1/24: Linear Programming & Sensitivity Analysis Review: –LP Requirements –Graphical solutions Using MS Excel for Linear Programming Sensitivity Analysis

Requirements of an LP Problem Objective Function –LP problems seek to maximize or minimize some quantity. Constraints –restrictions that limit the degree to which we can pursue our objective. Alternative courses of action. Linear equations or inequalities.

Graphical Solutions for LP Must be no more than 2 decision variables. Plot the limits of the constraints. Find the feasible region. Determine the optimized solution (using corner point method).

Graphical Solutions for LP: Corner Point Method Theory behind LP states that the optimal solution to any problem will lie at a corner point, or extreme point of the feasible region. Find result for each corner of the feasible region, pick the one that best satisfies the goal.

EX: The Brewery The microbrewery produces beer and ale. Beer sells for $5 per barrel, and ale for $2/barrel. Producing a barrel of beer requires 5 pounds of corn and 2 pounds of hops. Producing a barrel of ale requires 2 pounds of corn and 1 pound of hops. Every day, the microbrewery has 60 pounds of corn and 25 pounds of hops. Maximize the daily profit for the microbrewery.

EX: The Brewery

Max Z ( revenue ) = 5 x x 2 S. T.5 x x 2 <= 60 2 x x 2 <= 25 x 1, x 2 >= 0

EX: The Brewery Max Z ( revenue ) = 5 x x 2 S. T.5 x x 2 <= 60 2 x x 2 <= 25 x 1, x 2 >= ,12.5 0, 25 = $50 10, 5 = $60 12, 0 = $60

Spreadsheet Solutions for LP: MS Excel Formulate the optimization equation and the constraints. Build spreadsheet with formulas showing the relationships between the various elements. Using MS Excel Solver, define the cell to optimize, and the constraints on it. In “Options”, use “Assume Linear Model”

Using MS Excel for Linear Programming: the Answer Report Binding & non-binding constraints –Binding: limits the present solution –Non-binding: does not limit the present solution Slack: –How much extra capacity there is for a constraint

Using MS Excel for LP: Sensitivity Analysis: Adjustable Cells Reduced Cost –how much a coefficient would need to change to affect the optimal solution (distrust sign) Allowable increase & decrease –how much a coefficient can change without changing the optimal combination for the solution (but it can change the actual optimized value)

Using MS Excel for LP: Sensitivity Analysis: Constraints Shadow Price (also called Dual Price) –How much, per unit, the constraint affects the optimized value Allowable increase & decrease –In what range the shadow price holds true.

Using MS Excel for LP: Sensitivity Analysis All data in the sensitivity analysis are based on changing ONE THING AT A TIME.

Using MS Excel for LP: Troubleshooting Infeasibility: two possibilities: –An answer cannot be found that satisfies all constraints. –Part of the LP was input incorrectly. (More likely) Unboundedness: –An answer has no limit to its goal. –Part of the LP was input incorrectly. (See above)

Using MS Excel for LP: Example A laboratory orders pig heart valves for use in mechanical heart valves for humans. The 3 sizes ordered come from 3 suppliers in differing ratios. The lab needs 500 large, 300 medium, and 300 small valves per month. Each supplier can send no more than 500 valves total per month. The product mixes & cost/valve are below. Minimize the cost. $ / valve% large% med% small Supplier 1$52040 Supplier 2$ Supplier 3$36020

Using MS Excel for LP: Group Exercise A telemarketing company needs to contact 150 wives, 120 husbands, 100 single males, and 110 single females. A day call costs $2.00, and an evening call costs $5.00. No more than half the calls can be made at night. See the table for percentage of answers. No more than half the calls can be made at night. Minimize the cost. Who answers WifeHusbandSingle male Single female No one Day30%10% 40% Night30% 15%20%5%

Using MS Excel for LP: Group Exercise Go to lab to solve the problem using Excel. Print out main sheet, formulas, answer report, and sensitivity report. Discuss the meanings of the answer report & sensitivity report items.

Using MS Excel for LP: Homework Ch. 3 #25, 26, 27, 34 Along with working up an Excel Solver answer to the problems, answer all questions in the problems – type them on the spreadsheets, or add a worksheet to answer each problem’s questions. Turn in printouts of solved spreadsheets, formulas, and answers, and the Excel files by BEGINNING OF CLASS Thursday, January 31