B.5 Binary Indicator Variables Review Questions

Transcription

1 Lesson Topics Machine Use Indicators are binary variables where 0 indicates no use and 1 indicates positive use. They are part of a simple linear model of the fixed cost of machine use. Resource Allocation Problems with Fixed Costs trade off the advantage of using a variety of inputs to conform to fixed resources with the positive fixed cost of using each input. Product Mix Problems with Fixed Costs (1) trade off the advantage of producing a variety of goods to conform to fixed resources with the positive fixed cost of producing each good. Make or Buy Decisions with Fixed Costs (3) trade off the lower unit cost of producing a good yourself with the positive fixed cost of production. Relational Constraints (2) such as either project i or project j is completed or both project i and project j are completed can be written as linear constraints in binary indicator variables. Capital Budgeting (4) Problems maximize the net present value or net return from a selection of projects that each require a fixed amount of capital. Relational constraints are often imposed. 1

2 Product Mix with Fixed Costs Question. Muir Manufacturing produces three popular grades of commercial carpeting among its many other products. In the coming production period, Muir needs to decide how many rolls of each grade should be produced in order to maximize profit. Each roll of Grade X carpet uses 1 unit of synthetic fiber, requires 3 hours of production time, and needs 2 units of foam backing. Each roll of Grade Y carpet uses 2 units of synthetic fiber, requires 1 hour of production time, and needs 3 units of foam backing. Each roll of Grade Z carpet uses 3 units of synthetic fiber, requires 2 hours of production time, and needs 1 unit of foam backing. The selling price per roll of Grade X carpet is $2; the selling price per roll of Grade Y carpet is $3; and the selling price per roll of Grade Z carpet is $2. Muir has already paid for a fixed amount of inputs to be used in the coming production period. Muir has 6 units of synthetic fiber available for use; workers have been scheduled to provide up to 12 hours of production time; and the company has 24 units of foam backing available for use. Finally, Muir incurs a fixed setup cost of $2 if it produces any positive quantity of Grade X carpet; a fixed setup cost of $3 if it produces any positive quantity of Grade Y carpet; and a fixed setup cost of $4 if it produces any positive quantity of Grade Z carpet. Formulate a linear program for Muir to maximize its profit. But you need not compute an optimum. Tip: Your written answer should define the decision variables, carefully state which variables are continuous and which are binary, and formulate the objective and constraints. 2

3 Answer to Question: Continuous variables are used for the number of units made: Let X = the number of rolls of Grade X carpet to make Let Y = the number of rolls of Grade Y carpet to make Let Z = the number of rolls of Grade Z carpet to make Binary variables are used to indicate whether production is positive, and setup costs are incurred: Let SX = 1 if Grade X is produced; 0 if not Let SY = 1 if Grade Y is produced; 0 if not Let SZ = 1 if Grade Z is produced; 0 if not Objective: Maximize Profit 2X + 3Y + 2Z 2SX 3SY 4SZ Input Constraints: X + 2Y + 3Z < 6 (Synthetic fiber) 3X + Y + 2Z < 12 (Production time) 2X + 3Y + Z < 24 (Foam backing) Setup Constraints (given input constraints imply X < 6, Y < 12, Z < 24): X < 6SX Y < 12SY Z < 24SZ 3

4 Make or Buy with Fixed Costs Question. Danchuk Manufacturing produces a variety of classic automobiles, including a 1955 Chevy and a 1955 Thunderbird. Each car consists of three components that can be manufactured by Danchuk: a body, an interior, and an engine. Both cars use the same interior and engine, but different bodies. Danchuk s sales forecast indicates that 400 Chevys and 600 Thunderbirds will be needed to satisfy demand during the next year. Because only 2000 hours of in-house manufacturing time is available, Danchuk is considering purchasing some, or all, of the components from outside suppliers. If Denchuk manufactures a component in-house, it incurs a fixed setup cost as well as a variable manufacturing cost. The following table shows the setup cost, the manufacturing time per component, the manufacturing cost per component, and the cost to purchase each of the components from an outside supplier Manufacturing Cost per Unit (thousands of Purchase Cost per Unit (thousands of dollars) Component Setup Cost (thousands of dollars) Manufacturing Time per Unit (hours) dollars) Chevy Body Thunderbird Body Interior Engine Formulate a linear program for Denchuk to minimize total cost to meet the sales forecasts. Formulate the program, but you need not solve for the optimum. Tip: Your written answer should define the decision variables, carefully state which variables are continuous and which are binary, and formulate the objective and constraints. 4

5 Answer to Question: Continuous (or integer) variables are used for the number of units made: Let CB = the number of Chevy Bodies to make Let TB = the number of Thunderbird Bodies to make Let IN = the number of Interiors to make Let E = the number of Engines to make Continuous (or integer) variables are also used for the number of units purchased: Let CBP = the number of Chevy Bodies to purchase Let TBP = the number of Thunderbird Bodies to purchase Let INP = the number of Interiors to purchase Let EP = the number of Engines to purchase Binary variables are used to indicate whether production is positive, and setup costs are incurred: Let CBS = 1 if Chevy Bodies are produced; 0, if not. Let TBS = if Thunderbird Bodies are produced; 0, if not. Let INS = if Interiors are produced; 0, if not. Let ES = if Engines are produced; 0, if not. Objective: Minimize Total Cost 4CB + 2TB + 1IN + 4E (production costs) +5CBP + 3TBP + 2INP + 5EP (purchase costs) +100CBS + 90TBS + 30INS + 20ES (setup costs) Input Constraints: 3CB + 2TB + 1IN + 2E < 2000 (In-house hours) Sales Constraints: CB + CBP = 400 (manufactured or purchased bodies used for Chevys) TB + TBP = 600 (manufactured or purchased bodies used for Thunderbirds) IN + INP = 1000 (manufactured or purchased interiors used for both Chevys and Thunderbirds) 5

6 E + EP = 1000 (manufactured or purchased engines used for both Chevys and Thunderbirds) Setup Constraints (given sales constraints imply CB < 400, TB < 600, IN < 1000, and E < 1000): CB < 400CBS TB < 600TBS IN < 1000INS E < 1000ES 6

7 Make or Buy with Fixed Costs Question. Danchuk Manufacturing produces a variety of classic automobiles, including a 1955 Chevy and a 1955 Thunderbird. Each car consists of three components that can be manufactured by Danchuk: a body, an interior, and an engine. Both cars use the same engine, but different bodies and different interiors. Danchuk s sales forecast indicates that 300 Chevys and 500 Thunderbirds will be needed to satisfy demand during the next year. Because only 2000 hours of in-house manufacturing time is available, Danchuk is considering purchasing some, or all, of the components from outside suppliers. If Denchuk manufactures a component in-house, it incurs a fixed setup cost as well as a variable manufacturing cost. The following table shows the setup cost, the manufacturing time per component, the manufacturing cost per component, and the cost to purchase each of the components from an outside supplier Component Setup Cost (thousands of dollars) Manufacturing Time per Unit (hours) dollars) Chevy Body Thunderbird Body Chevy Interior Thunderbird Interior Engine Manufacturing Cost per Unit (thousands of Purchase Cost per Unit (thousands of dollars) Formulate a linear program for Denchuk to minimize total cost to meet the sales forecasts. But you need not compute an optimum. Tip: Your written answer should define the decision variables, carefully state which variables are continuous and which are binary, and formulate the objective and constraints.

8 Answer to Question: Continuous (or integer) variables are used for the number of units made: Let CB = the number of Chevy Bodies to make Let TB = the number of Thunderbird Bodies to make Let CI = the number of Chevy Interiors to make Let TI = the number of Thunderbird Interiors to make Let E = the number of Engines to make Continuous (or integer) variables are also used for the number of units purchased: Let CBP = the number of Chevy Bodies to purchase Let TBP = the number of Thunderbird Bodies to purchase Let CIP = the number of Chevy Interiors to purchase Let TIP = the number of Thunderbird Interiors to purchase Let EP = the number of Engines to purchase Binary variables are used to indicate whether production is positive, and setup costs are incurred: Let CBS = 1 if Chevy Bodies are produced; 0, if not. Let TBS = if Thunderbird Bodies are produced; 0, if not. Let CIS = 1 if Chevy Interiors are produced; 0, if not. Let TIS = if Thunderbird Interiors are produced; 0, if not. Let ES = if Engines are produced; 0, if not. Objective: Minimize Total Cost 4CB + 2TB + 2CI + 3TI + 4E (production costs) +5CBP + 3TBP + 3CIP + 4TIP + 5EP (purchase costs) +100CBS + 90TBS + 10CIS + 20TIS + 20ES (setup costs) Input Constraints: 3CB + 2TB + 1CI + 2TI + 2E < 2000 (In-house hours) 8

9 Sales Constraints: CB + CBP = 300 (manufactured or purchased bodies used for Chevys) TB + TBP = 500 (manufactured or purchased bodies used for Thunderbirds) CI + CIP = 300 (manufactured or purchased interiors used for Chevys) TI + TIP = 500 (manufactured or purchased interiors used for Thunderbirds) E + EP = 800 (manufactured or purchased engines used for both Chevys and Thunderbirds) Setup Constraints (given sales constraints imply CB < 300, TB < 500, CI < 300, TI < 500, and E < 800): CB < 300CBS TB < 500TBS CI < 300CIS TI < 500TIS E < 800ES 9

10 Make or Buy with Fixed Costs Question. Dell Inc. produces a variety of personal computers, including the OptiPlex brand and the Precision brand. Each brand consists of four components that can be manufactured by Dell: a motherboard, a monitor, a hard drive, and a case. Both brands use the same monitor and hard drive and case, but different motherboards. Dell s sales forecast indicates that 500 OptiPlexs and 300 Precisions will be needed to satisfy demand during the next month. Because only 2000 hours of in-house manufacturing time is available each month, Dell is considering purchasing some, or all, of the components from outside suppliers. If Dell manufactures a component in-house during a month, it incurs a fixed setup cost as well as a variable manufacturing cost. The following table shows the setup cost, the manufacturing time per component, the manufacturing cost per component, and the cost to purchase each of the components from an outside supplier Setup Cost (thousands of dollars) Manufacturing Time per Unit (hours) Manufacturing Cost per Unit (thousands of dollars) Purchase Cost per Unit (thousands of dollars) Component OptiPlex motherboard Precision motherboard Monitor Hard Drive Case Formulate a linear program for Dell to minimize total cost to meet the sales forecasts. But you need not compute an optimum. Tip: Your written answer should define the decision variables, carefully state which variables are continuous and which are binary, and formulate the objective and constraints. 10

11 Answer to Question: Continuous (or integer) variables are used for the number of units made: Let OM = the number of OptiPlex motherboards to make each month Let PM = the number of Precision motherboards to make each month Let M = the number of Monitors to make each month Let H = the number of Hard Drives to make each month Let C = the number of Cases to make each month Continuous (or integer) variables are also used for the number of units purchased: Let OMP = the number of OptiPlex motherboards to purchase each month Let PMP = the number of Precision motherboards to purchase each month Let MP = the number of Monitors to purchase each month Let HP = the number of Hard Drives to purchase each month Let CP = the number of Cases to purchase each month Binary variables are used to indicate whether production is positive, and setup costs are incurred: Let OMS = 1 if OptiPlex motherboards are produced; 0, if not Let PMS = 1 if Precision motherboards are produced; 0, if not Let MS = 1 if Monitors are produced; 0, if not Let HS = 1 if Hard Drives are produced; 0, if not Let CS = 1 if Cases are produced; 0, if not Objective: Minimize Total Cost 4OM + 2PM+ 1M + 5H + 3C (production costs) +5OMP + 3PMP+ 2MP + 1HP + 4CP (purchase costs) +40OMS + 50PMS+ 20MS + 10HS + 30CS (setup costs) Input Constraints: 3OM + 2PM+ 1M + 4H + 5C < 2000 (In-house hours) 11

12 Sales Constraints: OM + OMP = 500 (manufactured or purchased OptiPlex motherboards) PM + PMP = 300 (manufactured or purchased Precision motherboards) M +MP = 800 (manufactured or purchased Monitors used for both OptiPlexs and Precisions) H +HP = 800 (manufactured or purchased Hard Drives used for both OptiPlexs and Precisions) C +CP = 800 (manufactured or purchased Cases used for both OptiPlexs and Precisions) Setup Constraints (given sales constraints imply OM < 500, PM < 300, M < 800, H < 800, and C < 800): OM < 500OMS PM < 300PMS M < 800MS H < 800HS C < 800CS 12

13 Relational Constraints Question. Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year. The following table summarizes the number of engineers and the number of support personnel required for each project, and the expected profits for each project: Project Engineers Required Support Personnel Required Profit ($1,000,000s) Formulate an integer program that maximizes Tower's profit subject to the following management constraints: 1) Use no more than 175 engineers 2) Use no more than 150 support personnel 3) If either project 4 or project 6 is done, both must be done 4) Project 2 can be done only if project 1 is done 5) If project 5 is done, project 3 must not be done. And if project 3 is done, project 5 must not be done. 6) No more than three projects are to be done. Formulate the program, but you need not compute an optimum. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 13

14 Answer to Question: Define binary variables, where P i = 1 if project i is done, and = 0 if not. Max P P 2 + 2P P P P 6 s.t. 20P P P P P P 6 < P P P P P P 6 < 150 P 4 - P 6 = 0 P 1 - P 2 > 0 P 3 + P 5 < 1 P 1 + P 2 + P 3 + P 4 + P 5 + P 6 < 3 P i = 0 or 1 (Note: P 3 + P 5 < 1 says the same thing as If project 5 is done, project 3 must not be done. And if project 3 is done, project 5 must not be done.) 14

15 Relational Constraints Question. Grush Consulting has five projects to consider. Each will require time in the next two quarters according to the table below. Project Time in first quarter Time in second quarter Revenue A B C D E Revenue from each project is also shown. Formulate and solve a model whose solution would maximize revenue; meet the time budget of 25 in the first quarter and 20 in the second quarter; not do both projects C and D; and do project D only if project E is done. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 15

16 Answer to Question: Let A = 1 if project A is done, 0 otherwise; same for B, C, D, and E Max 12000A B C D E s.t. 5A + 3B + 7C + 2D + 15E 25 8A + 12B + 5C + 3D + 1E 20 C + D 1 D - E 0 16

17 Capital Budgeting Question. Frys Electronics is planning to expand its sales operation by offering new electronic appliances. The company has identified seven new product lines it can carry. Initial Floor Space Exp. Rate Product Line Invest. (Sq.Ft.) of Return 1. ProjectionTVs $ 6, % 2. Cell Phones $ 12, Plasma TVs $ 20, IPODs $14, DVD Players $15, PDAs $ 2, Computers $32, Frys will not stock Projection TVs unless they stock both Plasma TVs and IPODs. They will not stock both Cell Phones and PDAs. And they will stock DVD Players only if they stock Computers. Finally, the company wishes to introduce at most four new product lines. If the company has $45,000 to invest and 420 sq. ft. of floor space available, formulate an integer linear program for Metropolitan to maximize its overall expected return. But you need not compute an optimum. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 17

18 Answer to Question: Define the Decision Variables: x 1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on. Product line 1. Projection TVs Product line 2. Cell Phones Product line 3. Plasma TVs Product line 4. IPODs Product line 5. DVD Players Product line 6. PDAs Product line 7. Computers Define the Objective: Maximize total expected return. Max.181(6000)x (12000)x (20000)x (14000)x (15000)x (2000)x (32000)x 7 1. Constrain total investment by $45,000: 6000x x x x x x x 7 < 45, Constrain space by 420 square feet: 125x x x 3 +40x x x x 7 < Frys will not stock Projection TVs unless they stock both Plasma TVs and IPODs: x 3 > x 1 and x 4 > x 1. Another linear way to write that constraint is x 3 + x 4 > 2x 1, and a nonlinear way is x 3 x 4 > x Frys will not stock both Cell Phones and PDAs: x 2 + x 6 < 1 5. Frys will stock DVD Players only if they stock Computers: x 5 < x 7 6. Frys will introduce at most four new product lines: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 < 4 18

19 Capital Budgeting Question. Frys Electronics is planning to expand its sales operation by offering new electronic appliances. The company has identified seven new product lines it can carry. Product Line Investment ($) Floor Space (Square Feet) Expected Return (Percent) Projection TVs 6, Cell Phones 12, Plasma TVs 20, IPODs 14, DVD Players 15, PDAs 2, Computers 32, Frys will not stock Projection TVs unless they stock both Plasma TVs and IPODs. They will not stock both Cell Phones and PDAs. And they will stock DVD Players only if they stock Computers. Finally, the company wishes to introduce at most four new product lines. If the company has $45,000 to invest and 420 square feet of floor space available, formulate an integer linear program for Frys to maximize its overall expected return. But you need not compute an optimum. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 19

20 Answer to Question: Define the Decision Variables: x 1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on. Product line 1. Projection TVs Product line 2. Cell Phones Product line 3. Plasma TVs Product line 4. IPODs Product line 5. DVD Players Product line 6. PDAs Product line 7. Computers Define the Objective: Maximize total expected return. Max.16(6000)x (12000)x (20000)x (14000)x (15000)x (2000)x (32000)x 7 1. Constrain total investment by $45,000: 6000x x x x x x x 7 < 45, Constrain space by 420 square feet: 125x x x 3 +40x x x x 7 < Frys will not stock Projection TVs unless they stock both Plasma TVs and IPODs: x 3 > x 1 and x 4 > x 1. Another linear way to write that constraint is x 3 + x 4 > 2x 1, and a nonlinear way is x 3 x 4 > x Frys will not stock both Cell Phones and PDAs: x 2 + x 6 < 1 5. Frys will stock DVD Players only if they stock Computers: x 5 < x 7 6. Frys will introduce at most four new product lines: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 < 4 20

21 Capital Budgeting Question. Frys Electronics is planning to expand its sales operation by offering new electronic appliances. The company has identified seven new product lines it can carry. Initial Floor Space Exp. Rate Product Line Invest. (Sq.Ft.) of Return 1. ProjectionTVs $ 6, % 2. Cell Phones $ 12, Plasma TVs $ 20, IPODs $14, DVD Players $15, PDAs $ 2, Computers $32, If Frys stocks Projection TVs, then they stock both Plasma TVs and IPODs. They will not stock both Cell Phones and PDAs. And they will stock DVD Players if, and only if, they stock Computers. Finally, the company wishes to introduce at most four new product lines. If the company has $45,000 to invest and 420 sq. ft. of floor space available, formulate an integer linear program for Metropolitan to maximize its overall expected return. But you need not compute an optimum. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 21

22 Answer to Question: Define the Decision Variables: x 1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on. Product line 1. Projection TVs Product line 2. Cell Phones Product line 3. Plasma TVs Product line 4. IPODs Product line 5. DVD Players Product line 6. PDAs Product line 7. Computers Define the Objective: Maximize total expected return. Max.081(6000)x (12000)x (20000)x (14000)x (15000)x (2000)x (32000)x 7 1. Constrain total investment by $45,000: 6000x x x x x x x 7 < 45, Constrain space by 420 square feet: 125x x x 3 +40x x x x 7 < If Frys stocks Projection TVs, then they stock both Plasma TVs and IPODs: x 3 > x 1 and x 4 > x 1. Another linear way to write that constraint is x 3 + x 4 > 2x 1, and a nonlinear way is x 3 x 4 > x Frys will not stock both Cell Phones and PDAs: x 2 + x 6 < 1 5. Frys will stock DVD Players if, and only if, they stock Computers: x 5 = x 7 6. Frys will introduce at most four new product lines: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 < 4 22

23 Capital Budgeting Question. Newegg.com offers the best prices on computer parts, laptop computers, digital cameras, electronics and more with fast shipping. The company has identified seven new product lines it can carry. Each requires a specific investment and floor space at the warehouse. Product Line Investment Floor Space Expected Return ($) (Square Feet) (Percent) Internal Hard Drive 8, LCD Monitor 7, Laptop Computer 20, Portable DVD Player 14, Universal Remote Control 15, Flash Card 2, Computer Server 32, Newegg.com will not stock Portable DVD Players unless they stock both LCD Monitors and Flash Cards. They will not stock both Internal Hard Drives and Computer Servers. And they will stock LCD Monitors only if they stock Universal Remote Controls. Finally, the company will introduce at least three new product lines. If the company has $45,000 to invest and 420 square feet of floor space available, formulate an integer linear program for Newegg.com to maximize its overall expected return. But you need not compute an optimum. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 23

24 Answer to Question: Define the Decision Variables: x 1 = 1 if product line 1 is introduced, and = 0 otherwise. And so on. Product line 1. Internal Hard Drive Product line 2. LCD Monitor Product line 3. Laptop Computer Product line 4. Portable DVD Player Product line 5. Universal Remote Control Product line 6. Flash Card Product line 7. Computer Server Define the Objective: Maximize total expected return. Max.14(8000)x (7000)x (20000)x (14000)x (15000)x (2000)x (32000)x 7 1. Constrain total investment by $45,000: 8000x x x x x x x 7 < 45, Constrain space by 420 square feet: 120x x x 3 +40x x x x 7 < will not stock Portable DVD Players unless they stock both LCD Monitors and Flash Cards: x 2 > x 4 and x 6 > x 4. Another linear way to write that constraint is x 2 + x 6 > 2x 4. (And if we allowed nonlinear constraints x 2 x 6 > x 4.) 4. will not stock both Internal Hard Drives and Computer Servers: x 1 + x 7 < 1 5. will stock LCD Monitors only if they stock Universal Remote Controls: x 2 < x 5 6. will introduce at least three new product lines: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 > 3 24