Decision Modeling

1. Decision Modeling Linear Programming

2. Constrained Optimization • A constrained optimization model takes the form of a performance measure to be optimized over a range of feasible values of the decision variables. • The feasible values of the decision variables are determined by a set of constraints (usually inequalities). • Values of the decision variables must be chosen such that all constraints are satisfied while either maximizing or minimizing the desired objective function.

3. Linear Programs (LPs) • In a linear program (linear optimization model), the objective function and all constraints are linear • LPs can be solved very efficiently • LPs can contain tens, hundreds, or thousands of decision variables and constraints

4. Formulating LPs • Every LP has three important components: • Decision variables • Those resources controlled by the decision-maker • Constraints • Constraints are restrictions or requirements on the choice of solutions • Constraints stem from physical, economic or policy limitations or requirements • Objective function • A single performance measure to be minimized or maximized (e.g., maximize profit, minimize cost)

5. Formulating LPs • The formulation of an LP should follow these three steps: • Identify the decision variables • Identify the constraints • Which resources are in scarce supply? • Which factors limit your use of or access to resources? • Specify an objective function • What is a reasonable measure for a good decision?

6. Example Constraints • Investment decisions are restricted by the amount of available capital and by government regulations • Production decisions are limited by plant capacity and the availability of raw materials and components • Staffing and flight plans of an airline are restricted by the maintenance needs of the planes and the number of employees on hand • The use of a certain type of crude oil in producing gasoline is restricted by the characteristics of the gasoline (e.g., the octane rating, etc.)

7. Some Terminology max 56C + 40M (objective function) subject to 8C + 4M ≤ 1280 (constraint) coefficient } } LHS(left-handside) RHS(right-handside) variable inequality

8. Oak Products Example • Consider a small furniture manufacturer, Oak Products, Inc. • The company has two products: • Captain chairs • Mate chairs • Based on an economic forecast for the next week, Oak Products has determined that the company can sell as many chairs as it can produce at the current wholesale price • Oak Products needs to decide how many Captains and Mates to produce next week so as to maximize profit

9. Oak Products Example The following factors must be considered: • The unit profit contribution (price minus unit variable cost) is $56 for each Captain sold and $40 for each Mate • Each chair is assembled from long dowels, short dowels, legs, and one of two types of seats • Each Captain requires 8 long and 4 short dowels • Each Mate requires 4 long and 12 short dowels • There is a total inventory of 1280 long dowels and 1600 short dowels for next week’s production

10. Oak Products Example • The total inventory of legs is 760 units and each chair of either type uses 4 legs • The inventory of heavy seats (one for each Captain) and light seats (one for each Mate) is 140 and 120, respectively • Management has entered into an agreement with the union to manufacture a minimum of 100 chairs in any combination each week so as to guarantee the workers pay for a minimum number of hours per week

11. Oak Products Example • Oak Product’s problem is known as an optimal production plan problem • Formulate the decision problem as an LP • See board

12. Oak Products LP Formulation max 56C + 40M subject to 8C + 4M ≤ 1280 (long dowels) 4C + 12M ≤ 1600 (short dowels) 4C + 4M ≤ 760 (legs) C ≤ 140 (heavy seats) M ≤ 120 (light seats) C + M ≥ 100 (union agreement) C, M ≥ 0 (non-negativity)

13. Some LP Rules • Constraint LHS consists of as many terms as there are decision variables • Each term consists of a coefficient multiplied by a variable • Coefficient might be 0, in which case the term does not have to be displayed • Each variable appears only once on the LHS • Terms are separated by either + or - • Constraint RHS consists of a single constant (number) • There are no variables on the RHS • Objective function has the same structure as the constraint LHS

14. Mapping an LP into Excel • Each model variable will result in a spreadsheet column • Objective function and each constraint will result in a row of coefficients • Set aside one cell for each decision variable to hold its numerical value • Line up objective function coefficients and constraint coefficients in their respective column below the decision variable

15. Mapping an LP into Excel

16. Mapping an LP into Excel • Excel functions used: • SUMPRODUCT (inner product in linear algebra) • Absolute cell references (toggle with F4)

17. Evaluating an LP • A solution is a set of choices of values for the decision variables • Every solution that satisfies all of the constraints is called a feasible solution • A feasible solution that maximizes (minimizes) the objective function is called optimal • An optimal solution may have fractional (non-integer) values for some or all of the decision variables • If this is unacceptable, integer constraints must be added to the formulation • The interpretation of a solution may depend on whether or not the solution is allowed to be fractional

18. Using Solver in Excel • On-screen demonstration