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Chapter 8 Strategic Allocation of Resources (Linear Programming)

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A company makes 3 products: A, B and C. A B C Available A B C Available Profit 35 45 25 Labor Hrs 5 7 3 2000 hrs Fiberglass 18 25 12 7000 lbs At least 100 units each must be made of A, B, C How many A’s, B’s, and C’s should be produced in order to maximize total profits?

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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: 2000 7000 available: 2000 7000 Labor Fiberglass Profit Labor Fiberglass Profit make 100 A’s make 100 A’s make 100 C’s make 100 C’s remaining: remaining: How many B’s? 500 1800 3500 300 1200 2500 1200 4000 1200/7 = 171 4000/25 = 160 We run out of fiberglass 1st make 160 B’s 1120 4000 7200 remaining: 80 0 $13,200 Total Profit Optimal solution is $13,625 using LP (100 A, 100 B, 225 C) Difference of $425

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Linear Programming Formulation Max Z = 35A + 45B + 25C ST A = # of units of product A to produce B = # of units of product B to produce C = # of units of product C to produce 5A + 7B + 3C ≤ 2000 labor hours 18A + 25B + 12C ≤ 7000 fiberglass A ≥ 100 minimum A B ≥ 100 minimum B C ≥ 100 minimum C

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Linear Programming using Lindo software Max35 A + 45 B + 25 C Subject to 2)5 A + 7 B + 3 C <= 2000 3)18 A + 25 B + 12 C <= 7000 4)A >= 100 5)B >= 100 6)C >= 100 End LP Optimum found at step 4

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Objective Function Value 1) 13625.000 Variable Value Reduced Cost A100.000000.000000 A100.000000.000000 B100.000000.000000 B100.000000.000000 C225.000000.000000 C225.000000.000000 Row Slack or Surplus Dual Prices Row Slack or Surplus Dual Prices 2)125.000000.000000 3).000000 2.083333 4).000000 -2.500000 5).000000 -7.083333 6)125.000000.000000 No. Iterations = 4

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Ranges in which the basis is unchanged: Obj Coefficient Ranges Obj Coefficient Ranges VariableCurrent AllowableAllowable Coef IncreaseDecrease A35.000000 2.500000Infinity A35.000000 2.500000Infinity B45.000000 7.083333Infinity B45.000000 7.083333Infinity C25.000000 Infinity1.666667 C25.000000 Infinity1.666667 Righthand Side Ranges Righthand Side Ranges RowCurrent AllowableAllowable RHS IncreaseDecrease 2 2000.000000 Infinity 125.000000 3 7000.000000 500.000000 1500.000000 4 100.000000 83.333340 100.000000 5 100.000000 60.000000 100.000000 6 100.000000 125.000000 Inifinity

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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.

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Example Using Excel Solver LP Formulation: Max Z = 3P + 2.7R + 2.8L ST P + R + L < 27000capacity 3.5P + 2.9R + 2.6L < 55000barley 1.1P +.8R +.6L < 20000hops R – 2L > 02:1 ratio P > 4000minimum P requirement L > 4000minimum L requirement P + L < 12000maximum requirement

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Instructions for Using Excel to Solve LP Models 1. Set up spreadsheet like example in packet. (Z-value and LHS column should be formulas) 2. Select “Tools” on menu bar. Then select “Solver…”. 3. “Set Target Cell:” should be the cell of your Z-value formula. 4. Select “Min” or “Max”. 5. “By Changing Cells:” should be the range of cells for your decision variables values. 6. Select “Options…” 7. Check 2 boxes: “Assume Linear Model” and “Assume Non-Negative”. Then click “OK”. 8. Select “Add” to add constraints.

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9. In “Cell Reference:” box point to LHS formula of first constraint. Select. 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.

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ABCDEFG 1PRLObjective 2 Dec Vars 000 Value (Z) 3 Obj Coef 32.72.8=sumproduct(B3:D3,B$2:D$2) 4 5ConstraintsLHS<,=,>RHS 6capacity111=sumproduct(B6:D6,B$2:D$2)<27000 7barley3.52.92.6=sumproduct(B7:D7,B$2:D$2)<55000 8hops1.10.80.6=sumproduct(B8:D8,B$2:D$2)<20000 9 2:1 ratio 1-2=sumproduct(B9:D9,B$2:D$2)>0 10 min req. P 1=sumproduct(B10:D10,B$2:D$2)>4000 11 min req. L 1=sumproduct(B11:D11,B$2:D$2)>4000 12 max req. 11=sumproduct(B12:D12,B$2:D$2)<12000 =sumproduct(B3:D3,B2:D2) is equivalent to =B3*B2 + C3*C2 + D3*D2

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ABCDEFG 1PRLObjective 2 Dec Vars 40009761.9054880.952 Value (Z) 3 Obj Coef 32.72.852023.81 4 5ConstraintsLHS<,=,>RHS 6capacity11118642.86<27000 7barley3.52.92.655000<55000 8hops1.10.80.615138.1<20000 9 2:1 ratio 1-20>0 10 min req. P 14000>4000 11 min req. L 14880.952>4000 12 max req. 118880.952<12000

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Microsoft Excel 10.0 Answer Report Worksheet: [Book1]Sheet1 Report Created: 1/15/2003 9:35:20 AM Target Cell (Max) CellName Original Value Final Value $E$3 Obj Coef Value (Z) 052023.80952 Adjustable Cells CellName Original Value Final Value $B$2 Dec Vars P 04000 $C$2 Dec Vars R 09761.904762 $D$2 Dec Vars L 04880.952381 Constraints CellName Cell Value FormulaStatusSlack $E$6 capacity LHS 18642.85714$E$6<=$G$6 Not Binding 8357.142857 $E$7 barley LHS 55000$E$7<=$G$7Binding0 $E$8 hops LHS 15138.09524$E$8<=$G$8 Not Binding 4861.904762 $E$9 2:1 ratio LHS 0$E$9>=$G$9Binding0 $E$10 min req. P LHS 4000$E$10>=$G$10Binding0 $E$11 min req. L LHS 4880.952381$E$11>=$G$11 Not Binding 880.952381 $E$12 max req. LHS 8880.952381$E$12<=$G$12 Not Binding 3119.047619

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Microsoft Excel 10.0 Sensitivity Report Worksheet: [Book1]Sheet1 Report Created: 1/15/2003 9:35:20 AM Adjustable Cells FinalReducedObjectiveAllowableAllowable CellNameValueCostCoefficientIncreaseDecrease $B$2 Dec Vars P 4000030.4166666671E+30 $C$2 Dec Vars R 9761.90476202.70.4230769230.5 $D$2 Dec Vars L 4880.95238102.81E+300.379310345 Constraints FinalShadowConstraintAllowableAllowable CellNameValuePrice R.H. Side IncreaseDecrease $E$6 capacity LHS 18642.857140270001E+308357.142857 $E$7 barley LHS 550000.9761904765500018563.636367400 $E$8 hops LHS 15138.095240200001E+304861.904762 $E$9 2:1 ratio LHS 0-0.13095238102551.7241389034.482759 $E$10 min req. P LHS 4000-0.41666666740002114.2857144000 $E$11 min req. L LHS 4880.95238104000880.9523811E+30 $E$12 max req. LHS 8880.9523810120001E+303119.047619

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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.

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P = # of gallons of Premium ice cream to make L = # of gallons of Light ice cream to make Max Z = 100P + 100L ST.3P +.5L ≤ 140capacity of mixing machine 90P + 70L ≤ 28000max milk available Solution: P = 175; L = 175; Z = 35,000

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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.

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S = # tons of sand to add to mixture G = # tons of gravel to add to mixture Min Z = 6S + 8G ST S + G = 1500sand & gravel are 75% of 2000 S ≤ 800sand no more than 40% of 2000 G ≥ 600gravel at least 30% of 2000 Solution: S = 800; G = 700; Z = 10,400

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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.

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BF = # lbs beef to load in fore cargo hold BA = # lbs beef to load in aft cargo hold GF = # lbs grain to load in fore cargo hold GA = # lbs grain to load in aft cargo hold Max Z =.35 BF +.35BA +.12GF +.12 GA ST BF + GF ≤ 70000fore weight capacity – lbs BA + GA ≤ 90000aft weight capacity – lbs.2BF +.4GF ≤ 30000for volume capacity – cubic feet.2BA +.4GA ≤ 40000for volume capacity – cubic feet BF + BA ≤ 85000max beef available GF + GA ≤ 100000max grain available

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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.

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S = # jars apple Sauce to make J = # bottles apple Juice to make SA = $ for apple Sauce Advertising JA = $ for apple Juice Advertising Max Z = 1.45S + 1.75J -.6S -.85J – SA – JA ST S ≥.3(S + J)at least 30% apple sauce S ≤.6(S + J)no more than 60% apple sauce S ≤ 5000 + 3SAdon’t exceed demand for apple sauce J ≤ 4000 + 5JAdon’t exceed demand for apple juice.6S +.85J + SA + JA ≤ 16000budget

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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.

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12 variables(one for each time block) X1 = # of nurses starting at Midnight & working 8 hours X2 = “ 2am“ X3 = “ 4am“ X4 = “ 6am“ X5 = “ 8am“ X6 = “ 10am“ X7 = “ Noon“ X8 = “ 2pm“ X9 = “ 4pm“ X10 = “ 6pm“ X11 = “ 8pm“ X12 =“ 10pm“

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Min Z = X1 + X2 + X3 + X4 + X5 + X6 + ……. + X11 + X12 ST X1 + X10 + X11 + X12≥ 30midn – 2am X1 + X2 + X11 + X12≥ 202am – 4am X1 + X2 + X3 + X12≥ 404am – 6am X1 + X2 + X3 + X4≥ 506am – 8am X2 + X3 + X4 + X5≥ 608am – 10am X2 + X3 + X4 + X5≥ 608am – 10am X3 + X4 + X5 + X6≥ 8010am–Noon X3 + X4 + X5 + X6≥ 8010am–Noon X4 + X5 + X6 + X7≥ 80Noon – 2pm X4 + X5 + X6 + X7≥ 80Noon – 2pm X5 + X6 + X7 + X8≥ 702pm – 4pm X5 + X6 + X7 + X8≥ 702pm – 4pm X6 + X7 + X8 + X9≥ 704pm – 6pm X6 + X7 + X8 + X9≥ 704pm – 6pm X7 + X8 + X9 + X10≥ 606pm – 8pm X7 + X8 + X9 + X10≥ 606pm – 8pm X8 + X9 + X10 + X11≥ 508pm – 10pm X8 + X9 + X10 + X11≥ 508pm – 10pm X9 + X10 + X11 + X12≥ 5010pm – midn X9 + X10 + X11 + X12≥ 5010pm – midn

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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($) lb/Day Cost/lb($) Chicken200.20 Beef300.30 Pork150.50 Cereal Additive400.05 Each type of wiener has certain ingredient specifications, as follows. ________________________________________________________________________________ SpecificationsSelling Price/lb($) SpecificationsSelling Price/lb($) RegularNot more than 10% beef and pork combined Not less than 20% chicken$0.90 BeefNot less than 75% beef 1.25 All-MeatNo cereal additive Not more than 50% beef and pork combined 1.75 50% beef and pork combined 1.75 The firm wants to know the amount of wieners of each type to produce.

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14 variables(you could also formulate it with 11 variables) CR = # lbs Chicken ingredient in Regular wiener per day CB = # lbs Chicken ingredient in Beef wiener per day CM = # lbs Chicken ingredient in all-Meat wiener per day BR = # lbs Beef ingredient in Regular wiener per day BB = # lbs Beef ingredient in Beef wiener per day BM = # lbs Beef ingredient in all-Meat wiener per day PR = # lbs Pork ingredient in Regular wiener per day PB = # lbs Pork ingredient in Beef wiener per day PM = # lbs Pork ingredient in all-Meat wiener per day AR = # lbs Additive ingredient in Regular wiener per day AB = # lbs Additive ingredient in Beef wiener per day R = total lbs of Regular wiener B = total lbs of Beef wiener M = total lbs of all-Meat wiener

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Max Z = 0.90R + 1.25 B + 1.75 M -.2CR -.2CB -.2CM -.3BR -.3BB -.3BM -.5PR -.5PB -.5PM -.05AR -.05AB ST CR + BR + PR + AR = RR is sum of all ingredients in Regular CB + BB + PB + AB = BB is sum of all ingredients in Beef CM + BM + PM = MM is sum of all ingredients in Meat CR + CB + CM ≤ 200max Chicken ingredient available BR + BB + BM ≤ 300max Beef ingredient available PR + PB + PM ≤ 150max Pork ingredient available AR + AB ≤ 400max Additive ingredient available BR + PR ≤.1Rnot more than 10% BR+PR combined CR ≥.2Rnot less than 20% CR in Regular BB ≥.75Bnot less than 75% BB in Beef BM + PM ≤.5Mnot more than 50% BM+PM combined

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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 350. 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.)

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9 variables S1 = # tractors produced in month 1 using straight-time S2 = # tractors produced in month 2 using straight-time S3 = # tractors produced in month 3 using straight-time V1 = # tractors produced in month 1 using overtime V2 = # tractors produced in month 2 using overtime V3 = # tractors produced in month 3 using overtime C1 = # tractors carried in warehouse at end of month 1 C2 = # tractors carried in warehouse at end of month 2 C3 = # tractors carried in warehouse at end of month 3 sources of tractors = uses of tractors(for each month) production + beg.inv. = sales + end.inv.

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Min Z = 2000S1 + 2000S2 + 2000S3 + 3000V1 + 3000V2 + 3000V3 + 600C1 + 600C2 + 600C3 ST S1 + V1 + 0 = 250 + C1month 1: sources = uses S2 + V2 + C1 = 305 + C2month 2: sources = uses S3 + V3 + C2 = 350 + C3month 3: sources = uses 100S1 ≤ 29000straight-time capacity month 1 100S2 ≤ 29000straight-time capacity month 2 100S3 ≤ 29000straight-time capacity month 3 100V1 ≤ 2900overtime capacity month 1 100V2 ≤ 2900overtime capacity month 2 100V3 ≤ 2900overtime capacity month 3 C1 ≤ 40storage capacity month 1 C2 ≤ 40storage capacity month 2 C3 ≤ 40storage capacity month 3

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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 Mill Monthly Mill Destination Production Destination Production A B C D (cases) A B C D (cases) Mill 1$5.40 $6.20 $4.10 $4.90 15,000 Mill 2 4.00 7.10 5.60 3.90 10,000 Mill 3 4.50 5.20 5.50 6.10 15,000 Monthly Warehouse Requirement (cases) 9,000 9,000 12,000 10,000 How many cases of paper should be shipped per month from each mill to each warehouse to minimize monthly shipping costs?

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A1 = # of units shipped from Mill 1 to Destination A C3 = # of units shipped from Mill 3 to Destination C (12 variables) Min Z = 5.4A1 + 6.2B1 + 4.1C1 + 4.9D1 + 4.0A2 + 7.1B2 + 5.6C2 + 3.9D2 + 4.5A3 + 5.2B3 + 5.5C3 + 6.1D3 ST A1 + B1 + C1 + D1 ≤ 15000Mill 1 capacity A2 + B2 + C2 + D2 ≤ 10000Mill 2 capacity A3 + B3 + C3 + D3 ≤ 15000Mill 3 capacity A1 + A2 + A3 = 9000Destination A demand B1 + B2 + B3 = 9000Destination B demand C1 + C2 + C3 = 12000Destination C demand D1 + D2 + D3 = 10000Destination D demand

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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? Project Project 123 A876276 TeamB817664 C775470

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A1 = 1 if team A does project 1 = 0 if not(9 variables) = 0 if not(9 variables) Min Z = 87A1 + 62A2 + 76A3 + 81B1 + 76B2 + 64B3 + 77C1 + 54C2 + 70C3 ST A1 + A2 + A3 = 1 B1 + B2 + B3 = 1Each team does exactly one project C1 + C2 + C3 = 1 A1 + B1 + C1 = 1 A2 + B2 + C2 = 1Each project is done exactly once A3 + B3 + C3 = 1

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