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Chapter%202:%20An%20Introduction%20to%20Linear%20Programming

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Chapter 2: An Introduction to Linear Programming Instructor: Dr. Mohamed Mostafa * Solve for the Extreme Point at the Intersection of the Two Binding Constraints ... – PowerPoint PPT presentation

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Title: Chapter%202:%20An%20Introduction%20to%20Linear%20Programming


1
Chapter 2 An Introduction to Linear Programming
Instructor Dr. Mohamed Mostafa
2
Overview
  • Linear Programming Problem
  • Problem Formulation
  • A Simple Maximization Problem
  • Graphical Solution Procedure
  • Extreme Points and the Optimal Solution
  • A Simple Minimization Problem
  • Special Cases

2
3
Linear Programming
  • Linear programming has nothing to do with
    computer programming.
  • The use of the word programming here means
    choosing a course of action.
  • Linear programming is a problem-solving approach
    developed to help managers make decisions.

4
Linear Programming (LP) Problem
  • The maximization or minimization of some quantity
    is the objective in all linear programming
    problems.
  • All LP problems have
  • Constraints that limit the objective function
    value.
  • feasible solution satisfies all the problem's
    constraints.
  • optimal solution is the largest possible
    objective function value when maximizing (or
    smallest when minimizing).
  • A graphical solution method can be used to solve
    a linear program with two variables.

5
Linear Programming (LP) Problem
  • If both the objective function and the
    constraints are linear, the problem is referred
    to as a linear programming problem.
  • Linear functions are functions in which each
    variable appears in a separate term raised to the
    first power and is multiplied by a constant
    (which could be 0).
  • Linear constraints are linear functions that are
    restricted to be "less than or equal to", "equal
    to", or "greater than or equal to" a constant.

6
Which are NOT linear functions?
  1. -2x1 4x2 1x3 lt 80
  2. 3vx1 - 2x2 15
  3. 2x1x2 5x3 lt 17
  4. 4x1 7x3 12

6
7
Problem Formulation
  • Problem formulation or modeling is the process of
    translating a verbal statement of a problem into
    a mathematical statement.
  • Formulating models is an art that can only be
    mastered with practice and experience.
  • Every LP problem has some unique features, but
    most problems also have common features.
  • General guidelines for LP model formulation are
    illustrated on the slides that follow.

8
Guidelines for Model Formulation
  • Read and Understand the problem.
  • Describe the objective.
  • Describe each constraint.
  • Define the decision variables.
  • Write the objective in terms of the decision
    variables.
  • Write the constraints in terms of the decision
    variables.

9
Problem Statement
  • A Starbucks wants to maximize hourly profit on
    sales of lattes and cappuccinos. They make 5
    per latte and 7 per cappuccino.
  • In any given hour,
  • The latte frother can blend up to 6 cups per
    hour.
  • The maximum milk supply in each hour is 19 cups.
    Lattes require 2 cups, and cappuccinos take 3.
  • The lid station can provide a max of 8 lids per
    hour. Each latte and cappuccino must have a lid.
  • How can Starbucks maximize profit in each hour on
    sales of lattes and cappuccinos?

9
10
Objective Function
Max 5x1 7x2 s.t. x1
lt 6 2x1 3x2 lt 19
x1 x2 lt 8 x1 gt 0 and x2 gt 0
Regular Constraints
Non-negativity Constraints
11
Graphical Solution
  • Prepare a graph of the feasible solutions for
    each of the constraints.
  • Determine the feasible region that satisfies all
    the constraints simultaneously.
  • Type of constraint Feasible region (usually)
    will be
  • gt above/to the right
  • lt below/to the left
  • the line
  • Draw an objective function line.
  • Move parallel objective function lines toward
    larger objective function values without entirely
    leaving the feasible region.
  • Any feasible solution on the objective function
    line with the largest value is an optimal
    solution.

11
12
Starbucks Graphical Solution
x2
8 7 6 5 4 3 2 1
x1
1 2 3 4 5 6 7
8 9 10
13
Starbucks Graphical Solution
  • First Constraint Graphed

x2
8 7 6 5 4 3 2 1
x1 6
Shaded region contains all feasible points for
this constraint
(6, 0)
x1
1 2 3 4 5 6 7
8 9 10
14
  • Second Constraint Graphed

x2
8 7 6 5 4 3 2 1
(0, 6 1/3)
2x1 3x2 19
Shaded region contains all feasible points for
this constraint
(9 1/2, 0)
x1
1 2 3 4 5 6 7
8 9 10
15
  • Third Constraint Graphed

x2
(0, 8)
8 7 6 5 4 3 2 1
x1 x2 8
Shaded region contains all feasible points for
this constraint
(8, 0)
x1
1 2 3 4 5 6 7
8 9 10
16
  • Combined-Constraint Graph
  • Showing Feasible Region

x2
x1 x2 8
8 7 6 5 4 3 2 1
x1 6
2x1 3x2 19
Feasible Region
x1
1 2 3 4 5 6 7
8 9 10
17
  • Objective Function Line

x2
8 7 6 5 4 3 2 1
(0, 5)
Objective Function 5x1 7x2 35
(7, 0)
x1
1 2 3 4 5 6 7
8 9 10
18
Selected Objective Function Lines
x2
8 7 6 5 4 3 2 1
5x1 7x2 35
5x1 7x2 39
5x1 7x2 42
x1
1 2 3 4 5 6 7
8 9 10
19
  • Optimal Solution

x2
Maximum Objective Function Line 5x1 7x2 46
8 7 6 5 4 3 2 1
Optimal Solution (x1 5, x2 3)
x1
1 2 3 4 5 6 7
8 9 10
20
  • Solve for the Extreme Point at the Intersection
    of the Two Binding Constraints
  • 2x1 3x2 19
  • x1 x2 8
  • The two equations will give
  • x2 3
  • Substituting this into x1 x2 8 gives
    x1 5

Solve for the Optimal Value of the Objective
Function 5x1 7x2 5(5) 7(3)
46
21
Extreme Points and the Optimal Solution
  • The corners or vertices of the feasible region
    are referred to as the extreme points.
  • An optimal solution to an LP problem can be found
    at an extreme point of the feasible region.
  • When looking for the optimal solution, you do not
    have to evaluate all feasible solution points
    consider only the extreme points of the feasible
    region.

22
Extreme Points
x2
8 7 6 5 4 3 2 1
(0, 6 1/3)
5
(5, 3)
4
Feasible Region
(6, 2)
3
(0, 0)
(6, 0)
2
1
x1
1 2 3 4 5 6 7
8 9 10
23
x2
Max Z 5x1 7x2
(x1 , x2 ) Z
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
(0, 0) 0
1
5
(6, 0) 30
2
(6, 2) 44
3
(5, 3) 46
4
(6 1/3,0) 30 5/3
4
5
Feasible Region
3
1
2
x1
23
24
Problem
  • A woodcarving ??? ????? agency manufactures two
    types of wooden toys soldiers and trains. A
    soldier sells for 27 and uses 10 worth of raw
    materials. Each soldier that is manufactured
    increases his variable labor and overhead costs
    (Examples include rent, gas, electricity, and
    wages) by 14. A train sells for 21 and uses 9
    worth of raw materials. Each train built
    increases his variable labor and overhead costs
    by 10.
  • The manufacture of soldier and trains requires
    two types of skilled labor carpentry and
    finishing. A soldier requires 2 hours of
    finishing labor and 1 hour of carpentry labor. A
    train requires 1 hour of finishing and 1 hour of
    carpentry labor. Each week he can obtain all the
    raw material he needs but only 100 hours of
    finishing and 80 carpentry hours. Demand for
    trains is unlimited, but at most 40 soldiers are
    bought every week.
  • Formulate the problem that may maximize the
    companys weekly profit and solve it graphically.

25
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26
Problem
  • The following table summarizes the key facts
    about two products, A and B, and the resources Q,
    R, and S required to produce them. Formulate a
    linear programming model for this problem. The
    profit per unit is 3 for Product A and 2 for
    Product B.

Resource Resource Usage per Unit Produced Resource Usage per Unit Produced Amount of Available Resource
Resource Product A Product B Amount of Available Resource
Q 2 1 2
R 1 2 2
S 3 3 4
27
Assignment
  • The Primo Insurance Company is introducing two
    new product lines special risk insurance and.
    Mortgages ??? ????? The expected profit is 5 per
    unit on special risk insurance and 2 per unit on
    mortgages. Management wishes to establish sales
    quotas ??? ???????? for new product lines. The
    work req. are as follows. Formulate the LP model
    and solve it graphically.

Department Work-hours per unit Work-hours per unit Work hours Available
Department Special Risk Mortgage Work hours Available
Underwriting ???? ????????-???????? 3 2 2,400
Admin.??????? 0 1 800
Claims ?????????-??????? 2 0 1,200
28
Computer Solution Windows QM
Demo version available at http//wps.prenhall.co
m/bp_weiss_software_1/0,6750,91664-,00.html
29
(No Transcript)
30
Solving Graphically Minimization Problem
  • Prepare a graph of the feasible solutions for
    each of the constraints.
  • Determine the feasible region that satisfies all
    the constraints simultaneously.
  • Draw an objective function line.
  • Move parallel objective function lines toward
    smaller objective function values without
    entirely leaving the feasible region.
  • Any feasible solution on the objective function
    line with the smallest value is an optimal
    solution.

31
A Simple Minimization Problem
  • LP Formulation

Min 5x1 2x2 s.t. 2x1 5x2 gt
10 4x1 - x2 gt 12
x1 x2 gt 4 x1, x2 gt 0
32
Graphical Solution
  • Constraints Graphed

x2
6 5 4 3 2 1
Feasible Region
4x1 - x2 gt 12
x1 x2 gt 4
2x1 5x2 gt 10
x1
1 2 3 4 5 6
33
Objective Function Graphed
x2
Min 5x1 2x2
6 5 4 3 2 1
4x1 - x2 gt 12
x1 x2 gt 4
2x1 5x2 gt 10
x1
1 2 3 4 5 6
34
  • Solve for the Extreme Point at the Intersection
    of the Two Binding Constraints
  • 4x1 - x2 12
  • x1 x2 4
  • Adding these two equations gives
  • 5x1 16 or x1 16/5
  • Substituting this into x1 x2 4 gives
    x2 4/5

Solve for the Optimal Value of the Objective
Function 5x1 2x2 5(16/5)
2(4/5) 88/5
35
LP in Standard Form
  • A linear program in which all the variables are
    non-negative and all the constraints are
    equalities is said to be in standard form
  • To attain ????standard form you must
  • lt constraints Add slack variable ????? ?????? to
    constraint (coefficient of 0 in obj function)
  • gt constraints Subtract surplus ???? from
    constraint (coefficient of 0 in obj function)
  • constraints Add artificial variable ?????
    ????????? to constraint (coefficient of M in obj
    function)

36
  • Slack and surplus variables represent the
    difference between the left and right sides of
    the constraints. Slack is any unused resource,
    while surplus is the amount over some required
    minimum level.
  • The objective function coefficient for slack and
    surplus variables is equal to 0.
  • If slack/surplus variables are equal to 0 for a
    constraint, the constraint is said to be binding
    ??????.

37
Slack Variables (for lt constraints)
  • Max 5x1 7x2 0s1 0s2 0s3
  • s.t. x1 s1 6
  • 2x1 3x2 s2 19
  • x1 x2 s3 8
  • x1, x2 , s1 , s2 ,
    s3 gt 0

s1 , s2 , and s3 are slack variables
Example in Standard Form
38
Optimal Solution
x2
Third Constraint x1 x2 8
First Constraint x1 6
8 7 6 5 4 3 2 1
s3 0
s1 1
Second Constraint 2x1 3x2 19
Optimal Solution (x1 5, x2 3)
s2 0
x1
1 2 3 4 5 6 7
8 9 10
39
Surplus Variables
Minimization Example in Standard Form
Min 5x1 2x2 0s1 0s2 0s3 s.t.
2x1 5x2 - s1 10
4x1 - x2 - s2 12
x1 x2 - s3
4 x1, x2, s1,
s2, s3 gt 0
s1 , s2 , and s3 are surplus variables
40
LPs Special Case Alternative Optimal Solutions
Max 4x1 6x2 s.t. x1
lt 6 2x1 3x2 lt 18
x1 x2 lt 7 x1 gt 0 and x2
gt 0
41
Boundary constraint 2x1 3x2 lt 18 and objective
function Max 4x1 6x2 are parallel. All points
on line segment A B are optimal solutions.
x2
x1 x2 lt 7
7 6 5 4 3 2 1
Max 4x1 6x2
A
B
x1 lt 6
2x1 3x2 lt 18
x1
1 2 3 4 5 6 7
8 9 10
42
Infeasibility
Max 2x1 6x2 s.t. 4x1 3x2 lt 12
2x1 x2 gt 8
x1, x2 gt 0
43
  • There are no points that satisfy both
    constraints, so there is no feasible region (and
    no feasible solution).

x2
10
2x1 x2 gt 8
8
6
4x1 3x2 lt 12
4
2
x1
2 4 6 8 10
44
Unbounded Problem
Max 4x1 5x2 s.t. x1 x2 gt 5
3x1 x2 gt 8
x1, x2 gt 0
45
  • The feasible region is unbounded and the
    objective function line can be moved outward from
    the origin without bound, infinitely increasing
    the objective function.

x2
10
3x1 x2 gt 8
8
6
Max 4x1 5x2
4
x1 x2 gt 5
2
x1
2 4 6 8 10
46
Assignment
  • The Sanders Garden Shop mixes two types of grass
    seed into a blend. Each type of grass has been
    rated (per pound) according to its shade
    tolerance, ability to stand up to traffic, and
    drought resistance, as shown in the table. Type A
    seed costs 1 and Type B seed costs 2. If the
    blend needs to score at least 300 points for
    shade tolerance, 400 points for traffic
    resistance, and 750 points for drought
    resistance, how many pounds of each seed should
    be in the blend? How much will the blend cost?
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