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Linear Programming
LP (2003) 1

OPERATIONS RESEARCH: 343

1. LINEAR PROGRAMMING 2. INTEGER PROGRAMMING 3. GAMES

Books: Ð3Ñ IntroÞ to OR ÐF.Hillier & J. LiebermanÑ; Ð33Ñ OR ÐH. TahaÑ; Ð333Ñ IntroÞ to Mathematical Prog ÐF.Hillier & J. LiebermanÑ; Ð3@Ñ IntroÞ to OR ÐJ.Eckert & M. KupferschmidÑÞ

LP (2003) 2

LINEAR PROGRAMMING (LP)
LP is an optimal decision making tool in which the objective is a linear function and the constraints on the decision problem are linear equalities and inequalities. It is a very popular decision support tool: in a survey of Fortune 500 firms, 85% of the responding firms said that they had used LP. Example 1: Manufacturer Produces: Ingredients used in the production of A & C: Each ton of A requires: Each ton of C requires: Supply of X is limited to: Supply of Y is limited to: 1 ton of A sells for: 1 ton of C sells for: A (acid) and C (caustic) X and Y 2lb of X; 1lb of Y 1lb of X ; 3lb of Y 11lb/week 18lb/week £1000 £1000

Manufacturer wishes to maximize weekly value of sales of A & C. Market research indicates no more than 4 tons of acid can be sold each week. How much A & C to produce to solve this problem. The answer is a pair of numbers: x" Ðweekly production of AÑ, x# Ðweekly p.of CÑ There are many pairs of numbers Ðx" , x# Ñ: Ð0,0Ñ, Ð1,1Ñ, Ð3,5Ñ.... Not all pairs Ðx" , x# Ñ are possible weekly productions Ðex. x" œ 27, x# œ 2 are not possibleÑ Ð Ð27, 2Ñ is not a feasible set of production figuresÑ. The constraints on x" , x# are such that Ðx" , x# Ñ represent a possible set of production figures: The amount each product is produced is non-negative: The amount of ingredient X required to produce x" tons of A & x# tons of C is 2x"  x# . As X is limited to 11lb/week: The amount of ingredient Y required combined with the supply restriction: We cannot sell more than 4 tons of A/week:

x" 0 x # 0

2x"  x# Ÿ 11 x"  3x# Ÿ 18 x" Ÿ 4 Conversely any Ðx" , x# Ñ satisfying these

A possible set of production figures satisfies these
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