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 constraints. constraints are a possible set of production figures: see FIGURE 1

THE FEASIBLE REGION is the intersection of the shaded regions & is given by see FIGURE 2 . The feasible region ÐOPQRSÑ represents all pairs Ðx" , x# Ñ that satisfy the constraints. The corners (vertices) O,P,Q,R,S have a special significance [ O=(0,0), P=(0,6), Q=(3,5), R=(4,3), S=(4,0)].

LP (2003) 3

Associated with each feasible Ðx" , x# ) is a sales value of £1000 ‚ Ðx" x# Ñ. Since we wish to maximize this amount, our problem is: Maximize : Subject to x" x# 2x" x" x# 3x# Ÿ 11 Ÿ 18 Ÿ 4 oq objective function oq constraint oq constraint oq constraint oq constraint

x" x" , x# 0

This is called a LINEAR PROGRAM (LP): A problem of optimizing (maximizing or minimizing) a linear function subject to linear constraints. ÐLinear: no powers, exponentials or product termsÑ. PROPERTY Ð*Ñ: Observe that the set ÖO, P, Q, R, S× contains an optimal solution to our L.P. evaluate the objective function x" x# at these points: 0,6,8,7,4 Ê Q œ Ð3,5Ñ, x" œ 3, x# œ 5 is the optimal solution. see FIGURE 3 Note: The feasible region Ði.e. area described by the polygon OPQRSÑ lies entirely within that half of the plane for which x" x# Ÿ 8. Since 5 3 œ 8 no feasible point has a higher objective value than that of Q. Property Ð*Ñ holds if we replace x" x# by any linear function c" x" c# x# . e.g. to minimize 3x" x# over points in the polyhedron OPQRS we take the smallest point of 0, 6, 4, 9, 12 and find that P : x" œ 0, x# œ 6 is an optimal solution. The SIMPLEX ALGORITHM, to be described later, is an efficient method for finding an optimal vertex without necessarily examining all of them. Property Ð*Ñ does not imply that points other than vertices cannot be optimal. e.g. if we want to maximize 2x" x# then any point on the segment QR is optimal. 2. STANDARD LP FORM Any LP can be transformed into STANDARD FORM minimise subject to x0 = c1 x1 + c2 x2 + ... + cn x2 + ... + a1n xn x n...