1. Chapter 11: Hillier and Lieberman Dr. Hurley’s AGB 328 Course
Integer Programming
Chapter 11: Hillier and Lieberman
Dr. Hurley’s AGB 328 Course

2. Terms to Know
Integer Programming, Mixed Integer Programming, Binary Variable, Binary Integer Programming, Mutually Exclusive Alternatives, Contingent Decisions, Tours-of-duty Planning Problem, Rostering Problem, Auxiliary Binary Variable, Binary Representation

3. Terms to Know Cont.
Set Covering Problems, Set Partitioning Problems, Exponential Growth, LP Relaxation, Branching Tree, Branching Variable, Relaxation, Incumbent, Fathom, Descendants, Lagrangian Relaxation, Cutting Plane, Cut, Minimum Cover, Global Constraints

4. Applications of Binary Variables
Binary variables only allow two choices
This makes them suited for problems that are characterized by variables that can take on only two possibilities.
Examples:
Do a project or not do a project?
To hire or not to hire?
To build or not to build?
To Sell or not to sell?

5. Case Study: California Manufacturing Company (CMC)
The California Manufacturing Company is a company with factories and warehouses throughout California.
It is currently considering whether to build a new factory in Los Angeles and/or San Francisco.
Management is also considering building one new warehouse where a new factory has been recently built.
Should the CMC build factories and/or warehouses in Los Angeles and/or San Francisco?

6. Capital Needed (Millions)
Case Study: CMC Cont.
Binary Decision
Decision Variable
NPV (Millions)
Capital Needed (Millions)
Build a factory in Los Angeles
FLA $9 $6
Build a factory in San Francisco
FSF 5 3
Build a warehouse in Los Angeles
WLA 6
Build a warehouse in San Francisco
WSF 4 2
Building Money Available:
$10 million

7. Case Study: CMC Cont.
FLA, FSF, WLA,WSF are all binary variables which take on the value of 1 if the specific item is done and zero if it is not done.
We also need to make sure that at most one warehouse is built and it is built where a factory is built.

8. Mathematical Model for CMC

9. Innovative Uses for Binary Variables
Either-Or Constraint
A Subset of Constraints Must Hold
A Constraint that Needs a Single Value Out of Multiple Possibilities
A Fixed Cost Is Only Associated with a Positive Usage of a Variable
Binary Representation of General Integer Variables

10. Either-Or Constraint
Suppose you have a situation where you have two potential constraints, but only one of them can hold
To handle this issue, you can add to one constraint My1 to the RHS and to the other constraint M(1-y1) to the RHS where y1 is a binary variable
For example:
5x1+9x2 ≤ 24 + My1
8x1+6x2 ≤ 35 + M(1-y1)

11. A Subset of Constraints Must Hold
Suppose you have N constraints where only K of the constraints hold (K<N)
You would add to constraint i the amount Myi and one other constraint where 𝑖=1 𝑁 𝑦 𝑖 =𝑁−𝐾
For example:
5x1 + 12x2 ≤ 34 + My1
6x1 + 11x2 ≤ 43 + My2
7x1 + 10x2 ≤ 57 + My3
y1+ y2 + y3 = 1

12. A Constraint that Needs a Single Value Out of Multiple Possibilities
Suppose you have a constraint where the function can take on one out of many values
To handle this issue, you would multiply value i by a binary variable yi and sum them all together while adding one more constraint that 𝑖=1 𝑁 𝑦 𝑖 =1
For example:
5x1 +31x2 = 25 or 50 or 75
5x1 +31x2 = 25y1+ 50y2+ 75y3
y1+ y2 + y3 = 1

13. A Fixed Cost Is Only Associated with a Positive Usage of a Variable
Suppose you have the situation where your objective function only takes on a fixed cost ki if you use a corresponding variable xi > 0, if xi = 0 then ki = 0
To handle this issue you would add to the objective function kiyi and a new constraint xi ≤ Myi
For example, suppose that variable x1 has an associated set-up cost if you decide to use it

14. min 𝑥 1 , 𝑥 𝑥 𝑥 𝑦 1
Subject to:
9 𝑥 𝑥 2 ≤900
23 𝑥 𝑥 2 ≤750
𝑥 1 ≤𝑀 𝑦 1

15. Binary Representation of General Integer Variables
There are some times when you may want to represent a variable that is supposed to be an integer by transforming it into a binary representation
This can be done by defining the variable x = 𝑖=0 𝑁 2 𝑖 𝑦 𝑖

16. Goods Product Company Example
Objective: Maximize Profits
Decision Variables: Product 1, Product 2, and Product 3
Constraints:
Production time available for Plants 1 and 2
At most two out of the three products can be produced
Only one of the two plants can produce the new products

17. Data for Goods Product Company
Product 1 uses 3 hours of Plant 1 or 4 hours of Plant 2 per unit of production
You can sell up to 7 at a profit of $5,000
Product 2 uses 4 hours of Plant 1 or 6 hours of Plant 2 per unit of production
You can sell up to 5 at a profit of $7,000
Product 3 uses 2 hours of Plant 1 or 2 hours of Plant 2 per unit of production
You can sell up to 9 at a profit of $3,000
Plant 1 has 30 hours available while Plant 2 has 40 hours

18. Mathematical Model for Goods Product Company
max 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 , 𝑥 5 , 𝑥 6 , 𝑥 𝑥 1 +7 𝑥 2 + 3𝑥 3
Subject to:
𝑥 1 −𝑀 𝑥 4 ≤0
𝑥 2 −𝑀 𝑥 5 ≤0
𝑥 3 −𝑀 𝑥 6 ≤0
𝑥 4 + 𝑥 5 + 𝑥 6 ≤2
3𝑥 1 + 4𝑥 2 +2 𝑥 3 −𝑀 𝑥 7 ≤30
4𝑥 1 + 6𝑥 2 +2 𝑥 3 +𝑀 𝑥 7 ≤30+𝑀
0≤𝑥 1 ≤7, 0≤𝑥 2 ≤5, 0≤𝑥 3 ≤9
𝑥 4 , 𝑥 5 , 𝑥 6 , 𝑥 7 ∈ 0,1
We will examine the spreadsheet model in class

19. Supersuds Corporation Example
Objective: Maximize Profits
Decision Variables: Number of TV spots for Product 1, Product 2, and Product 3
Constraints:
Number of TV spots allocated to the three products cannot be more than five
Major Issue: Proportionality Assumption is violated

20. Data for Supersuds
If any of the products does not buy a TV spot, it will not get any profit
If Product 1 buys 1, 2 , or 3 TV spot(s), its profitability would be 1, 3, or 3 respectively
If Product 2 buys 1, 2 , or 3 TV spot(s), its profitability would be 0, 2, or 3 respectively
If Product 3 buys 1, 2 , or 3 TV spot(s), its profitability would be -1, 2, or 4 respectively

21. Mathematical Model 1 for Supersuds
max 𝑥 11 , 𝑥 12 , 𝑥 13 , 𝑥 21 , 𝑥 22 , 𝑥 23 , 𝑥 31 , 𝑥 32 , 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 23 − 𝑥 𝑥 𝑥 33
Subject to:
𝑥 11 + 𝑥 12 + 𝑥 13 ≤1
𝑥 21 + 𝑥 22 + 𝑥 23 ≤1
𝑥 31 + 𝑥 32 + 𝑥 33 ≤1
𝑥 𝑥 𝑥 13 + 𝑥 𝑥 𝑥 23 + 𝑥 𝑥 𝑥 33 ≤1
𝑥 11 , 𝑥 12 , 𝑥 13 , 𝑥 21 , 𝑥 22 , 𝑥 23 , 𝑥 31 , 𝑥 32 , 𝑥 33 ∈ 0,1
We will examine the spreadsheet model in class

22. Mathematical Model 2 for Supersuds
max 𝑥 11 , 𝑥 12 , 𝑥 13 , 𝑥 21 , 𝑥 22 , 𝑥 23 , 𝑥 31 , 𝑥 32 , 𝑥 𝑥 𝑥 𝑥 22 + 𝑥 23 − 𝑥 𝑥 𝑥 33
Subject to:
𝑥 12 − 𝑥 11 ≤0
𝑥 13 − 𝑥 12 ≤0
𝑥 22 − 𝑥 21 ≤0
𝑥 23 − 𝑥 22 ≤0
𝑥 32 − 𝑥 31 ≤0
𝑥 33 − 𝑥 32 ≤0
𝑥 11 + 𝑥 12 + 𝑥 13 + 𝑥 21 + 𝑥 22 + 𝑥 23 + 𝑥 31 + 𝑥 32 + 𝑥 33 =5
𝑥 11 , 𝑥 12 , 𝑥 13 , 𝑥 21 , 𝑥 22 , 𝑥 23 , 𝑥 31 , 𝑥 32 , 𝑥 33 ∈ 0,1
We will examine the spreadsheet model in class

23. Wyndor Case Revisited Two new products have been developed:
An 8-foot glass door
A 4x6 foot glass window
Wyndor has three production plants
Production of the door utilizes Plants 1 and 3
Production of the window utilizes Plants 2 and 3
Objective is to find the optimal mix of these two new products.

24. Wyndor Case Revisited Cont.
Production Time Used for Each Unit Produced
Plant
Doors
Windows
Available Per Week 1
1 hour
0 hour
4 hour 2
2 hour
12 hour 3
3 hour
18 hour
Unit Profit
$300
$500

25. Wyndor Case Revisited Cont.

26. Changing Wyndor to Account for Setup Costs
Suppose that two changes are made to the original Wyndor problem:
If Wyndor chooses to produce doors, it must pay a one time set-up cost of $700, while if Wyndor produces windows it must pay a set-up cost of $1,300.
We want to restrict the doors and windows to be integer values.

27. Wyndor’s Mathematical Model With Set-Up Costs

28. Changing Wyndor to Account for Mutually Exclusive Products
Suppose Wyndor decides that it only wants to produce doors or windows rather than both.
This implies that either doors have to be zero or windows have to be zero.

29. Wyndor’s Mathematical Model With Mutually Exclusive Products

30. Changing Wyndor to Account for Either-Or Constraints
Suppose the company is trying to decide whether to build a new up-to-date plant that will be used to replace plant 3.
This implies that Wyndor wants to examine the profitably of using plant 4 versus plant 3.

31. Wyndor’s Data with Either/Or Constraint
Production Time Used for Each Unit Produced
Plant
Doors
Windows
Available Per Week 1
1 hour
0 hour
4 hour 2
2 hour
12 hour 3
3 hour
18 hour 4
2 hours
4 hours
28 hours
Unit Profit
$300
$500

32. Wyndor’s Mathematical Model With Either/Or Constraint

33. The Challenges of Rounding
It may be tempting to round a solution from a non-integer problem, rather than modeling for the integer value.
There are three main issues that can arise:
Rounded Solution may not be feasible.
Rounded solution may not be close to optimal.
There can be many rounded solutions
The solution to the LP-relaxation shown on the graph is approximately (3.8, 4.9).
None of the possible rounded solutions, (3, 4), (4, 4), (3, 5), or (4, 5), are even feasible.
The optimal solution at (1, 3) is not even close to the LP-relaxation solution.
There are 230, or approximately 1 billion rounded solutions to a problem with 30 variables that are non-integer.

34. Algorithms For Solving BIPs and MIPs
Branch-and-Bound Algorithm
Branch-and-Cut Algorithm

35. Branch-and-Bound Algorithm
This algorithm has three main steps in an iteration:
Branching
This is where you create two new sub-problems
Bounding
For each sub-problem, apply the simplex method to the LP relaxation
Fathoming
This where you are deciding whether to dismiss the sub-problem from further consideration

36. Fathoming Test
Test 1: Its bound is less than Z*, where Z* represents the value you get from the objective function for the current best incumbent solution
Test 2: The LP relaxation problem has no solution
Test 3: The optimal solution for the LP relaxation problem is binary/integer

37. Branch-and-Cut Algorithms
These algorithms primarily rely upon:
Problem preprocessing
Fixing variables that you know upfront must be a particular variable
Eliminating redundant constraints
Tightening constraints in a way to reduce the feasible region
Generating cutting planes
This reduces the feasible region for the LP relaxation which does not reduce the feasible solutions
Clever branch-and-bound techniques

38. In-Class Activity (Not Graded)
Create a spreadsheet model for each of the Wyndor problems previously discussed