Jochen Könemann - Mathematicshwolkowi/henry/teaching/w11/370.w...Jochen Könemann, March 14, 2007 CO 370 – Deterministic Operations Research Models - p. 15/34 Gandhi Cloth Company

Jochen Könemann, March 14, 2007 CO 370 – Deterministic Operations Research Models - p. 1/34

C&O 370: Deterministic OR Models

Integer Programming – Modeling

Jochen Könemannhttp://www.math.uwaterloo.ca/∼jochen

http://www.math.uwaterloo.ca/~jochen


● What is an IP?

● Visualization

Stockco Investing

Fixed Charge Problems

Facility Location

Either-Or Constraints


What is an IP?

■ Linear programming allows for fractional values in solutions.Fractional values are not meaningful in many applications!

■ Sometimes, we are lucky and an LP model has integral basicfeasible solutions. But this is certainly not always the case.

■ A pure integer program looks like this

max cT x

s.t. Ax ≤ b

x ≥ 0

x integer


● What is an IP?

● Visualization

Stockco Investing


Facility Location



Visualization

■ The important difference to linear programming is that thesolution space is not any more convex:

max 3x1 + 2x2

s.t. x1 + 2x2 ≤ 6

x1, x2 ≥ 0


● What is an IP?

● Visualization

Stockco Investing

● Stockco: Investment Example

● Two out of Four

● Implications

● Exclusions


Facility Location



Stockco Investing


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location



Stockco: Investment Example

■ Stockco is considering 4 investments:

Investment NPV Initial Cash Outflow

1 $16,000 $5,0002 $22,000 $7,0003 $12,000 $4,0004 $8,000 $3,000

■ Stockco has $14, 000 in cash available■ Want to maximize total NPV given available cash


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location




■ Variable xi ∈ {0, 1} has value 1 if we invest in investmentoption i and 0 otherwise

■ Total NPV obtained by Stockco:

$16000x1 + $22000x2 + $12000x3 + $8000x4

■ Total amount invested

$5000x1 + $7000x2 + $4000x3 + $3000x4

and this needs to be at most $14000.


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location




■ {0, 1}-IP model looks like:

max 16x1 + 22x2 + 12x3 + 8x4

s.t. 5x1 + 7x2 + 4x3 + 3x4 ≤ 14

x1, x2, x3, x4 ∈ {0, 1}


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location



Two out of Four

■ Stockco can invest in at most two of the four investmentoptions.

How do you model that?■ This is enforced by the constraint

x1 + x2 + x3 + x4 ≤ 2.

Remember: The xi are 0, 1-variables! All positive variableshave value 1. There can be at most two of those.


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location



Two out of Four


max 16x1 + 22x2 + 12x3 + 8x4

s.t. 5x1 + 7x2 + 4x3 + 3x4 ≤ 14

x1 + x2 + x3 + x4 ≤ 2

x1, x2, x3, x4 ∈ {0, 1}


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location



Implications

■ If Stockco invests into investment option 2 they must alsoinvest in investment option 1?

■ How do we model this using linear constraints?■ Claim: The constraint

x2 ≤ x1

forces x1 = 1 whenever x2 = 1.■ Proof: x2 = 1 implies x1 must be at least 1. Since

x1 ∈ {0, 1} this means that x1 must take on value 1.


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location



Implications


max 16x1 + 22x2 + 12x3 + 8x4

s.t. 5x1 + 7x2 + 4x3 + 3x4 ≤ 14

x1 + x2 + x3 + x4 ≤ 2

x2 ≤ x1

x1, x2, x3, x4 ∈ {0, 1}


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location



Exclusions

■ If Stockco invests into investment option 2 they cannot investin investment option 4?

■ How do we model this using linear constraints?■ Claim: The constraint

x2 + x4 ≤ 1

forces x4 = 0 whenever x2 = 1.■ Proof: x2 = 1 implies x4 must be at most 0. Since

x4 ∈ {0, 1} this means that x4 must take on value 0.


● What is an IP?

● Visualization

Stockco Investing


● Two out of Four

● Implications

● Exclusions


Facility Location



Exclusions


max 16x1 + 22x2 + 12x3 + 8x4

s.t. 5x1 + 7x2 + 4x3 + 3x4 ≤ 14

x1 + x2 + x3 + x4 ≤ 2

x2 ≤ x1

x2 + x4 ≤ 1

x1, x2, x3, x4 ∈ {0, 1}


● What is an IP?

● Visualization

Stockco Investing


● Gandhi Cloth Company

● Resource constraints

● Fixed Charge

● Big-M Constraints

● Gandhi’s Profit

● IP Formulation

Facility Location





● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



Gandhi Cloth Company

■ Gandhi produces shirts, shorts, and pants.◆ Manufacturing each product requires renting a specific on

a weekly basis (fixed cost)◆ Producing a product uses resources labour and cloth◆ Producing a product incurs cost for labour and cloths

(variable cost)◆ Each product has a sales price

No Product Sales P. Fixed Var Labour(h) Cloth(m2)

1 Shirt 12 200 6 3 42 Shorts 8 150 4 2 33 Pants 15 100 8 6 4

■ Have 150h of labour and 160m2 of cloth available eachweek.


● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



Resource constraints

■ Variables are no surprise: How much of each product isproduced?

x1, x2, x3 non-negative integers

■ Company has at most 150h of labour available each week:

3x1 + 2x2 + 6x3 ≤ 150

■ Also have 160m2 of cloth each week:

4x1 + 3x2 + 4x3 ≤ 160


● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



Fixed Charge

■ The rental cost for production equipment for product i occursonly if we produce products of type i.It is independent of the produced quantity. This is called afixed charge.

■ Want to express the following:

We need to pay equipment rental for product i if we produceproduct i.

■ By implication: Introduce a new 0, 1-variable yi that hasvalue 1 if xi > 0.

■ Let M1, M2, M3 be large numbers:

xi ≤ Mi · yi

forces yi = 1 if xi > 0.


● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



Fixed Charge

■ Let M1, M2, M3 be large numbers:

xi ≤ Mi · yi

forces yi = 1 if xi > 0.■ Constraints like this are often called big-M constraints.■ How do we choose Mi? Remember, yi is a 0, 1-variable.■ Need to have xi ≤ Mi in any feasible solution!■ Ex.: How many shirts can we produce per week at most?■ Gandhi has 150h of labour available each week. It takes 3h

of labour to produce a shirt: x1 is at most 50!

Can choose M1 = 50.


● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



Big-M Constraints

■ Gandhi has 160 m2 of cloth available each week. It takes4m2 of cloths to produce a shirt: x1 is at most 40!

Can choose M1 = 40.■ A word of caution: Big-M constraints lead to week linear

programming relaxations.

Linear programming relaxations are used in IP solvers.

Choose Mi as small as possible!■ In the same way we derive that we can choose:

M1 = 40

M2 = 53

M3 = 25


● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



Big- M Constraints

■ In summary, this leads to the following constraints:

x1 ≤ 40y1

x2 ≤ 53y2

x3 ≤ 25y3

■ What is the profit of Gandhi?■ Ex.: Shirts.

Consequences of producing x1 shirts:◆ Fixed cost of 200 for machine rental◆ Variable costs of 6 per shirt◆ Revenue from sales of 12 per shirt

■ Can you express Gandhi’s profit for shirts using x1 and y1?


● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



Gandhi’s Profit

■ Ex.: Shirts.

Consequences of producing x1 shirts:◆ Fixed cost of 200 for machine rental◆ Variable costs of 6 per shirt◆ Revenue from sales of 12 per shirt

■ Can you express Gandhi’s profit for shirts using x1 and y1?■ Producing x1 shirts leads to profit

(12 − 6)x1 − 200y1 = 6x1 − 200y1

■ Similar for shorts and pants:

4x2 − 150y2

7x3 − 100y3


● What is an IP?

● Visualization

Stockco Investing




● Fixed Charge



● IP Formulation

Facility Location



IP Formulation

max 6x1 + 4x2 + 7x3 − 200y1 − 150y2 − 100y3

s.t. 3x1 + 2x2 + 6x3 ≤ 150

4x1 + 3x2 + 4x3 ≤ 160

x1 ≤ 40y1

x2 ≤ 53y2

x3 ≤ 25y3

y1, y2, y3 ∈ {0, 1}

x1, x2, x3 ≥ 0 and integer.


● What is an IP?

● Visualization

Stockco Investing


Facility Location

● Introduction

● Reachability

● Enforce Reachability

● IP Formulation



Facility Location


● What is an IP?

● Visualization

Stockco Investing


Facility Location

● Introduction

● Reachability


● IP Formulation



Introduction

■ The county of Kilroy has 6 cities. We want to build firestations in some of these cities.

■ Objectives:◆ Each city should be reachable from a fire station within 15

minutes◆ It costs money to build fire stations! We therefore want to

build as few as possible.■ Distance-Map:


● What is an IP?

● Visualization

Stockco Investing


Facility Location

● Introduction

● Reachability


● IP Formulation



Reachability

■ Can reach city 1 only from cities 1 and 2 within 15 minutes.■ Cities and possible fire stations for them

City Fire Stations City Fire Stations

1 1,2 4 3,4,52 1,2,6 5 4,5,63 3,4 6 2,5,6


● What is an IP?

● Visualization

Stockco Investing


Facility Location

● Introduction

● Reachability


● IP Formulation



Enforce Reachability

City Fire Stations City Fire Stations

1 1,2 4 3,4,52 1,2,6 5 4,5,63 3,4 6 2,5,6

■ For each city i we need to have a fire station in a city that canreach i within 15 minutes. How can we do this with an IP?

■ Have a 0, 1-variable xi that is 1 if we build a fire station in cityi and 0 otherwise.

■ Reachability constraint for city 3:

x3 + x4 ≥ 1


● What is an IP?

● Visualization

Stockco Investing


Facility Location

● Introduction

● Reachability


● IP Formulation



IP Formulation

min x1 + x2 + x3 + x4 + x5 + x6

s.t. x1 + x2 ≥ 1

x1 + x2 + x6 ≥ 1

x3 + x4 ≥ 1

x3 + x4 + x5 ≥ 1

x4 + x5 + x6 ≥ 1

x2 + x5 + x6 ≥ 1

xi ∈ {0, 1} for all 1 ≤ i ≤ 6


● What is an IP?

● Visualization

Stockco Investing


Facility Location


● Introduction

● Logical Constraints

● Either or for Small Cars

● Either or for Other Types




● What is an IP?

● Visualization

Stockco Investing


Facility Location


● Introduction





Introduction

■ Dorian Auto manufactures three types of cars: compact,mid-size and large

■ Producing a car requires steel and labour and selling a caryields a certain profit:

Resource Compact Mid-size Large

Steel 1.5 tons 3 tons 5 tonsLabour 30h 25h 40h

Profit 2,000 3,000 4,000

■ Dorian has 6,000 tons of steel and 60,000h of labouravailable

■ For production of a type of car to be economically feasibleDorian needs to produce at least 1000 cars of that type.


● What is an IP?

● Visualization

Stockco Investing


Facility Location


● Introduction





Introduction

■ Almost looks like a standard production problem: variablesare xs, xm, xl for production levels of small, mid-size andlarge cars.

■ Dorian’s profit is

2, 000 · xs + 3, 000 · xm + 4, 000 · xl

■ Resource constraints are also straight-forward.

Steel: 1.5 · xs + 3 · xm + 5 · xl ≤ 6, 000

Labour: 30 · xs + 25 · xm + 40 · xl ≤ 60, 000

■ But then we also want either xj ≤ 0 or xj ≥ 1000 for allj ∈ {s, m, l}. How?


● What is an IP?

● Visualization

Stockco Investing


Facility Location


● Introduction





Logical Constraints

■ Sounds like a logical constraint... either I produce a certaintype of car or I don’t.

■ Once again, use 0, 1-variables! Introduce variable yj ∈ {0, 1}for all j ∈ {s, m, l}.

Let yj = 1 if Dorian produces cars of type j and 0 otherwise.■ Example: Small cars. How many small cars can Dorian

produce at most?■ Building a small car uses 1.5 tons of steel and 30 hours of

labour. Dorian has 6,000 tons of steel and 60,000h of labouravailable. Therefore:

1.5xs ≤ 6, 000 and

30 · xs ≤ 60, 000

Therefore xs ≤ min{4000, 2000}.


● What is an IP?

● Visualization

Stockco Investing


Facility Location


● Introduction





Either or for Small Cars

■ Building a small car uses 1.5 tons of steel and 30 hours oflabour. Dorian has 6,000 tons of steel and 60,000h of labouravailable. Therefore:

1.5xs ≤ 6, 000 and

30 · xs ≤ 60, 000

Therefore xs ≤ min{4000, 2000}.■ Choose Ms = 2000 and add constraint

xs ≤ Ms · ys

■ If Dorian produces small cars (ys = 1) then this constraint isalways satisfied!

■ If Dorian does not produce small cars (ys = 0) then alsoxs = 0 and the constraint holds as well.


● What is an IP?

● Visualization

Stockco Investing


Facility Location


● Introduction





Either or for Small Cars

■ Also want: If Dorian produces small cars (ys = 1) then atleast 1000 cars are produced.

■ One way of doing that is like this

xs ≥ 1, 000 · ys

■ This is trivially satisfied if Dorian does not manufacture smallcars and therefore ys = 0.


● What is an IP?

● Visualization

Stockco Investing


Facility Location


● Introduction





Either or for Other Types

■ We also get that Dorian produces at most 2,000 mid-sizeand at most 1,200 large cars.

■ Full production model looks like:

max 2, 000 · xs + 3, 000 · xm + 4, 000 · xl

s.t. 1.5 · xs + 3 · xm + 5 · xl ≤ 6, 000

30 · xs + 25 · xm + 40 · xl ≤ 60, 000

xs ≤ 2000 · ys

xs ≥ 1000 · ys

xm ≤ 2000 · ym

xm ≥ 1000 · ym

xl ≤ 1200 · yl

xl ≥ 1000 · yl

yj ∈ {0, 1}, xj ≥ 0 and integer, for all j ∈ {s, m, l}


Documents

Jochen Könemann - Mathematicshwolkowi/henry/teaching/w11/370.w...Jochen Könemann, March 14, 2007 CO 370 – Deterministic Operations Research Models - p. 15/34 Gandhi Cloth Company