MA3264 Mathematical ModellingLecture 8

Chapter 7

Discrete Optimization Modelling

Example page 238

How many* tables and how many* bookcases should a carpenter make each week to maximize profit? He realizes a profit of $25 per table and $30 per bookcase. He has 600 feet of lumber per week and 40 hours of labor per week. Each table requires 20 feet of lumber and 5 hours of labor. Each bookcase requires 30 feet of lumber and 4 hours of labor. He has signed contracts to deliver 4 tables and 2 bookcases every week.

How many* : need not be integers since part of a table or bookcase can be made in a week

Mathematical Formulation

Maximize

decisionvariables

21 3025 xx objectivefunction Subject to

6003020 21 xx4045 21 xx41 x22 x

constraints

Mathematical Formulation

Maximize

decisionvariables

21 3025 xx objectivefunction Subject to

6003020 21 xx4045 21 xx41 x22 x

constraints

Question what real world entity does each decision variable represent ?

Mathematical Formulation

Maximize

decisionvariables

21 3025 xx objectivefunction Subject to

6003020 21 xx4045 21 xx41 x22 x

constraints

Question what real world entity does the objective function represent ?

Mathematical Formulation

Maximize

decisionvariables

21 3025 xx objectivefunction Subject to

6003020 21 xx4045 21 xx41 x22 x

constraints

Question what real world entity does each constraint represent ?

Mathematical Formulation

Maximize

decisionvariables

21 3025 xx objectivefunction Subject to

6003020 21 xx4045 21 xx41 x22 x

constraints

Question what real world entities have been abstracted out of this formulation ?

Solution

Maximize

decisionvariables

21 3025 xx objectivefunction Subject to

6003020 21 xx4045 21 xx41 x22 x

constraints

Question can we set the derivatives = 0 to solve this maximization problem ?

Constraints

6003020 21 xx4045 21 xx

41 x22 x

constraints

Question what are the regions consisting of all 1x

2x

),( 21 xx that satisfypoints

the 1st , 2nd ,3rd, 4th constraint ?

Constraint Line

1x

2x

)20,0()0,30(

Question What equation describes the doted line ?

Constraint Line

6003020 21 xx

1x

2x

)20,0()0,30(

Answer The equation above describes the dotted line.

Constraint Region

6003020 21 xx

1x

2x

)20,0()0,30(

Question What region is described by the inequality above ?

Constraint Region

6003020 21 xx

1x

2x

)20,0()0,30(

THIS RED REGION

Feasible Region

6003020 21 xx4045 21 xx

41 x22 x

constraints

Question what region consisting of all 1x

2x

),( 21 xx that satisfypoints

all four constraints ?

Feasible Region

41 x

6003020 21 xx

1x

2x

)20,0(

)0,30(

)10,0(

)4,0(

)8,0(22 x

4045 21 xx is the red region

Feasible Region

1x)2,4( )2,4.6(

)5,4(

2x

is both closed and bounded

Objective Function

1x)2,4( )2,4.6(

)5,4(

2x

is continuous on the feasible region

21 3025 xx

Objective Function

1x)2,4( )2,4.6(

)5,4(

2x

must have a maximum at some point p in the feasible region

21 3025 xx

not necessarily

unique

Objective Function

1x)2,4( )2,4.6(

)5,4(

2x

A similar argument applied to edges (to be shown using visualizer) shows that f has a maximum at a vertex of the feasible region the simplex method

2121 3025),(f xxxx

Suggested Reading

Introduction and Section 7.1 overview of discrete optimization modelling p. 236- 249

Linear Programming 1: Geometric Solutions p. 250-259

Tutorial 8 Due Week 20–24 Oct

Problem 1. Page 245 Problem 1

Problem 2. Page 245 Problem 2

Problem 3. Page 258 Problem 3

Problem 4. Page 259 Problem 4, a

Problem 5. Page 259 Problem 4, b

Problem 6. Page 259 Problem 4, c