Formulation of LP Problems Introductory Examples Low complexity Straight forward Farmer Jones Example Example: Book Problem 3.1-1 Farmer Jones must determine how many acres of corn and wheat to plant each year. An acre of wheat yields 25 bushels of wheat and requires 10 hours of labor per week. An acre of corn yields 10 bushels of corn and requires 4 hours of labor per week. All wheat can be sold at $4 a bushel, and all corn can be sold at $3 a bushel. Seven acres of land and 40 hours per week of labor are available. Government regulations require that at least 30 bushels of corn be produced during the current year. Let X1 be the number of acres of corn planted and X2 be the number of acres of wheat planted. Using these decision variables: 1. Formulate an LP whose solution will tell Farmer Jones how to maximize the total revenue from wheat and corn. 2. Solve using MS Excel. (Work at home the following problems, as practice.) 3. Solve the problem using the graphical method, determine which constraints are biding, which are non-biding, and identify all extreme points. 4. Solve using Lingo. Reddy Mikks Example Example: Taha 1998 Book Problem 2.1 Reddy Mikks produces both interior and exterior paints from two raw materials, M1 and M2. The table provides basic data of the problem. A market survey indicates that the daily demand for interior paint cannot exceed that for exterior paint by more than 1 ton. Also, the maximum daily demand for interior paint is 2 tons. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit. 1. Formulate as an LP. 2. Solve using MS Excel. (Work at home the following problems, as practice.) 3. Solve the problem using the graphical method, determine: a. optimal solution, including objective function value b. which constraints are biding, at the optimal solution c. which are non-biding, at the optimal solution d. identify all extreme points e. determine which of the following solution is the best feasible: (1,4) or (2,2) or (3,1.5) or (2,1) or (2,-1) f. For the feasible solution (2,2), determine the unused amounts of raw material Tons of raw material per ton of Exterior paint Maximum daily availability (tons) Interior paint M1 6 4 24 M2 1 2 6 Profit per ton ($1000) 5 4 Diet Problem “Diet Problem” Type Formulation THE “DIET PROBLEM” IS A COST MINIMIZATION PROBLEM THAT IS SUBJECT TO CONSTRAINTS THAT GUARANTEE A “BALANCED” COMPOSITION OF THE VARIABLES. IN THIS CASE, THIS PROBLEMS ARE MOTIVATED AS FINDING THE COMBINATION OF FOODS THAT MINIMIZE COST SUBJECT TO NUTRITIONAL CONSTRAINTS. Diet Problem - Nutrition Example • Your diet requires that all the food you get come from one of the four “basic food groups”. Type of Food Calories Chocolate (oz.) Sugar (oz.) Fat (oz.) Brownie 400 3 2 3 Chocolate Ice Cream (scoop) 200 2 2 4 Cola (bottle) 150 0 4 1 Pineapple Cheesecake (piece) 500 0 4 5 • At present, the following four foods are available for consumption: • brownies, chocolate ice cream, cola and pineapple cheesecake. • Each brownie costs 50 ¢, each scoop of ice cream costs 20 ¢, each bottle of cola costs 30 ¢, and each piece of pineapple cheesecake costs 80 ¢. • Each day, you must ingest at least 500 calories, 6 oz. of chocolate, 10 oz. of sugar, and 8 oz. of fat. • The nutritional content per unit of each food is given by the following table. • It is desired to satisfy nutritional needs at the lowest cost. Diet Problem - Alloy Composition An industrial recycling center uses two scrap aluminum metals, A and B, to produce a special alloy. Scrap A contains 6% aluminum, 3% silicon, and 4% carbon. Scrap B has 3% aluminum, 6% silicon , and 3% carbon. The cost per ton for scraps A and B are $100 and $80, respectively. The specifications of the special alloy require that (1) the aluminum content must be at least 3% and at most 6%, (2) the silicon content must lie between 3% and 5%, and (3) the carbon content must be between 3% and 7%. Determine the optimum mix of the scraps that should be used in producing 1000 tons of the alloy. Work-Scheduling Problem Type Formulation Work-Scheduling Problem” Type Formulation • Many applications of linear programming involve determining the minimum-cost method for satisfying workforce requirements. Work-Scheduling Problem - Example Find the minimum number of hospital employees that will satisfy the following requirements. Time Period No. Workers Needed 12 a.m. – 4 a.m. 5 4 a.m. – 8 a.m. 7 8 a.m. – 12 p.m. 15 12 p.m. – 4 p.m. 8 4 p.m. – 8 p.m. 15 8 p.m. – 12 a.m. 9 Note: Union contract requires that all employees work for 8 consecutive hours. Capital Budgeting Capital Budgeting Problem • Capital budgeting problems are problems that involve the allocation of limited funds in different projects. • Usually, they involve the maximization of the ‘return of the investment’ or in the maximization of the ‘net present value’. • The ‘net present value’(NPV), in our case, will refer to the concept that a dollar ($) invested today will yield a rate r in a year from now. That is, $1 invested today will be worth (1+r)k , k years from now. (r is the annual interest rate.) Capital Budgeting Problem - Example • Star Oil is considering investing in five different investment-opportunities. • Star Oil has $40 Million to invest in Year 1 and $20 Million to invest in Year 2. • The company can invest a fraction of the quantity required per investment given that the NPV will also be fractioned by the same percent. For example, if the company decides to invest in half Investment 1, then on Year 1 they have a corresponding 0.5*(11) cash outflow and on Year 2 a cash outflow of 0.5*(3) and a NPV yield of 0.5*(13). • The following table summarizes the investment opportunities. • Assume that any left-over cash from Year 1 cannot be invested on Year 2. Investment 1 2 3 4 5 Time 0 (Year 1) Cash Outflow ($M) 11 53 5 5 29 Time 1 (Year 2) Cash Outflow ($M) 3 6 5 1 34 NPV 13 16 16 14 39 Blending • Situations in which various inputs must be blended in some desired proportion to produce goods for sale are often amenable to linear programming analysis. Blending Problem • Some examples of how linear programming has been used to solve blending problems. • Blending various types of crude oils to produce different types of gasoline and other outputs • Blending various chemicals to produce other chemicals • Blending various types of metal alloys to produce various types of steels Blending Problem – Eli Daisy Example • Eli Daisy uses chemicals 1 and 2 to produce two drugs. • Drug 1 must be at least 70% chemical 1, and drug 2 must be at least 60 % chemical 2. • Up to 40 oz. of drug 1 can be sold at $6.5 per ounce; up to 30 oz. of drug 2 can be sold at $5 per oz. • Up to 45 oz. of chemical 1 can be purchased at $6 per ounce, and up to 40 oz. of chemical 2 can be purchased at $4 per oz. • Formulate an LP that maximizes Daisy’s profits! Production Process Production Process This types of problems relates outputs from later stages of a process to outputs of earlier stages. Production Process – Chemco Example • Chemco produces three products: 1, 2 and 3. • Each pound of raw material cost $25 and produces 3 oz. of Product 1 and 1 oz. of Product 2. It cost $1 to process a pound of raw material and takes 2 hours of labor. • Each ounce of Product 1 can be used in one of three ways: • Sold for $10/oz • Processed into 1oz. of Product 2; requires 2 hrs. labor and costs $1 • Processed into 1 oz. of Product 3; requires 3 hrs. labor and costs $2 • Each ounce of Product 2 can be used in one of two ways: • Sold for $20/oz. • Processed into 1 oz. of Product 3; requires 1 hrs. labor and costs $6 • Product 3 is sold for $30/oz. • The maximum that can be sold of Product 1, 2 and 3 is 5000, 5000 and 3000 respectively. • A maximum of 25,000 hours of labor are available. Multi-period Decision Multi-period Decision Problems • All problems discussed previously have been static, or one-period, models. • Linear programming can also be used to determine optimal decisions in multiperiod, or dynamic models. • Dynamic models arise when the decision maker makes decisions at more than one point in time. • In a dynamic model, decisions made during the current period influence decisions made during future periods. Multiperiod Decision Problems: Inventory Model • Example: James Beerd Cakes • James Beerd bakes two cakes: cheesecakes and black forest cakes • During a month he can bake at most 65 cakes. • It cost $0.50 to hold a cheesecake (per month) and $0.40 a black forest cake. • The costs for baking and demand per month are given next. • Formulate as a cost minimization LP! Month 1 Item Month 2 Month 3 Deman d Cost/Cake ($) Deman d Cost/Cake ($) Deman d Cost/Cake ($) Cheesecake 40 3.00 30 3.40 20 3.80 Black Forest 20 2.50 30 2.80 10 3.40 Multi-period Decision Problems: Work-Scheduling • Example: Rental Laptops • An Insurance company believes that it will require the following number of laptops per month: January, 9 ; February, 5; March, 7; April, 9; May, 10; June, 5. • Computers can be rented for the following periods of time: 1-month for $200, 2-month for $350 and 3-month for $450. • Formulate an LP that can be used to minimize the cost of renting computers.
