Types of IP Models All-integer linear programs Mixed integer linear programs (MILP) Binary integer linear programs, mixed or all integer: some or all of the decision variables can only assume 0 or 1 values A LP relaxation of an IP model is the solution that occurs when the integer restrictions are relaxed or ignored.

Properties of IP Solutions The value of the optimal solution to an IP model yields a value that is the same or less desirable than the value of the optimal solution to its LP relaxation. In maximization problems, the IP optimal solution will be lower; in minimization cases, the IP solution will be higher.

Integer Programming & Solver To perform Integer Programming, add constraints in Solver that require specific decision variables (changing cells) to be Integer (int) or Binary (bin) Solver will perform the Branch and Bound technique to identify the optimal solution within a specified tolerance.

Capital Budgeting Problems A decision maker has several potential projects or investment alternatives Projects or investments require different amounts of resources (which are limited) and generate different cash flows for the company Cash flows are often converted into a NPV as input into the model There may be dependency relationships between some of the projects or investment alternatives The problem is to identify which set of projects or investments to select in order to achieve the maximum possible NPV

Branch and Bound Technique Solve a LP relaxation of the IP model. For max problems, the solution is an Upper Bound for the IP solution. For min problems, the solution is a Lower Bound for the IP solution. If solution is all integer, done. Otherwise, set the other bound equal to the objective value that would occur if only the integer variables identified in the solution were used.

Branch and Bound Technique (Continued) Create two different subproblems by adding a constraint to the problem, forcing a non-integer value to be a given integer. Solve these problems one at a time. If a solution is all integer or if it is outside the bounds of the first problem, you are done creating more problems in this direction. If a solution is within the bounds and still not all integer, create two more problems by adding another constraint forcing another variable to be an integer.

Branch and Bound Technique (Continued II) When all the necessary sub-problems have been created and solved, identify the solution which achieves the objective from all the sub- problems which resulted in all integer solutions.

Solver’s Tolerance Option Solver allows you to set a tolerance % option to determine how close the identified solution must be to the true optimal solution before Solver can stop searching for a better solution. By default, the suboptimality tolerance factor is 5%. This means that the solution found by Solver is within 5% of the optimal solution. It may actually be the optimal solution but there is no guarantee to that! To guarantee the optimal solution, you must set the tolerance % to 0. This will result in much longer solution times however when you do this.

Example of IP Applications Capital Budgeting Models Municipal Bond Underwriting Purchase Ordering Models: Fixed-Charge or Fixed Cost problems Fixed Order Quantities & Quantity Discounts Minimum Order Size Constraints Workforce Scheduling Problems

Mutually Exclusive Events When two alternatives (A and B) cannot both be picked, force Solver to choose at most one by including the constraint: X A + X B <= 1 Similiarly choosing no more than k out of n events requires adding the constraint: X A + X B + ……+ X n <= k

Dependent Decisions When one alternative (B) cannot be used unless another alternative is also picked (A), add the constraint: X B <= X A

Minimum Purchase Size Constraints Assume that a particular investment opportunity that you are interested in requires a minimum investment of $25,000. You do not wish to invest more than $200,000 in any particular investment. How would you model this as a constraint for Solver?

Let Y i =0 if you choose not to invest in option i, 1 if you choose to invest in option i. Let X i =amount of money you choose to invest in i. There are two constraints required: X i <= Y i (this forces Y i to be 1 if you put money into this option) X i >=25000 Y i (this forces X i to meet minimum amount if you put money into this option, because Y i will be 1)

Investment Problem You have currently identified 20 stocks as possible investments. You have $100,000 available. Their potential returns have been estimated and you would like to invest your money so as to maximize the expected return on your new portfolio.

Each stock requires a minimum purchase of $5,000 and you do not want to invest more than $20,000 in any one particular stock. You would like to invest in at least 8 different stocks. In order to control your risk, you do not want to put more than $60,000 in the the first 10 stocks listed. To invest in stock 5 you must also invest some money in stock 16.