Presentation on theme: "Strategic Allocation of Resources (Linear Programming)"— Presentation transcript:
1 Strategic Allocation of Resources (Linear Programming) Chapter 8Strategic Allocation of Resources(Linear Programming)
2 A company makes 3 products: A, B and C. A B C AvailableProfitLabor Hrs hrsFiberglass lbsAt least 100 units each must be made of A, B, CHow many A’s, B’s, and C’s should be produced in order to maximize total profits?
3 Incorrect Strategy: make as much as possible of the most profitable product (B), so make as little as possible of the other products (100 A’s and 100 C’s)available:Labor Fiberglass Profitmake 100 A’smake 100 C’sremaining:How many B’s?1200/7= 1714000/25= 160We run out offiberglass 1stmake 160 B’sremaining:$13,200 Total ProfitOptimal solution is $13,625 using LP (100 A, 100 B, 225 C) Difference of $425
4 Linear Programming Formulation A = # of units of product A to produceB = # of units of product B to produceC = # of units of product C to produceMax Z = 35A + 45B + 25CST5A + 7B + 3C ≤ labor hours18A + 25B + 12C ≤ fiberglassA ≥ minimum AB ≥ minimum BC ≥ minimum C
5 Linear Programming using Lindo software Max 35 A B CSubject to2) 5 A B C <= 20003) 18 A B C <= 70004) A >= 1005) B >= 1006) C >= 100EndLP Optimum found at step 4
6 Objective Function Value 1)Variable Value Reduced CostABCRow Slack or Surplus Dual Prices2)3)4)5)6)No. Iterations = 4
7 Ranges in which the basis is unchanged: Obj Coefficient RangesVariable Current Allowable AllowableCoef Increase DecreaseA InfinityB InfinityC InfinityRighthand Side RangesRow Current Allowable AllowableRHS Increase DecreaseInfinityInifinity
8 Example Using Excel Solver 10. A local brewery produces three types of beer: premium, regular, and light. The brewery has enough vat capacity to produce 27,000 gallons of beer per month. A gallon of premium beer requires 3.5 pounds of barley and 1.1 pounds of hops, a gallon of regular requires 2.9 pounds of barley and .8 pounds of hops, and a gallon of light requires 2.6 pounds of barley and .6 pounds of hops. The brewery is able to acquire only 55,000 pounds of barley and 20,000 pounds of hops next month. The brewery’s largest seller is regular beer, so it wants to produce at least twice as much regular beer as it does light beer. It also wants to have a competitive market mix of beer. Thus, the brewery wishes to produce at least 4000 gallons each of light beer and premium beer, but not more than 12,000 gallons of these two beers combined. The brewery makes a profit of $3.00 per gallon on premium beer, $2.70 per gallon on regular beer, and $2.80 per gallon on light beer. The brewery manager wants to know how much of each type of beer to produce next month in order to maximize profit.
9 Example Using Excel Solver LP Formulation:Max Z = 3P + 2.7R + 2.8LSTP + R + L < capacity3.5P + 2.9R + 2.6L < barley1.1P + .8R + .6L < hopsR – 2L > :1 ratioP > minimum P requirementL > minimum L requirementP + L < maximum requirement
10 Instructions for Using Excel to Solve LP Models Set up spreadsheet like example in packet (Z-value and LHS column should be formulas)Select “Tools” on menu bar. Then select “Solver…”.“Set Target Cell:” should be the cell of your Z-value formula.Select “Min” or “Max”.“By Changing Cells:” should be the range of cells for your decision variables values.Select “Options…”Check 2 boxes: “Assume Linear Model” and “Assume Non-Negative”. Then click “OK”.Select “Add” to add constraints.
11 9. In “Cell Reference:” box point to LHS formula of first constraint 9. In “Cell Reference:” box point to LHS formula of first constraint. Select <, =, or >. Click on “Constraint:” box and point to RHS value of first constraint. Click “Add” for next constraint or “OK” if finished.10. Repeat Step 9 for each other constraint.11. Select “Solve”.12. If it worked okay you should get the message “Solver found a solution. All constraints and optimality conditions are satisfied.” If you do not get this message you should modify your formulation or check for mistakes.13. In the Solver Results window under “Reports” click on “Answer”. Then hold down the ‘Ctrl’ button while you click on “Sensitivity”. Then click “OK”.14. Print your final worksheet showing the new values, print the Answer Report and print the Sensitivity Report.
18 Microsoft Excel 10.0 Answer Report Worksheet: [Book1]Sheet1Report Created: 1/15/2003 9:35:20 AMTarget Cell (Max)CellNameOriginal ValueFinal Value$E$3Obj Coef Value (Z)Adjustable Cells$B$2Dec Vars P4000$C$2Dec Vars R$D$2Dec Vars LConstraintsCell ValueFormulaStatusSlack$E$6capacity LHS$E$6<=$G$6Not Binding$E$7barley LHS55000$E$7<=$G$7Binding$E$8hops LHS$E$8<=$G$8$E$92:1 ratio LHS$E$9>=$G$9$E$10min req. P LHS$E$10>=$G$10$E$11min req. L LHS$E$11>=$G$11$E$12max req. LHS$E$12<=$G$12
19 Microsoft Excel 10.0 Sensitivity Report Worksheet: [Book1]Sheet1Report Created: 1/15/2003 9:35:20 AMAdjustable CellsFinalReducedObjectiveAllowableCellNameValueCostCoefficientIncreaseDecrease$B$2Dec Vars P400031E+30$C$2Dec Vars R2.70.5$D$2Dec Vars L2.8ConstraintsShadowConstraintPriceR.H. Side$E$6capacity LHS27000$E$7barley LHS550007400$E$8hops LHS20000$E$92:1 ratio LHS$E$10min req. P LHS$E$11min req. L LHS$E$12max req. LHS12000
20 1. The Ohio Creek Ice Cream Company is planning production for next week. Demand for Ohio Creek premium and light ice cream continue to outpace the company’s production capacities. Ohio Creek earns a profit of $100 per hundred gallons of premium and $100 per hundred gallons of light ice cream. Two resources used in ice cream production are in short supply for next week: the capacity of the mixing machine and the amount of high-grade milk. After accounting for required maintenance time, the mixing machine will be available 140 hours next week. A hundred gallons of premium ice cream requires .3 hours of mixing and a hundred gallons of light ice cream requires .5 hours of mixing. Only 28,000 gallons of high-grade milk will be available for next week. A hundred gallons of premium ice cream requires 90 gallons of milk and a hundred gallons of light ice cream requires 70 gallons of milk.
21 P = # of gallons of Premium ice cream to make L = # of gallons of Light ice cream to makeMax Z = 100P + 100LST.3P + .5L ≤ 140 capacity of mixing machine90P + 70L ≤ max milk availableSolution: P = 175; L = 175; Z = 35,000
22 2. The Sureset Concrete Company produces concrete in a continuous process. Two ingredients in the concrete are sand, which Sureset purchases for $6 per ton, and gravel, which costs $8 per ton. Sand and gravel together must make up exactly 75% of the weight of the concrete. Furthermore, no more than 40% of the concrete can be sand, and at least 30% of the concrete must be gravel. Each day 2,000 tons of concrete are produced.
23 S = # tons of sand to add to mixture G = # tons of gravel to add to mixtureMin Z = 6S + 8GSTS + G = sand & gravel are 75% of 2000S ≤ sand no more than 40% of 2000G ≥ gravel at least 30% of 2000Solution: S = 800; G = 700; Z = 10,400
24 3. A ship has two cargo holds, one fore and one aft 3. A ship has two cargo holds, one fore and one aft. The fore cargo hold has a weight capacity of 70,000 pounds and a volume capacity of 30,000 cubic feet. The aft hold has a weight capacity of 90,000 pounds and a volume capacity of 40,000 cubic feet. The shipowner has contracted to carry loads of packaged beef and grain. The total weight of the available beef is 85,000 pounds; the total weight of the available grain is 100,000 pounds. The volume per mass of the beef is 0.2 cubic foot per pound, and the volume per mass of the grain is 0.4 cubic foot per pound. The profit for shipping beef is $0.35 per pound, and the profit for shipping grain is $0.12 per pound. The shipowner is free to accept all or part of the available cargo; he wants to know how much meat and grain to accept in order to maximize profit.
25 BF = # lbs beef to load in fore cargo hold BA = # lbs beef to load in aft cargo holdGF = # lbs grain to load in fore cargo holdGA = # lbs grain to load in aft cargo holdMax Z = .35 BF + .35BA + .12GF GASTBF + GF ≤ fore weight capacity – lbsBA + GA ≤ aft weight capacity – lbs.2BF + .4GF ≤ for volume capacity – cubic feet.2BA + .4GA ≤ for volume capacity – cubic feetBF + BA ≤ max beef availableGF + GA ≤ max grain available
26 4. The White Horse Apple Products Company purchases apples from local growers and makes applesauce and apple juice. It costs $0.60 to produce a jar of applesauce and $0.85 to produce a bottle of apple juice. The company has a policy that at least 30% but not more than 60% of its output must be applesauce.The company wants to meet but not exceed the demand for each product. The marketing manager estimates that the demand for applesauce is a maximum of 5,000 jars, plus an additional 3 jars for each $1 spent on advertising. The maximum demand for apple juice is estimated to be 4,000 bottles, plus an additional 5 bottles for every $1 spent to promote apple juice. The company has $16,000 to spend on producing and advertising applesauce and apple juice. Applesauce sells for $1.45 per jar; apple juice sells for $1.75 per bottle. The company wants to know how many units of each to produce and how much advertising to spend on each in order to maximize profit.
27 S = # jars apple Sauce to make J = # bottles apple Juice to makeSA = $ for apple Sauce AdvertisingJA = $ for apple Juice AdvertisingMax Z = 1.45S J - .6S - .85J – SA – JASTS ≥ .3(S + J) at least 30% apple sauceS ≤ .6(S + J) no more than 60% apple sauceS ≤ SA don’t exceed demand for apple sauceJ ≤ JA don’t exceed demand for apple juice.6S + .85J + SA + JA ≤ budget
28 5. Dr. Maureen Becker, the head administrator at Jefferson County Regional Hospital, must determine a schedule for nurses to make sure there are enough nurses on duty throughout the day. During the day, the demand for nurses varies. Maureen has broken the day into 12 two-hour periods. The slowest time of the day encompasses the three periods from 12:00 A.M. to 6:00 A.M., which, beginning at midnight, require a minimum of 30, 20, and 40 nurses, respectively. The demand for nurses steadily increases during the next four daytime periods. Beginning with the 6:00 A.M. – 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses are required for these four periods, respectively. After 2:00 P.M. the demand for nurses decreases during the afternoon and evening hours. For the five two-hour periods beginning at 2:00 P.M. and ending at midnight, 70, 70, 60, 50, and 50 nurses are required, respectively. A nurse reports for duty at the beginning of one of the two-hour periods and works eight consecutive hours (which is required in the nurses’ contract). Dr. Becker wants to determine a nursing schedule that will meet the hospital’s minimum requirements throughout the day while using the minimum number of nurses.
31 _____________________________________________ lb/Day Cost/lb($) 6. The Donnor meat processing firm produces wieners from four ingredients: chicken, beef, pork, and a cereal additive. The firm produces three types of wieners: regular, beef, and all-meat. The company has the following amounts of each ingredient available on a daily basis._____________________________________________lb/Day Cost/lb($)ChickenBeefPorkCereal AdditiveEach type of wiener has certain ingredient specifications, as follows.________________________________________________________________________________Specifications Selling Price/lb($)Regular Not more than 10% beef and pork combinedNot less than 20% chicken $0.90Beef Not less than 75% beefAll-Meat No cereal additiveNot more than50% beef and pork combinedThe firm wants to know the amount of wieners of each type to produce.
32 14 variables (you could also formulate it with 11 variables) CR = # lbs Chicken ingredient in Regular wiener per dayCB = # lbs Chicken ingredient in Beef wiener per dayCM = # lbs Chicken ingredient in all-Meat wiener per dayBR = # lbs Beef ingredient in Regular wiener per dayBB = # lbs Beef ingredient in Beef wiener per dayBM = # lbs Beef ingredient in all-Meat wiener per dayPR = # lbs Pork ingredient in Regular wiener per dayPB = # lbs Pork ingredient in Beef wiener per dayPM = # lbs Pork ingredient in all-Meat wiener per dayAR = # lbs Additive ingredient in Regular wiener per dayAB = # lbs Additive ingredient in Beef wiener per dayR = total lbs of Regular wienerB = total lbs of Beef wienerM = total lbs of all-Meat wiener
33 Max Z = 0.90R + 1.25 B + 1.75 M - .2CR - .2CB - .2CM - .3BR - .3BB - .3BM - .5PR - .5PB - .5PM - .05AR - .05ABSTCR + BR + PR + AR = R R is sum of all ingredients in RegularCB + BB + PB + AB = B B is sum of all ingredients in BeefCM + BM + PM = M M is sum of all ingredients in MeatCR + CB + CM ≤ 200 max Chicken ingredient availableBR + BB + BM ≤ 300 max Beef ingredient availablePR + PB + PM ≤ 150 max Pork ingredient availableAR + AB ≤ max Additive ingredient availableBR + PR ≤ .1R not more than 10% BR+PR combinedCR ≥ .2R not less than 20% CR in RegularBB ≥ .75B not less than 75% BB in BeefBM + PM ≤ .5M not more than 50% BM+PM combined
34 7. The Jane Deere Company manufactures tractors in Provo, Utah 7. The Jane Deere Company manufactures tractors in Provo, Utah. Jeremiah Goldstein, the production planner, is scheduling tractor production for the next three months. Factors that Mr. Goldstein must consider include sales forecasts, straight-time and overtime labor hours available, labor cost, storage capacity, and carrying cost. The marketing department has forecasted that the number of tractors shipped during the next three months will be 250, 305, and Each tractor requires 100 labor hours to produce. In each month 29,000 straight-time labor hours will be available, and company policy prohibits overtime hours from exceeding 10% of straight-time hours. Straight-time labor cost rate is $20 per hour, including benefits. The overtime labor cost rate is 150% (time-and-a-half) of the straight-time rate. Excess production capacity during a month may be used to produce tractors that will be stored and sold during a later month. However, the amount of storage space can accommodate only 40 tractors. A carrying cost of $600 is charged for each month a tractor is stored (if not shipped during the month it was produced). Currently, no tractors are in storage.How many tractors should be produced in each month using straight-time and using overtime in order to minimize total labor cost and carrying cost? Sales forecasts, straight-time and overtime labor capacities, and storage capacity must be adhered to. (Tip: During each month, all “sources” of tractors must exactly equal “uses” of tractors.)
35 9 variablesS1 = # tractors produced in month 1 using straight-timeS2 = # tractors produced in month 2 using straight-timeS3 = # tractors produced in month 3 using straight-timeV1 = # tractors produced in month 1 using overtimeV2 = # tractors produced in month 2 using overtimeV3 = # tractors produced in month 3 using overtimeC1 = # tractors carried in warehouse at end of month 1C2 = # tractors carried in warehouse at end of month 2C3 = # tractors carried in warehouse at end of month 3sources of tractors = uses of tractors (for each month)production + beg.inv. = sales + end.inv.
37 Destination Production A B C D (cases) 8. MadeRite, a manufacturer of paper stock for copiers and printers, produces cases of finished paper stock at Mills 1, 2, and 3. The paper is shipped to Warehouses A, B, C, and D. The shipping cost per case, the monthly warehouse requirements, and the monthly mill production levels are:Monthly MillDestination ProductionA B C D (cases)Mill 1 $ $ $ $ ,000Mill ,000Mill ,000Monthly WarehouseRequirement (cases) 9, , , ,000How many cases of paper should be shipped per month from each mill to each warehouse to minimize monthly shipping costs?
38 A1 = # of units shipped from Mill 1 to Destination A C3 = # of units shipped from Mill 3 to Destination C(12 variables)Min Z = 5.4A B C D A B2+ 5.6C D A B C D3STA1 + B1 + C1 + D1 ≤ Mill 1 capacityA2 + B2 + C2 + D2 ≤ Mill 2 capacityA3 + B3 + C3 + D3 ≤ Mill 3 capacityA1 + A2 + A3 = Destination A demandB1 + B2 + B3 = Destination B demandC1 + C2 + C3 = Destination C demandD1 + D2 + D3 = Destination D demand
39 9. A company has three research projects that it wants to do, and has three research teams that can do the projects. Any team could do any project but can only do one project. Some teams are better skilled at certain projects and could do them at lower costs. The estimated cost of each team doing each project (in $,000s) is shown below. Which team should do which project?Project1 2 3ATeam BC
40 A1 = 1 if team A does project 1 = 0 if not (9 variables)Min Z = 87A1 + 62A2 + 76A3 + 81B1 + 76B2 + 64B3+ 77C1 + 54C2 + 70C3STA1 + A2 + A3 = 1B1 + B2 + B3 = 1 Each team does exactly one projectC1 + C2 + C3 = 1A1 + B1 + C1 = 1A2 + B2 + C2 = 1 Each project is done exactly onceA3 + B3 + C3 = 1