operations research - linprog.pdf

OPERATIONS RESEARCH: 343Linear Programming

Panos [email protected], Room: 347

(slides courtesy of Herbert Wiklicky)

Imperial College

London

OPERATIONS RESEARCH: 343 – p.1/21

Overview

1. Linear Programming


Overview


2. Game Theory


Overview


2. Game Theory

3. Integer Programming


Some Books

Course Notes:http://www.doc.ic.ac.uk/∼pp500 (for LP)http://www.doc.ic.ac.uk/∼br (for IP & Games)

H.Taha:Operation Research

F.Hillier & J.Lieberman:Introduction to Operation Research

F.Hillier & J.Lieberman:Introduction to Mathematical Programming

J.G.Eckert & M.Kupferschmid:Introduction to Operation Research


Linear Programming (LP)

Optimal Decision Tool



Optimal Decision ToolLinear Objective Function




Linear Constraints




Linear ConstraintsEqualities and Inequalities




Linear ConstraintsEqualities and Inequalities

popular decision toolsurvey of Fortune 500 firms:

85% responding said they use LP


Example 1

Manufacturer produces: A (acid) and C (caustic).Ingredients used producing A and C are: X and Y.


Example 1


Each ton of A requires: 2lb of X; 1lb of Y

Each ton of C requires: 1lb of X ; 3lb of Y


Example 1




Supply of X limited to: 11lb/week

Supply of Y limited to: 18lb/week


Example 1






A sells for: £1000/ton

C sells for: £1000/ton


Example 1








Market research: max 4 tons of A/week can be sold.


Example 1








Market research: max 4 tons of A/week can be sold.

Maximize weekly value of sales of A and C.


Example 1 (cont)

How much A and C to produce?


Example 1 (cont)


Answer: A pair of numbers

x1 (weekly prod. of A)

x2 (weekly prod. of C)


Example 1 (cont)





There are many pairs of (x1, x2):e.g. (0, 0), (1, 1), (3, 5), . . .


Example 1 (cont)





There are many pairs of (x1, x2):e.g. (0, 0), (1, 1), (3, 5), . . .

Not all (x1, x2) possible weekly productions:e.g. x1 = 27, x2 = 2 are not possible

i.e. (27, 2) not feasible.


Example 1 (cont)

Amount A produced is non-negative:


Example 1 (cont)

Amount A produced is non-negative: x1 ≥ 0


Example 1 (cont)


x1

x2


Example 1 (cont)


x2

x1

x1 = 0


Example 1 (cont)

Amount C produced is non-negative:


Example 1 (cont)

Amount C produced is non-negative: x2 ≥ 0


Example 1 (cont)


x1

x2


Example 1 (cont)


x2

x1x2 = 0


Example 1 (cont)

X required for x1 tons of A & x2 tons of C is: 2x1 + x2.


Example 1 (cont)

X required for x1 tons of A & x2 tons of C is: 2x1 + x2.X limited to 11lb/week:


Example 1 (cont)

X required for x1 tons of A & x2 tons of C is: 2x1 + x2.X limited to 11lb/week: 2x1 + x2 ≤ 11


Example 1 (cont)


x1

x2


Example 1 (cont)


x2

x1

2x1 + x2 = 11


Example 1 (cont)

Y amount required & supply restriction:


Example 1 (cont)

Y amount required & supply restriction: x1 + 3x2 ≤ 18


Example 1 (cont)


x1

x2


Example 1 (cont)


x2

x1

x1 + 3x2 = 18


Example 1 (cont)

Cannot sell more than 4 tons of A/week:


Example 1 (cont)

Cannot sell more than 4 tons of A/week: x1 ≤ 4


Example 1 (cont)


x1

x2


Example 1 (cont)


x2

x1

x1 = 4


Feasible Region (FR)

FEASIBLE REGION (FR)

x1O

x2




x1O

x2

S




x1O

x2

S

P




x1O

x2

S

PQ

R


Example 1 (cont)

The Feasible Region (OPQRS) representsALL FEASIBLE PAIRS (x1, x2)


Example 1 (cont)


Corners (Vertices) O,P,Q,R,S have special significance

O=(0,0), P=(0,6), Q=(3,5), R=(4,3), S=(4,0)


Example 1 (cont)



O=(0,0), P=(0,6), Q=(3,5), R=(4,3), S=(4,0)

feasible (x1, x2) ⇒ sales value:

£1000 × (x1 + x2)


Example 1 (cont)



O=(0,0), P=(0,6), Q=(3,5), R=(4,3), S=(4,0)

feasible (x1, x2) ⇒ sales value:

£1000 × (x1 + x2)

LP problem: MAXIMIZE this amount


Example 1 (Summary)

Maximize:

x1 + x2


Example 1 (Summary)

Maximize:

x1 + x2

Subject to:

2x1 + x2 ≤ 11x1 + 3x2 ≤ 18

x1 ≤ 4x1, x2 ≥ 0


Example 1 (Summary)

Maximize:

x1 + x2 ◭ objective function

Subject to:

2x1 + x2 ≤ 11x1 + 3x2 ≤ 18

x1 ≤ 4x1, x2 ≥ 0


Example 1 (Summary)

Maximize:

x1 + x2 ◭ objective function

Subject to:

2x1 + x2 ≤ 11 ◭ constraintx1 + 3x2 ≤ 18 ◭ constraint

x1 ≤ 4 ◭ constraintx1, x2 ≥ 0 ◭ constraint


Linear Programming

LINEAR PROGRAM (LP):

optimizing (maximizing or minimizing) linear functionsubject to linear constraints.


Linear Programming

LINEAR PROGRAM (LP):

optimizing (maximizing or minimizing) linear functionsubject to linear constraints.

(Linear: no powers, exponentials, product terms).


Example 1 (cont)

PROPERTY (*):

The set {O,P,Q,R, S} contains LP solution


Example 1 (cont)

PROPERTY (*):


Evaluate objective x1 + x2 at O,P,Q,R, S.


Example 1 (cont)

PROPERTY (*):



O=(0,0) P=(0,6) Q=(3,5) R=(4,3) S=(4,0)0 6 8 7 4


Example 1 (cont)

PROPERTY (*):



O=(0,0) P=(0,6) Q=(3,5) R=(4,3) S=(4,0)0 6 8 7 4

⇒ Q = (3, 5) has maximal value

x1 = 3, x2 = 5 optimal.


Example 1 (cont)

x1O

Q

Px2

R

S

x1 + x2 > 8

x1 + x2 < 8


Property (*)

Feasible Region (FR): area in polygon OPQRSlies entirely within that half plane for which:

x1 + x2 ≤ 8


Property (*)

Feasible Region (FR): area in polygon OPQRSlies entirely within that half plane for which:

x1 + x2 ≤ 8

Since 5 + 3 = 8no feasible point has higher objective value than Q.


Property (*)

PROPERTY (*) holds for any linear function:c1x1 + c2x2.


Property (*)


For Example:Minimise: 3x1 − x2 over polyhedron OPQRS


Property (*)



O=(0,0) P=(0,6) Q=(3,5) R=(4,3) S=(4,0)0 -6 4 9 12


Property (*)



O=(0,0) P=(0,6) Q=(3,5) R=(4,3) S=(4,0)0 -6 4 9 12

P: x1 = 0, x2 = 6 is optimal.


Simplex Algorithm

SIMPLEX ALGORITHM

efficient method for finding optimal vertexwithout necessarily examining all


Simplex Algorithm

SIMPLEX ALGORITHM


PROPERTY (*) does not imply that points other thanvertices cannot be optimal.


Simplex Algorithm

SIMPLEX ALGORITHM


PROPERTY (*) does not imply that points other thanvertices cannot be optimal.

e.g. maximise 2x1 + x2 (on OPQRS):any point on segment QR optimal


operations research - linprog.pdf

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