BA 555 Practical Business Analysis Agenda Linear Programming (LP) Examples Integer Linear Programming Sensitivity Analysis

Decision-making under Uncertainty Decision-making under uncertainty entails the selection of a course of action when we do not know with certainty the results that each alternative action will yield. This type of decision problems can be solved by statistical techniques along with good judgment and experience. Example: buying stocks/mutual funds.

Decision-making under Certainty Decision-making under certainty entails the selection of a course of action when we know the results that each alternative action will yield. This type of decision problems can be solved by linear/integer programming technique. Example: A company produces two different auto parts A and B. Part A (B) requires 2 (2) hours of grinding and 2 (4) hours of finishing. The company has two grinders and three finishers, each of which works 40 hours per week. Each Part A (B) brings a profit of $3 ($4). How many items of each part should be manufactured per week?

Steps in Quantifying and Solving a Decision Problem Under Certainty Formulate a mathematical model: Define decision variables, State an objective, State the constraints. Input the model to a LP/ILP solver, e.g., LINDO or EXCEL Solver. Obtain computer printouts and perform sensitivity analysis. Report optimal strategy.

Example 6 Blending (p.66) Ajax Fuels, Inc., is developing a new additive for airplane fuels. The additive is a mixture of three ingredients: A, B, and C. For proper performance, the total amount of additive (amount of A + amount of B + amount of C) must be at least 10 ounces per gallon of fuel. However, because of safety reasons, the amount of additive must not exceed 15 ounces per gallon of fuel. The mix or blend of the three ingredients is critical. At least 1 ounce of ingredient A must be used for every ounce of ingredient B. The amount of ingredient C must be greater than one-half the amount of ingredient A. If the costs per ounce for ingredients A, B, and C are $0.10, $0.03, and $0.09, respectively, find the minimum-cost mixture of A, B, and C for each gallon of airplane fuel.

American Steel Company Material and shipping costs processing cost shipping cost Butte: supply grade A ore up to 1000 tons Cheyenne: supply grade B ore up to 2000 tons Pittsburgh: process up to 700 tons of ore with a lower processing cost. Youngstown: process up to 1500 tons of ore with a higher Mexico Taiwan Korea Japan High-grade steel A:B = 1:2 Low-grade steel A:B = 1:3 Purchasing Blending Logistic

Example 7 Staff Scheduling (p.67)

Variation: Staff Scheduling SHIFT M T W R F SA SU 1 X1 2 X2 3 X3 4 X4 5 X5 6 A B Req’ 17 13 15 19 14 16 11 Different schedules Different benefits Full-time vs. part-time Etc.

Staff Scheduling at the USPS

Example 9 Multi-period Financial Planning (p.69) Cash Flow at Time Period # Investment 1 2 3 A –1 0.5 B C 1.2 D 1.9 E 1.5 Constraints: Balance cash inflow and cash outflow at all time periods.

Dedication / Cash Flow Matching Dedication or cash flow matching is a technique used to fund known liabilities in the future. The intent is to form a portfolio of assets such that cash inflows will exactly offset the cash outflows of the liabilities.

Integer LP: Example (p.75) An airline owns an aging fleet of Boeing 727 airplanes. It is considering a major purchase of up to 17 new Boeing 757 and 767 jets. The decision must take into account numerous cost and capability factors, including the following: the airline can finance up to $400 million in purchases; each Boeing 757 will cost $35 million, while each Boeing 767 will cost $22 million; at least one-third of the planes purchased should be the longer-ranged 757; the annual maintenance budget is to be no more than $8 million; the annual maintenance cost is estimated to be $800,000 for a 757, and $500,000 for a 767; and each 757 can carry 125,000 passengers per year, while each 767 can fly 81,000 passengers annually. Formulate this problem as an integer programming problem to maximize the annual passenger-carrying capacity.

Integer LP: Example 11 (p.75) BINARY decision variables

Common Types of Logical Constraints (p.76) Mutual Exclusion: either A or B, but not both, can be implemented. write down a constraint to express the restriction that, due to limited budget, either “New Machinery” project or “New Product Research” project can be implemented, but not both. Conditional: if B is implemented, then A must be implemented also. write down a constraint to express the restriction that if the plant is expanded, the warehouse must also be expanded. Corequisite: if B is implemented, then A must be implemented also, and vice versa. write down a constraint to express the restriction that if the plant is expanded, then “New Machinery” project must be carried out and vice versa. k out of n: at most/least k out of n alternatives can be implemented. at most 3 projects have to be implemented next year. at least 2 projects have to be implemented next year.

Binary Decision Variables: Example 12 (p.76) Kilroy Manufacturing is establishing plants in six cities (cities 1-6). The company must decide in which of these cities to procure space for storing inventory. The company wants to ensure that at least one inventory site is within 150 miles of each plant. The distance required to drive between the cities are shown in the table below: Formulate an ILP model to determine how many inventory sites must be built, and where they should be located.

Binary Decision Variables (Example 12, p.76)

Sensitivity Analysis (p.70) How will a change in a coefficient of the objective function affect the optimal solutions? How will a change in the right-hand-side value for a constraint affect the optimal solution?

Glossary (p.77) SLACK: A variable added to the left-hand side of a less-than-or-equal-to () constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount of unused resource. SURPLUS: A variable subtracted from the left-hand side of a greater-than-or-equal-to () constraint to convert the constraint into an equality. The value of this variable can usually be interpreted as the amount over and above some required minimum level. BINDING/NON-BINDING: A constraint is binding if it is satisfied as a strict equality in the optimal solution; otherwise, it is nonbinding. If a constraint is binding, its corresponding slack/surplus variable equals to 0.

Slack/Surplus/Binding/Non-Binding (p.70)

Sensitivity Analysis (p.70) How will a change in a coefficient of the objective function affect the optimal solutions? How will a change in the right-hand-side value for a constraint affect the optimal solution?

Range of Optimality (p.70) The range of values over which an objective function coefficient may vary without causing any change in the values of the decision variables in the optimal solution.

Range of Feasibility (p.70) The range of values over which an objective function coefficient may vary without causing any change in the values of the decision variables in the optimal solution.

Reduced Cost (p.70) The amount by which an objective function coefficient would have to improve (increase for a maximization problem, decrease for a minimization problem), before it would be possible for the corresponding variable to assume a positive value in the optimal solution.