Supplement I

Strategic Allocation of Resources

(Linear Programming)

A company makes 3 products: A, B and C.

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?

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

Labor Fiberglass Profit

make 100 A’s

make 100 C’s

remaining:

How many B’s?

Linear Programming using Lindo software

Max 35 A + 45 B + 25 C

Subject to2) 5 A + 7 B + 3 C <= 20003) 18 A + 25 B + 12 C <=

70004) A >= 1005) B >= 1006) C >= 100

EndLP Optimum found at step 4

Objective Function Value

1) 13625.000

Variable Value Reduced Cost A 100.000000 .000000 B 100.000000 .000000 C 225.000000 .000000

Row Slack or Surplus Dual Prices2) 125.000000 .0000003) .000000 2.0833334) .000000 -2.5000005) .000000 -7.0833336) 125.000000 .000000

No. Iterations = 4

Ranges in which the basis is unchanged:

Obj Coefficient RangesVariable Current Allowable Allowable

Coef Increase Decrease A 35.000000 2.500000 Infinity B 45.000000 7.083333 Infinity C 25.000000 Infinity 1.666667

Righthand Side RangesRow Current Allowable Allowable

RHS Increase Decrease2 2000.000000 Infinity 125.0000003 7000.000000 500.000000 1500.0000004 100.000000 83.333340 100.0000005 100.000000 60.000000 100.0000006 100.000000 125.000000 Inifinity

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 3.6 pounds of barley and 1.2 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.

Example Using Excel Solver

LP Formulation:

Max Z = 3P + 2.7R + 2.8L

ST

P + R + L < 27000 capacity

3.5P + 2.9R + 2.6L < 55000 barley

1.1P + .8R + .6L < 20000 hops

R – 2L > 0 2:1 ratio

P > 4000 minimum P requirement

L > 4000 minimum L requirement

P + L < 12000 maximum requirement

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.

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.

A B C D E F G

1 P R L Objective

2 Dec Vars 0 0 0 Value (Z)

3 Obj Coef 3 2.7 2.8 =sumproduct(B3:D3,B$2:D$2)

4

5 Constraints LHS <,=,> RHS

6 capacity 1 1 1 =sumproduct(B6:D6,B$2:D$2) < 27000

7 barley 3.5 2.9 2.6 =sumproduct(B7:D7,B$2:D$2) < 55000

8 hops 1.1 0.8 0.6 =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. 1 1 =sumproduct(B12:D12,B$2:D$2) < 12000

=sumproduct(B3:D3,B2:D2) is equivalent to =B3*B2 + C3*C2 + D3*D2

A B C D E F G

1 P R L Objective

2 Dec Vars 4000 9761.905 4880.952 Value (Z)

3 Obj Coef 3 2.7 2.8 52023.81

4

5 Constraints LHS <,=,> RHS

6 capacity 1 1 1 18642.86 < 27000

7 barley 3.5 2.9 2.6 55000 < 55000

8 hops 1.1 0.8 0.6 15138.1 < 20000

9 2:1 ratio 1 -2 0 > 0

10 min req. P 1 4000 > 4000

11 min req. L 1 4880.952 > 4000

12 max req. 1 1 8880.952 < 12000

Microsoft Excel 10.0 Answer Report

Worksheet: [Book1]Sheet1

Report Created: 1/15/2003 9:35:20 AM

Target Cell (Max)

Cell Name Original Value Final Value

$E$3 Obj Coef Value (Z) 0 52023.80952

Adjustable Cells

Cell Name Original Value Final Value

$B$2 Dec Vars P 0 4000

$C$2 Dec Vars R 0 9761.904762

$D$2 Dec Vars L 0 4880.952381

Constraints

Cell Name Cell Value Formula Status Slack

$E$6 capacity LHS 18642.85714 $E$6<=$G$6 Not Binding 8357.142857

$E$7 barley LHS 55000 $E$7<=$G$7 Binding 0

$E$8 hops LHS 15138.09524 $E$8<=$G$8 Not Binding 4861.904762

$E$9 2:1 ratio LHS 0 $E$9>=$G$9 Binding 0

$E$10 min req. P LHS 4000 $E$10>=$G$10 Binding 0

$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

Microsoft Excel 10.0 Sensitivity Report

Worksheet: [Book1]Sheet1

Report Created: 1/15/2003 9:35:20 AM

Adjustable Cells

    Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$2 Dec Vars P 4000 0 3 0.416666667 1E+30

$C$2 Dec Vars R 9761.904762 0 2.7 0.423076923 0.5

$D$2 Dec Vars L 4880.952381 0 2.8 1E+30 0.379310345

Constraints

    Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$E$6 capacity LHS 18642.85714 0 27000 1E+30 8357.142857

$E$7 barley LHS 55000 0.976190476 55000 18563.63636 7400

$E$8 hops LHS 15138.09524 0 20000 1E+30 4861.904762

$E$9 2:1 ratio LHS 0 -0.130952381 0 2551.724138 9034.482759

$E$10 min req. P LHS 4000 -0.416666667 4000 2114.285714 4000

$E$11 min req. L LHS 4880.952381 0 4000 880.952381 1E+30

$E$12 max req. LHS 8880.952381 0 12000 1E+30 3119.047619

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.

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.

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.

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.

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.

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($)

Chicken 200 .20Beef 300 .30Pork 150 .50Cereal Additive 400 .05

Each 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%

1.25All-Meat No cereal additive

Not more than50% beef and pork combined

1.75

The firm wants to know the amount of wieners of each type to produce in order to minimize cost.

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.)

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

Destination Production

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 15,000Monthly 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?

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

1 2 3

A 87 62 76

Team B 81 76 64

C 77 54 70