Linear Programming Technical Supplement 1 Linear Programming Copyright © 2014 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

Learning Objectives  Explain the importance of optimization to operations management.  Demonstrate how to develop linear programming models.  Show how linear programming models can be solved using EXCEL. 4 Demonstrate 0-1, transportation, and assignment models. TS1-2

Concepts OPTIMIZATION - find best possible answer OBJECTIVE - what it is you want to accomplish OBJECTIVE FUNCTION - measure of attainment of objective Objectives in Linear Programming TS1-3

LP Model optimize {max or min} objective function (if min, maximize the negative) subject to limits (constraints) ( ) for i=1,m Program Structure TS1-4

Assumptions of LP Linearity - all functions linear no diminishing returns, no economies of scale Certainty - all coefficients ( ) are known constants no distributions Continuity - solution liable to contain fractional values for Programming Requirements TS1-5

Production Mix Example Decision - how many cans to produce each day Variables – H&B, JHB, LB, JLB, JP Objective - maximize profit Objective function - how it is measured 0.21H&B+0.20JHB+0.10LB+0.15JLB+0.10JP Limits – cans, ham, beans, max, min Example TS1-6

Model Max0.21 H&B JHB LB JLB JP s.t.1 H&B + 1 JHB + 1 LB + 1 JLB + 1 JP  H&B + 3 JHB  H&B + 9 JHB + 14 LB + 12 JLB  H&B  5000H&B  JHB  1000JHB  4000 LB  1000LB  6000 JLB  2000JLB  4000 JP  0JP  1000 TS1-7

Constraints Ham function  30,000 oz/day Bean function  100,000 oz/day Can function  10,000 cans/day highly predictable, or would be uncertain Constraints TS1-8

Applications Resource allocation blending, product mix, inventory budget allocation, cash flow Planning & scheduling Transportation Assignment Types of Programs TS1-9

Issues Reduced cost - effect of change in contribution coefficients –will the optimal solution change? Dual price (shadow price; marginal value) –economic interpretation of resource (constraint) value Sensitivity Analysis TS1-10

Reduced Cost Amount a nonbasic decision variable needs to be improved before it would be attractive enough to include in the optimal solution Relates to Nonbasic - not included in the set of m variables in simplex solution –therefore, solution value = 0 Reduced Cost TS1-11

Example VariableValueReduced cost Objective coefficient Allowable increase Allowable decrease H&B Infinity0.01 JHB Infinity LB Infinity JLB Infinity JP infinity0.1 TS1-12

Reduced Cost Example H&B current profit 0.21, basic (in solution) –If price increases, no change (in solution) –If price decreases 0.01, another solution better JHB current profit 0.20, in solution but at minimum –INSUFFICIENT –Reduced cost –0.01 –Would need >= 0.21 to make worthwhile producing more Reduced Cost Explanation TS1-13

Reduced Cost Example Cj analysis APPLIES IF ONLY ONE CHANGE IN MODEL COEFFICIENTS If more than one Cj changes, can apply 100% RULE NO CHANGE IN OPTIMAL SOLUTION F: sum of ratios D/[allowable D] over all decision variables is <= 1.0 –note D is change Reduced Cost Explanation TS1-14

100% Rule Example If all profit rates dropped by.005 DallowableD/allowable H&B: JHB: infinity0 LB:-.005-infinity0 JLB:-.005-infinity0 JP: SO NO CHANGE.55 < 1 PROFIT would drop (still better than any other) Reduced Cost Explanation TS1-15

100% rule A ONE-WAY test: if sum > 1.0, doesn’t prove anythingif all drop.01 DallowableD/allowable H&B: JHB:-.01 -infinity0 LB:-.01-infinity0 JLB:-.01-infinity0 JP: SO DON’T KNOW1.1 > 1 Reduced Cost Explanation TS1-16

Dual Price Analysis DUAL PRICE: rate of change in objective function per one unit increase in right-hand-side Range analysis: NOT FOR OPTIMALITY, but for APPLICABILITY OF DUAL PRICE –how much right-hand-side could vary before DUAL PRICE would change Dual Price TS1-17

Dual Prices VariableValueDual priceLimitAllowable increase Allowable decrease Cans Infinity13112 Ham Infinity3444 Beans TS1-18

Dual Price Example ECONOMIC INTERPRETION IF Beans increased 1 unit, NEW SOLUTION: objective function increase of $0.023 per ounce/day IF Ham increased 1 unit, NO CHANGE IN OPTIMAL SOLUTION have 3444 spare units per day IF Cans increased 1 unit, NO CHANGE IN SOLUTION: have 13,111 spare cans per day Dual Price Explanation TS1-19

Dual Range Analysis How much RIGHT-HAND-SIDE can change before DUAL PRICE changes Cans: if drop (to 10888), starts to make a difference Ham: if drops 3444 (to 26556), starts to make a difference Beans: same dual price unless increases 7750 (to 107,750) or drops 8000 (to 92,000) assumes ONE CHANGED parameter Dual Price Explanation TS1-20

Dual Range Analysis Beans: if could acquire more beans, would be willing to pay up to per ounce –Currently paying $0.05/ounce –Value is more Not suggesting paying more IT WOULD BE ECONOMICALLY WORTHWHILE Dual Price Explanation TS1-21

Sensitivity Analysis Can gauge the stability of the solution ONLY CAN CONSIDER ONE CHANGED PARAMETER AT A TIME –except for 100% rule DUAL PRICE can show impact on objective of changing right-hand-side Sensitivity Analysis TS1-22

Types of Models All integer –require round numbers in solution –how many trucks to buy, buildings to build Zero-one –binary (incur cost or don’t; fund project or don’t) Mixed –some variables continuous; others integer or 0-1 Non-Continuous Models TS1-23

Linear Programming Summary 1 LP is a powerful decision tool – identifies optimal solutions to complex problems 2 Wide application given that underlying assumptions are met 3 Program structure and sensitivity analysis provide information that helps decision makers understand the problem, not just the solution TS1-24