1. Applied Statistical and Optimization Models
Topic 05: 0-1 Integer Programming

2. Objectives
Identify the solution to a 1-0 linear programming model that allows for integers 1-0 solutions.
Solve fixed cost problems
Solve 0-1 linear programming problems using Excel solver.

3. 0-1 Integer Linear Programming
Consider the following example, which is inspired by Joshua Emmanuel’s Integer Programming youtube video (May 8, 2016) available here:
(accessed January 31, 2018)
A professor can choose between four projects for the months January, February, and March. These projects are: Finishing a research project (R), Preparing lecture notes (L), Applying for a consulting project (C), or providing a service to the department by contributing to the strategic planning department (S). Each project has a payoff measured in bonus points for the annual review process.
The following Table summarizes the payoffs and available time budget in hours (It’s the same numbers like Emmanuel’ problem, just different context).

4. 0-1 Integer Linear Programming
Research (R)
Lecture (L)
Consulting (C)
Service (S)
Payoff (Objective function)
217
125 88
109
Time constraints (hours)
Maximum Time Budget
January 58 44 26 23
120
February 25 29 13 17 80
March 43 95

5. 0-1 Integer Linear Programming
The maximization problem can therefore be written as
Max Bonus Points = 217R + 125L + 88C + 109S
subject to R + 44L + 26C + 23S ≤ R + 29L + 13C + 17S ≤ R + 25L + 23C + 29S ≤ xi = [0,1]

6. 0-1 Integer Linear Programming
In Excel, the maximization problem can therefore be written as

7. 0-1 Integer Linear Programming
Add to the constraint box the fact that the selection variables are binary.
In Excel, the 0-1 Integer problem is specified as

8. 0-1 Integer Linear Programming
In Excel, the optimum solution to the 0-1 Integer problem is then

9. 0-1 Integer Linear Programming
Consider the following example, which is inspired by Joshua Emmanuel’s Integer Programming youtube video (May 8, 2016) available here:
(accessed January 31, 2018)
This is the second example in the video.

10. 0-1 Integer Linear Programming
A firm receives an order to produce 1000 cookies. The objective is to minimize costs. The firm can choose between 3 machines, which have different fixed and variable costs and different capacities.
Notations: xi = Quantities produced with machines i = [1,2,3] yi = Selection of machines i = [1,2,3] – This is a 0-1 variable
Constraints
Machine 1
Machine 2
Machine 3
Variable cost/unit
2.39
1.99
2.99
Fixed cost
300
250
400
Capacity
550
600

11. 0-1 Integer Linear Programming
The Fixed Cost Problem Specified in Excel

12. 0-1 Integer Linear Programming
The Fixed Cost Problem specified Min Z = 2.39x x x3 +300y y y s.t x1 + x2 + x3 = 1,000 x1 ≤ 400y1  x y1 ≤ 0 x2 ≤ 550y2  x y2 ≤ 0 x3 ≤ 400y3  x y3 ≤ 0

13. 0-1 Integer Linear Programming
The Fixed Cost Problem Specified in Excel’s Solver

14. 0-1 Integer Linear Programming
The Fixed Cost Problem Solved in Excel’s Solver

15. What you should have learned
Advance your linear programming skills in Excel, understanding especially the nature of 0-1 integer linear programming
Run Excel solver with even more confidence