Readings Readings Chapter 7 Integer Linear Programming.

Readings Readings Chapter 7 Integer Linear Programming

Overview Overview

Overview 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 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 trade off the lower unit cost of producing a good yourself with the positive fixed cost of production. Relational Constraints 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 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.

Tool Summary Tool Summary Use binary variables to indicate whether an activity, such as a production run, is undertaken. Write a multiple-choice constraint: The sum of two or more binary variables equals 1, so any feasible solution choose one variable to equal 1. Write a mutually-exclusive constraint: The sum of two or more binary variables is at most 1, so any feasible solution chooses at most one variable to equal 1. All variables could equal 0. Write a conditional constraint: An inequality constraint so that one binary variable cannot equal unless certain other binary variables also equal 1. Write a corequisite constraint: An equality constraint of binary variables, so are either both 0 or both 1.

Machine Indicators Machine Indicators

Machine Indicators Overview 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. For example, consider a standard Resource Allocation Problem with Machines, where the maximum machine resource available is 20. Suppose fixed setup costs of using a machine restrict machine use X so “either X = 0 or 6 < X < 20”. But “either X = 0 or 6 < X < 20” is not a linear constraint. To make “either X = 0 or 6 < X < 20” a linear constraint, introduce binary variable Y and replace the standard resource constraint X < 20 with “6Y < X < 20Y”. The linear constraints “6Y < X < 20Y” can be satisfied in one of two ways. On the one hand, if Y = 0, then the constraints read “6(0) < X < 20(0)” and so are satisfied only by X = 0. On the one hand, if Y = 1, then the constraints read “6(1) < X < 20(1)” and so are satisfied only by 6 < X < 20.

Machine Indicators Question: Ji-Yang Plastic makes plastic parts in Taiwan used in automobiles and computers. One of its major contracts involves the production of plastic printer cases for a computer company’s portable printers. The printer cases can be produced on two injection molding machines. The M-100 machine has a production capacity of 25 printer cases per hour, and the M-200 machine has a production capacity of 40 printer cases per hour. Both machines use the same chemical to produce the printer cases; the M-100 uses 40 pounds of raw material per hour, and the M-200 uses 50 pounds per hour.

Machine Indicators The computer company asked Ji-Yang to produce as many of the cases as possible during the upcoming week; it will pay $18 for each case. However, next week is a regularly scheduled vacation period for most of Ji-Yang’s production employees. During this time, annual maintenance is performed on all equipment. Because of the downtime for maintenance, the M-100 is only available for at most 15 hours, and the M-200 for at most 10 hours.

Machine Indicators The supplier of the chemical used in the production process informed Ji-Yang that a maximum of 1000 pounds of the chemical material will be available for next week’s production; the cost for this raw material is $6 per pound. In addition to the raw material cost, Ji-Yang estimates that the hourly cost of operating the M-100 and the M-200 are $50 and $75, respectively.

Machine Indicators However, because of the high setup cost on both machines, management requires that, if a machine is used at all, it must be used for at least 5 hours. (That last constraint makes the fixed cost of producing with a machine equal to the cost of using the machine for 5 hours.)

Define decision variables:
Machine Indicators Answer: Define decision variables: M1 = number of hours spent on the M-100 machine M2 = number of hours spent on the M-200 machine Total revenue = 18(25) M1 + 18(40) M2 = 450 M M2 Total cost = 6(40) M1 + 6(50) M M M2 = 290 M M2

Mixed binary formulation Define objective:
Machine Indicators Mixed binary formulation Define objective: Maximize (profit = revenue-cost) 160 M M2 Define constraints: M1 < (M-100 maximum) M2 < (M-200 maximum) 40M M2 < 1000 (Raw Material) M1 > 5 U1 (M-100 minimum, if M1 > 0) M2 > 5 U2 (M-200 minimum, if M2 > 0) 15U1 > M1 (so U1 = 1 if M1 > 0) 10U2 > M2 (so U2 = 1 if M2 > 0) M1, M2 > 0 and U1, U2 binary (0 or 1)

Mixed binary formulation
Machine Indicators Mixed binary formulation

Use M1 for 12.5 hours, and M2 for 10 hours.
Machine Indicators Mixed binary solution Use M1 for 12.5 hours, and M2 for 10 hours.

Resource Allocation with Fixed Cost

Overview Resource Allocation Problems with Fixed Cost trade off the advantage of using a variety of inputs to conform to fixed resources with the positive fixed cost of using each input. The simplest way to model those fixed costs in a linear program is with binary (0 or 1) variables used as indicators, where 0 values indicate no production and 1 indicates positive production. For example, consider a standard Resource Allocation Problem, where for one of the inputs, the maximum resource available is 10, and the cost of using X is 3X. Now suppose there is an added fixed setup costs of 15. To model those fixed costs in a linear program, introduce binary variable Y, replace the resource supply constraint X < 10 with “X < 10Y”, and add the cost term 15Y so total cost becomes 3X + 15Y. The constraint “X < 10Y” can be satisfied in one of two ways. On the one hand, if X = 0, then the constraint reads “0 < 10Y” and so can be satisfied by Y = 0, and so total cost = 3X. On the one hand, if X > 0, then the constraint “X < 10Y” can only be satisfied by Y = 1, and so total cost = 3X + 15.

Question: A businessman is considering opening a small specialized trucking firm. To make the firm profitable, it is estimated that it must have a daily trucking capacity of at least 12 tons. Two types of trucks are appropriate for the specialized operation. Their characteristics and costs are summarized in the table below. Note that each large truck require 2 drivers for long-haul trips. There are 8 potential drivers available and there are facilities for at most 4 trucks.

Truck Cost (tons) Needed
Resource Allocation with Fixed Cost The businessman's objective is to minimize the total cost outlay for trucks. Capacity Drivers Truck Cost (tons) Needed Small $10, Large $30, Formulate and graphically solve a linear programming model for this problem. How many of each type of truck should be used? Reformulate the problem if the trucking firm incurs a fixed setup cost of $2 if it uses any positive quantity of small trucks, and a fixed setup cost of $4 if it uses any positive quantity of large trucks?

Let S = the number of small trucks used
Resource Allocation with Fixed Cost Answer: Let S = the number of small trucks used Let L = the number of large trucks used Min 10000S L s.t. 2S + 4L > 12 (capacity constraint) S + 2L < (driver constraint) S + L < (facility constraint) S, L > 0 A graph of the feasible set and iso-value lines shows the optimal solution occurs where the first and third constraints bind (the second constraint is redundant). Solving the binding form of those two constraints yields the optimal solution: S = 2, L = 2.

Question: How would the formulation change if the trucking firm incurs a fixed setup cost of $2 if it uses any positive quantity of small trucks, and a fixed setup cost of $4 if it uses any positive quantity of large trucks?

Let SS = 1 if Small Trucks are used; 0 if not
Resource Allocation with Fixed Cost Answer: Let SS = 1 if Small Trucks are used; 0 if not Let SY = 1 if Large Trucks are used; 0 if not Objective: Min 10000S L + 2SS + 4SY Input Constraints: 2S + 4L > 12 (capacity constraint) S + 2L < (driver constraint) S + L < (facility constraint) S, L > 0 Setup Constraints (given input constraints imply S < 4, L < 4): S < 4SS L < 4SL

Product Mix with Fixed Cost

Overview Product Mix Problems with Fixed Cost trade off the advantage of producing a variety of goods to conform to fixed resources with the positive fixed cost of producing each good. The simplest way to model those fixed costs in a linear program is with binary (0 or 1) variables used as indicators, where 0 values indicate no production and 1 indicates positive production. For example, consider a standard Product Mix Problem, where for one of the outputs, the maximum production demanded is 10, and the cost of producing X is 3X. Now suppose there is an added fixed setup costs of 15. To model those fixed costs in a linear program, introduce binary variable Y, replace the demand constraint X < 10 with “X < 10Y”, and add the cost term 15Y so total cost becomes 3X + 15Y. The constraint “X < 10Y” can be satisfied in one of two ways. On the one hand, if X = 0, then the constraint reads “0 < 10Y” and so can be satisfied by Y = 0, and so total cost = 3X. On the one hand, if X > 0, then the constraint “X < 10Y” can only be satisfied by Y = 1, and so total cost = 3X + 15.

Formulate a linear program to maximize profit.
Product Mix with Fixed Cost Question: Iron Elegance seeks to maximize profit by making two products from steel. It just received this month's allocation of 19 pounds of steel. It takes 2 pounds of steel to make a unit of product 1, and 3 pounds of steel to make a unit of product 2. The physical plant has the capacity to make at most 6 units of product 1, and at most 8 units of total product (product 1 plus product 2). Product 1 sells for price 7, has marginal cost 2, and fixed cost 3. Product 2 sells for price 8, has marginal cost 1, and fixed cost 2. Formulate a linear program to maximize profit.

Let x1 and x2 denote this month's production level of
Make or Buy with Fixed Cost Answer: A mixed integer linear program can be set up to solve this problem. Binary variables are used to indicate whether or not we setup to produce the two products. Let x1 and x2 denote this month's production level of product 1 and product 2. y1 = 1 if x1 > 0; y1 = 0 if not y2 = 1 if x2 > 0; y2 = 0 if not

The total monthly profit =
Product Mix with Fixed Cost The total monthly profit = (profit per unit of product 1) x (monthly production of product 1) + (profit per unit of product 2) x (monthly production of product 2) = (7-2)x1 + (8-1)x2 - 3y1 - 2y2 Maximize total monthly profit: Max 5x1 + 7x2 - 3y1 - 2y2

Here is a mathematical formulation of constraints. The total amount of steel used during monthly production = (steel used per unit of product 1) x (monthly production of product 1) + (steel used per unit of product 2) x (monthly production of product 2) = 2x1 + 3x2 That quantity must be less than or equal to the allocated 19 pounds of steel (the inequality < in the constraint below assumes excess steel can be freely disposed; if disposal is impossible, then use equality =) : 2x1 + 3x2 < 19 The constraint that the physical plant has the capacity to make at most 6 units of product 1 is formulated x1 < 6 The constraint that the physical plant has the capacity to make at most 8 units of total product (product 1 plus product 2) is x1 + x2 < 8

Given resource constraint 2x1 + 3x2 < 19 implies x1 < 19/2 and x2 < 19/3, setup indicators x1 < (19/2) y1 x2 < (19/3) y2

Adding the non-negativity and binary constraints complete the formulation. Objective function Max x1 + 7x2 - 3y1 - 2y2 s.t. x < 6 2x1 + 3x2 < 19 x1 + x2 < 8 x1 < (19/2) y1 x2 < (19/3) y2 x1 > 0 and x2 > 0 y1 = 0 or 1 y2 = 0 or 1 Standard constraints Setup constraints Non-negativity Binary

The Management Scientist can solve this mixed integer linear program of 2 binary variables Yi and 2 continuous variables Xi.

Make or Buy with Fixed Cost

Overview Make or Buy Decisions with Fixed Cost trade off the lower unit cost of producing a good yourself with the positive fixed cost of production. The simplest way to model those fixed costs in a linear program is with binary (0 or 1) variables used as indicators, where 0 values indicate no production and 1 indicates positive production. For example, consider a standard Make or Buy Decision, where the maximum production demanded is 10, and the cost of producing X is 3X. Now suppose there is an added fixed setup costs of 15. To model those fixed costs in a linear program, introduce binary variable Y, replace the demand constraint X < 10 with “X < 10Y”, and add the cost term 15Y so total cost becomes 3X + 15Y. The constraint “X < 10Y” can be satisfied in one of two ways. On the one hand, if X = 0, then the constraint reads “0 < 10Y” and so can be satisfied by Y = 0, and so total cost = 3X. On the one hand, if X > 0, then the constraint “X < 10Y” can only be satisfied by Y = 1, and so total cost = 3X + 15.

Question: Sony produces remote controllers for both televisions and DVD players. Each controller consists of three subassemblies that are made by Sony --- a base, a cartridge, and a keypad. Both controllers use the same base subassembly, but different cartridge and keypad subassemblies. Sony’s sales forecast is that 7000 TV controllers and 5000 DVD controllers will be needed to satisfy demand during the upcoming Christmas season. Because only 500 hours of in-house manufacturing time is available, Sony considers buying some, or all, of the subassemblies from outside suppliers.

If Sony makes a subassembly, it incurs a fixed setup cost as well as a variable manufacturing cost. For each subassembly, the following are the fixed setup costs, the manufacturing time, the manufacturing costs, and the purchase costs: Subassembly Setup Cost ($) Manufacturing Time per Unit (min.) Manufacturing Cost per Unit ($) Purchase Cost per Unit ($) Base 1000 0.9 0.40 0.65 TV cartridge 1200 2.2 2.90 3.45 DVD cartridge 1900 3.0 3.15 3.70 TV keypad 1500 0.8 0.30 0.50 DVD keypad 1.0 0.55 0.70

Formulate a linear program to determine the optimal production and purchase of each subassembly.

SB = 1 if bases are produced; 0 if not
Make or Buy with Fixed Cost Answer: A mixed integer linear program can be set up to solve this problem. Binary variables are used to indicate whether or not we setup to produce the subassemblies. SB = 1 if bases are produced; 0 if not STVC = 1 if TV cartridges are produced; 0 if not SVCRC = 1 if VCR cartridges are produced; 0 if not STVP = 1 if TV keypads are produced; 0 if not SVCRP = 1 if VCR keypads are produced; 0 if not

Continuous variables are used for the number of units made or purchased. BM = Number of bases made BP = Number of bases purchased TVCM = Number of TV cartridges made TVCP = Number of TV cartridges purchased VCRCM = Number of VCR cartridges made VCRCP = Number of VCR cartridges purchased TVPM = Number of TV keypads made TVPP = Number of TV keypads purchased VCRPM = Number of VCR keypads made VCRPP = Number of VCR keypads purchased

0.4BM+2.9TVCM+3.15VCRCM+0.3TVPM+0.55VCRPM (purchase cost)
Make or Buy with Fixed Cost Minimize cost (manufacturing cost) 0.4BM+2.9TVCM+3.15VCRCM+0.3TVPM+0.55VCRPM (purchase cost) +0.65BP+3.45TVCP+3.7VCRCP+0.5TVPP+0.7VCRPP (setup cost) +1000SB+1200STVC+1900SVCRC+1500STVP+1500SVCRP

Demand constraints 1) BM+BP=12000 2) TVCM+TVCP=7000
Make or Buy with Fixed Cost Demand constraints 1) BM+BP=12000 2) TVCM+TVCP=7000 3) VCRCM+VCRCP=5000 4) TVPM+TVPP=7000 5) VCRPM+VCRPP=5000

Manufacturing time constraint (in minutes)
Make or Buy with Fixed Cost Manufacturing time constraint (in minutes) 6) 0.9BM+2.2TVCM+3VCRCM+0.8TVPM+1VCRPM < 30000 Setup indicators (given demand constraints) 7) BM-12000SB < 0 8) TVCM-7000STVC < 0 9) VCRCM-5000SVCRC < 0 10) TVPM-7000STVP < 0 11) VCRPM-5000SVCRP < 0

The Management Scientist can solve this mixed integer linear program of 5 binary variables and 10 continuous variables.

Relational Constraints

Overview Relational Constraints 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.

Binary variables can allow mathematical formulations of some intricate verbal descriptions. Let xi and xj represent binary variables designating whether projects i and j have been completed (1 means completed, 0 means not completed). The constraint “either project i or project j is completed” is formulated xi + xj > 1. In particular, xi + xj > 1 if both projects are completed (both variables equal 1) That is the standard use of “or” in quantitative or mathematical work. It is the inclusive “or”. For another example, if you are told “your friend is either at the snack bar or on the tennis court”, then you are told the truth if your friend is on the tennis court.

The relation “xi + xj = 1” is sometimes called the exclusive “or”.
Relational Constraints The constraint “either project i or project j is completed, but not both” is formulated xi + xj = 1. In particular, xi + xj = 1 is false if both projects are completed (both variables equal 1). The relation “xi + xj = 1” is sometimes called the exclusive “or”. There is debate over whether the English word “or” is inclusive or exclusive. Some say “you may have coffee or tea” is an example of an exclusive “or”. But it turns out that “or” is not a logical or at all because “you may have coffee or tea” is not considered true if you can only have coffee. In this class, “or” will only mean the inclusive “or”.

Summary Relational Constraints
Let xi be a binary variables that is 1 if project i is done; xj likewise. xi + xj = 1 means “either project i or project j is done, but not both” xi + xj < 1 means “project i and project j are not both done” xi + xj < 1 means “project i and project j will not both be done” xi + xj > 1 means “at least one of project i or project j is done” xi < xj means “if project i is done, then project j is done” xi > xj means “project i is done if project j is done” xi < xj means “project i is done only if project j is done” xi = xj means “project i is done if, and only if, project j is done” xi < xj means “project i will not be done unless project j is done” xi < xj means “project i will not be done unless project i and project j are done”

Question: Springer Verlag, a publisher of college textbooks, must decide which new books to adopt and publish next year. The books under consideration are described in the first column below, along with their expected three-year sales in the second column: Book Sales John Susan Monica Business Math X Finite Math X General Statistics X Mathematical Statistics X Business Statistics X Finance X X

Three individuals in the company can be assigned to these projects, all of whom have varying amounts of time available. John has 60 days, Susan has 52 days, and Monica has 43 days. The days required by each person to complete each project are showing in the third, fourth, and fifth columns above. For example, if the business calculus book is published, it will require 30 days of John’s time and 40 days of Susan’s time. (X means 0 time.)

Springer Verlag will not publish more than two statistics books in a single year. And one of the math books must be published, but not both. And the business math will be published only if the finance is published. And the general statistics will be published if the finite math is published. And the mathematical statistics and the finance will not both be published. Formulate a model to decide which books should be published to maximize sales.

where 1 is the business calculus book, 2 is the finite math book, … .
Relational Constraints Answer: where 1 is the business calculus book, 2 is the finite math book, … . Here is a binary programming model for maximizing projected sales (thousands of units) subject to the restrictions mentioned.

Max 20x1 + 32x2 17x3 10x4 25x5 18x6 s.t. 12x1 9x2 16x3 7x4 8x5 < 60 John 40x1 24x2 40 Susan 30x3 24x4 16x5 14x6 40 Monica x3 x4 x5 2 No. of Stat Books x1 x2 = 1 Math Book - x6 0 Only if 0 If 1 Not both

Capital Budgeting Capital Budgeting

Capital Budgeting Overview Capital Budgeting 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.

Initial Floor Space Exp. Rate
Capital Budgeting Question: Fry’s 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. TV/VCRs $ 6, % 2. TVs , 3. Projection TVs , 4. VCRs , 5. DVD Players 15, 6. Video Games , 7. Home Computers ,

Capital Budgeting Fry's has decided that they should not stock projection TVs unless they stock either TV/VCRs or TVs. Also, they will not stock both VCRs and DVD players, and they will stock video games if they stock TVs. Finally, the company wishes to introduce at least three 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 Fry's to maximize its overall expected return.

xj Capital Budgeting Define the Decision Variables
= 1 if product line j is introduced; = 0 otherwise. Product line 1 = TV/VCRs Product line 2 = TVs Product line 3 = Projection TVs Product line 4 = VCRs Product line 5 = DVD Players Product line 6 = Video Games Product line 7 = Home Computers xj

Define the Objective Function Maximize total expected return:
Capital Budgeting Define the Objective Function Maximize total expected return: Max .081(6000)x (12000)x (20000)x (14000)x (15000)x (2000)x (32000)x7 Invest. Space Return 1. $ 6, % , , , , , ,

Do not stock both VCRs and DVD players: x4 + x5 < 1
Capital Budgeting Constrain total investment by $45,000: x1 + 12x2 + 20x3 + 14x4 + 15x5 + 2x6 + 32x7 < 45 Constrain space by 420 square feet: 125x1 +150x2 +200x3 +40x4 + 40x5 + 20x x7 < 420 Stock projection TVs only if stock TV/VCRs or TVs: x1 + x2 > x3 or x1 + x2 - x3 > 0 Do not stock both VCRs and DVD players: x4 + x5 < 1 Stock video games if they stock TV's: x2 - x6 < 0 Introduce at least 3 new lines: x1 + x2 + x3 + x4 + x5 + x6 + x7 > 3 Invest. Space Return 1. $ 6, % , , , , , ,

TV/VCRs, Projection TVs, and DVD Players Do Not Introduce:
Capital Budgeting Interpretation: Introduce: TV/VCRs, Projection TVs, and DVD Players Do Not Introduce: TVs, VCRs, Video Games, and Home Computers Total Expected Return: $4,261 Surplus: $4,000 Investment 55 square feet of space

BA 452 Quantitative Analysis
End of Lesson B.5

Readings Readings Chapter 7 Integer Linear Programming.

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