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

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.

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: https://www.youtube.com/watch?v=-3my1TkyFiM (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).

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

0-1 Integer Linear Programming The maximization problem can therefore be written as Max Bonus Points = 217R + 125L + 88C + 109S subject to 58R + 44L + 26C + 23S ≤ 120 25R + 29L + 13C + 17S ≤ 80 43R + 25L + 23C + 29S ≤ 95 xi = [0,1]

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

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

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

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: https://www.youtube.com/watch?v=-3my1TkyFiM (accessed January 31, 2018) This is the second example in the video.

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

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

0-1 Integer Linear Programming The Fixed Cost Problem specified Min Z = 2.39x1 + 1.99x2 +2.99x3 +300y1 + 250y2 + 400y3 s.t. x1 + x2 + x3 = 1,000 x1 ≤ 400y1  x1 - 400y1 ≤ 0 x2 ≤ 550y2  x2 - 550y2 ≤ 0 x3 ≤ 400y3  x3 - 600y3 ≤ 0

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

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

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