available time, respectively. Each E-9 requires 10 hours in department A and 20 hours in department B, each E- 10 requires 13 hours in department A and 18 hours in department B whereas each F-9 requires 15 hours in department A and 10 hours in department B. In order to honor the labor contract, the total labor hours used in testing finished products can not fall more than 10% below the agreed goal of 150 hours. (This is independent of the processing requirements in departments A and B). Each E-9 requires 30 hours of testing, each E-10 requires 35, and each F-9 requires 10. Finally in order to maintain the current market position, management feels that it is necessary to build at least one F-9 for every three E-9 s produced and one E-10 for every four E-9 s produced. Formulate as an LP. 8

The World-Wide Grocery Store Company u The World-Wide Grocery Store Company, in preparation for the upcoming holiday season, has just purchased the following quantities of nuts: almonds 6000 lbs, Brazil 7500 lbs, filberts 7500 lbs, pecans 4000 lbs, walnuts 7500 lbs. They would like to package these nuts in one-pound bags and presently considering producing a regular mix (consisting of 10% pecans, 15% almonds, 25% filberts, 25% Brazil, 25% walnuts), a deluxe mix (20% of each type of nut), and individual bags of each type of nut. The profit figures for each bag they produce are as follows: 9

Direct Marketing, Inc. (recording avail. ) u For a telephone survey, a marketing research group needs to contact (annoy!) at least 150 wives, 120 husbands, 100 single adult males, and 110 single adult females. It costs $2 to make a daytime call, (due to higher labor costs) $5 to make an evening call and $3 to make a weekend call. Table below summarizes historical response rates. u Person % of Daytime % of Evening % of Weekend Responding Calls u Wife 30 10 Husband 10 30 15 u Single male 15 20 5 u Single female 10 15 10 u None (hang up) 35 60 u Because of limited staff, at most half of all phone calls can be evening calls and at least one fourth of all calls must be made during the weekend. What should the company do? Formulate as an LP. 10

Maintenance at a… (recording avail. ) Let xi = the # of employees starting their shift at time period i 11

Riverside oil… 12

Taco-Viva Problem Summary u Taco-Viva needs to establish a sinking fund to pay $800, 000 in building costs for a new restaurant in the next 6 months. u Payments of $250, 000 are due at the end of months 2 and 4, and a final payment of $300, 000 is due at the end of month 6. u The following investments may be used. Investment A B C D Available in Month 1, 2, 3, 4, 5, 6 1, 3, 5 1, 4 1 Months to Maturity 1 2 3 6 Yield at Maturity 1. 8% 3. 5% 5. 8% 11. 0% 13

Taco-Viva: Graphical view…a perspective… B 1 | Beginning of 1 A 1 | 2 | 3 | 4 | 5 | 6 | End of 6 Months Now, let’s develop the formulation… 14

Taco-Viva with Risk u Assume risk ratings of 1=lowest, 10=highest u Ratings for: A=1, B=3, C=8, D=6 u Determine an investment plan where weighted average risk <= 5. 15

The Pelletier Corp. 16

Suggested practice problems and homework u More formulations? #34 Newspaper recycling Suggested problems for practice: u Chap 3: u 6 th ed: #21, 22, 23, 24, 27, 28, 29, 32, 40 u Int’l ed: #29, 27, 25, 23, 28, 26, 24, 38, 31 First, formulate on your own or with your group, u Second, setup and solve via Analytic Solver (ASPE), u Third, check with the provided solutions. u Group Exercises: u Case 3. 1 Putting the link in the supply chain u The Carolinian Textiles - Part (a) u 17