MTD KWT 06 Linear Programming

  

Quantitative Analysis

Quantitative Analysis

for Management for Management

  Linear Programming

Linear Programming

  

Models:

Models:

  Linear Programming Linear Programming

  Problem Problem 

  1. Tujuan adalah maximize or minimize

variabel dependen dari beberapa kuantitas

variabel independen (fungsi tujuan).

  

  2. Batasan-batasan yang diperlukan guna mencapai tujuan.

   Tujuan dan Batasan dinyatakan dalam persamaan linear.

  

Basic Assumptions of Linear

Basic Assumptions of Linear

  

Programming

Programming

   Certainty  Proportionality  Additivity  Divisibility  Nonnegativity

  Flair Furniture Company Flair Furniture Company

  Data - Data - Table 7.1 Table 7.1

Hours Required to Produce One Unit

  Department

  X ₁ Tables

  X ₂ Chairs Available Hours This Week Carpentry Painting/Varnishing

  4

  2

  3

  1 240 100 Profit/unit $7 $5

  Flair Furniture Company Flair Furniture Company

Table 7.1 Table 7.1 Data - Data - STEP 1:

   Objective: Maximize:  

  X 

  X  

  STEP 2:   

  X  X  (carpentry ) Constraints:

      

  X  X  (painting & varnishing )  

  Flair Furniture Company Flair Furniture Company

  Feasible Region Feasible Region STEP 3: Plot Constraints

  120 100 rs ai Painting/Varnishing h

  80 C f o

  60 er b

  40 m Carpentry u Feasible N

  20 Region 20 40 60 80 100

  Flair Furniture Company Flair Furniture Company

  Isoprofit Lines Isoprofit Lines 120 STEP 4: Plot Objective Function

  100 rs Painting/Varnishing ai h

  80 C

  7X + 5X = 210

  1

  2 f o

  7X + 5X = 420

  60

  1

  2 er b

  40 Carpentry m u N

  20 20 40 60 80 100 Number of Tables

  Flair Furniture Company Flair Furniture Company

  Optimal Solution Optimal Solution 120 Corner Points

  Corner Points 100 rs Painting/Varnishing ai h

  2

80 C f

  Solution o

  60 er (X = 30, X = 40)

  1

  2 b

  40 m Carpentry u

3 N

  20

  1 20 40 60 80 100

  4

  Test Corner Point Solutions Test Corner Point Solutions

  Point 1) (0,0) => 7(0) + 5(0) = $0 Point 2) (0,100) => 7(0) + 5(80) = $400 Point 3) (30,40) => 7(30) + 5(40) = $410 Point 4) (50,0) => 7(50) + 5(0) = $350

  Solve Equations Simultaneously Solve Equations Simultaneously

  4X1 + 3X2 <= 240

  2X1 + 1X2 <= 100 X1 = 60 - 3/4 X2 X1 = 50 - 1/2 X2 60 - 3/4 X2 = 50 - 1/2 X2 60 - 50 = 3/4 X2 - 1/2 X2 10 = 1/4 X2 40 = X2; so, 4X1 + 3(40) = 240

  4X1 = 240 - 120 X1 = 30 To get X1 & X2 values for Point 3:

  

Special Cases in LP

Special Cases in LP

   Infeasibility  Unbounded Solutions  Redundancy  Degeneracy  More Than One Optimal Solution

  A Problem with No Feasible A Problem with No Feasible

  

Solution

Solution

  X ₂

  8

6 Region Satisfying 3rd Constraint

  4

  2

  2

  4

  6

  8 X ₁ Region Satisfying First 2 Constraints

  A Solution Region That is A Solution Region That is

  

Unbounded to the Right

Unbounded to the Right

  X ₂ X ₁

  15

  10

  5 5 10 15 Feasible Region

  X ₁

5 X

   >

  ₂ < 10

  X ₁ +

  2X

₂

> 10

  

A Problem with a

A Problem with a

  Redundant Constraint Redundant Constraint

  X ₂

  30 Redundant

  2X + X < 30 _{1 2}

  25 Constraint

  X ₁ < 25

  20

  15

• +

  X
₁ X < 20 ₂

  10 Feasible

  5 Region

  X ₁

  5

  10

   15

   20

   25

   30

  

An Example of Alternate

An Example of Alternate

  Optimal Solutions Optimal Solutions

  8

  7 Optimal Solution Consists of All A

  Combinations of X and X Along

  6

  1

  2 the AB Segment

  5 Isoprofit Line for $8

  4

  3 Isoprofit Line for $12

  2 B Overlays Line Segment

  1 AB

  1

  2

  3

  4

  5

  6

  7

  8

  

Marketing Applications

Marketing Applications

  Media Selection - Win Big Gambling Club Medium Audience Cost Maximum Reached Per Ads Per Per Ad Ad($) Week TV spot (1 minute) 5,000 800

  12 Daily newspaper 8,500 925

  5 (full-page ad) Radio spot (30 2,400 290

  25 seconds, prime time) Radio spot (1 minute, 2,800 380

  20 afternoon)

  Win Big Gambling Club Win Big Gambling Club Maximize :       

  X X

  X X    

  Subject to : 

  X 12 (max TV spots/week )

  1  

  X ( max newspaper ads/week) 

  X ( -   max 30 sec. radio spots/week ) 

  X   (max 1 min. radio spots/week ) - 

       X  X  X  X  ( weekly

      ad budget)

   

  X X  (min radio spots/week )  

    290X 380X 1800 (max radio expense)

  3

  4

  

Manufacturing Applications

Manufacturing Applications

  Production Mix - Fifth Avenue

Variety Selling Monthly Monthly Material Material

of Tie Price per Contract Demand Required Require

  Tie ($) Minimum per Tie ments (Yds) All silk 6.70 6000 7000 0.125 100% silk All 3.55 10000 14000 0.08 100% polyester polyester Poly- 4.31 13000 16000 0.10 50% cotton poly/50% blend 1 cotton Poly- 4.81 6000 8500 0.10 30% cotton - poly/70% blend 2 cotton

  

Fifth Avenue

Fifth Avenue

  Maximize :

  4.08X 

  3.07X 

  

3.56X 

  4.00X

  1

  2

  3

  4 Subject to :   .

  X   (yards of silk) 

   .  X    . X    . X   (yards polyester)   

   .  X    . X   (yards cotton)  

  X   (contract min, silk) X   (contract max, silk) 

   X  (contract min, all polyester)

   

  X  (contract max, all polyester) 

   

  X  (contract min, blend 1) X  (contract max, blend 1) 

  

  3 X  (contract min, blend 2) 

   X  (contract max, blend 2)

   

  

Manufacturing Applications

Manufacturing Applications

  Truck Loading - Goodman Shipping Item Value ($) Weight (lbs) 1 22,500 7,500 2 24,000 7,500 3 8,000 3,000 4 9,500 3,500 5 11,500 4,000 6 9,750 3,500 _3,500

  Goodman Shipping Goodman Shipping   

  Maximize load value :  X  X  X 

  X    

     X 

  X  

  Subject to :     

  X  X  X  X 

  X     

     X  (Capacity)

   

   X  

   X  

   X  

   X  

   X  

   

  X  

  

Flair Furniture Company

Flair Furniture Company

  

Hours Required to Produce One Unit

Available

  X ₁ X ₂ Department Hours This Tables Chairs Week Carpentry

  4 3 240 Painting/Varnishing

  2 1 100 Profit/unit $7 $5     

  X X (carpentry )  

  Constraints:     

  X X (painting & varnishing )  

  Maximize:  

  Objective: X 

  X  

  

Flair Furniture Company's

Flair Furniture Company's

  

Feasible Region & Corner Points

Feasible Region & Corner Points

  X ₂ s

  100

  ir

  B = (0,80)

  ha

   80 C

  f  X   X  

   

   60

   o er

  C = (30,40)

  b

   40

  m Feasible  X   X  

    u Region

   20 N D = (50,0)

  20

  40

  60 80 100

  X Number of Tables

  

Flair Furniture - Adding

Flair Furniture - Adding

  Slack Variables Slack Variables Constraints: Constraints:   

  X  X  (carpentry )  

     X  X  (painting & varnishing )

    Constraints with Slack Variables Constraints with Slack Variables

       

  X X S (carpentry )   

       

  X X S (painting & varnishing )   

  Objective Function Objective Function  X  

  X  

  Objective Function with Slack Variables Objective Function with Slack Variables  X   X   S   S

     

  Flair Furniture’s Initial Simplex Flair Furniture’s Initial Simplex

  

Tableau

Tableau

  Profit Real Slack per

  Production Constant Variables Variables

  Unit Mix Column Columns Columns

  Column Column Profit per

  C _j

  $7 $5 $0 $0

  unit row Solution

  X ₁ X S S Quantity _{2 1 2} Mix Constraint

  $0 S ₁

  2

  1 1 100

  equation rows

  $0 S ₂

  4

  3 1 240

  Gross

  $0 $0 $0 $0 $0

  profit row Z _j

  Net C - Z $7 $5 $0 $0 $0 _{j j}

  

Pivot Row, Pivot Number Identified

Pivot Row, Pivot Number Identified

in the Initial Simplex Tableau

in the Initial Simplex Tableau

  C _j Solution Mix

  X ₁ X ₂ S ₁ S ₂ Quantity

  $7 $5 $0 $0

  2

  1

  1

  4

  3

  $0 1 $0 $0 $0 $0 $7 $5 $0 $0 $0

  S ₁ S ₂ Z _j C _j - Z _j

  100 240 $0 $0

  Pivot row Pivot number Pivot column

  

Completed Second Simplex Tableau

Completed Second Simplex Tableau

for Flair Furniture

for Flair Furniture

  C _j Solution Mix

  X ₁ X ₂ S ₁ S ₂ Quantity

  $7 $5 $0 $0 1 1/2 1/2 1 -2

  $0 1 $7 $7/2 $7/2 $0 $0 $3/2 -$7/2 $0 $7

  X ₁ S ₂ Z _j C _j - Z _j

  50

  40 $350

  Pivot Row, Column, and Number

Pivot Row, Column, and Number

  

Identified in Second Simplex Tableau

Identified in Second Simplex Tableau

  C _j Solution Mix

  X ₁ X ₂ S ₁ S ₂ Quantity

  $7 $5 $0 $0 1 1/2 1/2 1 -2

  $0 1 $7 $7/2 $7/2 $0 $0 $3/2 -$7/2 $0 $7

  X ₁ S ₂ Z _j C _j - Z _j

  50

  40 $350 (Total Profit)

  Pivot row Pivot number Pivot column

  Calculating the New Calculating the New

  ( )

  1 1/2 1/2

  30

  1 3/2 -1/2

  = - x = - x

  ( )

  Corresponding number in new X ₂ row

  ( )

  Number above pivot number

   row

  X

  X ₁

  Number in old

  )

   row

  X ₁

  Number in new

  1 Row Row for Flair’s Third Tableau for Flair’s Third Tableau (

  1

  X

  50 (1/2) (1/2) (1/2) (1/2) (1/2) (0) (1) (-2) (1) (40) = - x = - x = - x = - x

  

Final Simplex Tableau for

Final Simplex Tableau for

the Flair Furniture Problem

the Flair Furniture Problem

  C _j Solution Mix

  X ₁ X ₂ S ₁ S ₂ Quantity

  $7 $5 $0 $0 1 3/2 -1/2 1 -2

  $5 1 $7 5 $1/2 $3/2 $0 $0 -$1/2 -$3/2 $7

  X ₁ X ₂ Z _j C _j - Z _j

  30

  40 $410

  Simplex Steps for Simplex Steps for

Maximization

  

Maximization

  1. Choose the variable with the greatest positive C _j - Z _j to enter the solution.

  2. Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-to-pivot-column ratio.

  3. Calculate the new values for the pivot row.

  4. Calculate the new values for the other row(s).

  5. Calculate the C _j and C _j - Z _j values for this tableau. If there are any C _j - Z _j values greater than zero, return to Step 1.

  Surplus & Artificial Variables Surplus & Artificial Variables Constraints Constraints   

  X Constraints-Surplus & Artificial Variables Constraints-Surplus & Artificial Variables Objective Function Objective Function

  X      : Min

  X X

  MA MA S

  : Min    

  X  

  X X

       

  X X

          

  X X A S

       A

         

        

  X   

  X X

  X X

      

  

Objective Function-Surplus & Artificial Variables

Objective Function-Surplus & Artificial Variables

  

Simplex Steps for

Simplex Steps for

  Minimization

Minimization

  1. Choose the variable with the greatest negative C - Z to _{j j} enter the solution.

  2. Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-to-pivot-column ratio.

  3. Calculate the new values for the pivot row.

  4. Calculate the new values for the other row(s).

  5. Calculate the C and C - Z values for this tableau. If there _{j j j} are any C - Z values less than zero, return to Step 1. _{j j}

  5 X

  1 2 -2 100 M A

  Infeasibility Infeasibility C j 5 8 0 M M Solution Mix

  X

  1 X

  2 S

  1 S

  2 A

  1 A

  2 Qty

  Special Cases Special Cases

  1 1 0 -2 3 -1 200

  8 X

  2

  1

• -1 -1

  2

  1

  20 Z j 5 8 -2 31-M -21-M M 1800+20M C j -Z j

2 M-31 2M+21

  

Special Cases

Special Cases

  C j

  6

9 Solution Mix

  X

1 X

  

2

S

  1 S

  2 Qty

  

Unboundedness

Unboundedness

9 X

  30 S

  2 -2 -1

  1

  10 Z j -9

  9 18 270 C j -Z j 15 -18 Pivot Column

  2

  2 -1

  1

  Special Cases Special Cases

  Degeneracy Degeneracy C

  5

  8

  2 j Solution

  X X

  X S S S Qty

  1

  2

  3

  1

  2

  3 Mix

  8 X 1/4

  1 1 -2

  10

  2 S 4 1/3 -1

  1

  20

  2 S

  2 2 2/5

  1

  10

  3 Z

  2

  8

  8

  16

  80 j C -Z

  3

  6

  16 j j Pivot Column

1 X

2 X

  1

  12 C j -Z j

  2

  2

  3

  3 Z j

  1

  2 1 1/2

  6 S

  1

  

Special Cases

Special Cases

  2 3/2

  2 Qty

  1 S

  2 S

  X

  

2

Solution Mix

  3

  C j

  

Multiple Optima

Multiple Optima

• -2

  

Sensitivity Analysis

Sensitivity Analysis

  

High Note Sound Company

High Note Sound Company

         

       

      

  X X

  X X

  X X : to Subject : Max

  

Sensitivity Analysis

Sensitivity Analysis

  

High Note Sound Company

High Note Sound Company

  

Simplex Solution

Simplex Solution

  

High Note Sound Company

High Note Sound Company

  C j 50 120 Solution Mix

  X

1 X

  

2

S

  1 S

  2 Qty 120

  X

  2 1/2 1 ¼

  20 S

  2 5/2 -1/4

  1

  40 Z j 60 120 30 2400 C j -Z j

• -30

  

Simplex Solution

Simplex Solution

  

High Note Sound Company

High Note Sound Company

  C j 50 120 Solution Mix

  X

1 X

  

2

S

  1 S

  2 Qty 120

  X

  2 1/2 1 ¼

  20 S

  2

5/2 -1/4

  1

  40 Z j 60 120 30 2400 C j -Z j

• -10 -30

  Nonbasic Objective Function Nonbasic Objective Function

  

Coefficients

Coefficients

  C j 50 120 Solution Mix

  X

1 X

  2 S

  1 S

  2 Qty 120

  X

  2 1/2 1 ¼

  20 S

  2

5/2 -1/4

  1

  40 Z j 60 120 30 2400 C j -Z j

• -10 -30

  Basic Objective Function Basic Objective Function

  Coefficients Coefficients C j 50 120 Solution Mix

  X

1 X

  2 S

  1 S

  2 Qty 120+

  X

  2 1/2 1 ¼ 0

  20 S

  2 5/2 -1/4 1

  40 Z j 60+1/2 120+ 30+1/4 0 2400+20

  C j -Z j -10-1/2

  0 -30-1/4 0

  

Simplex Solution

Simplex Solution

  

High Note Sound Company

High Note Sound Company

  C j 50 120 Solution Mix

  X

1 X

  

2

S

  1 S

  2 Qty 120

  X

  2 1/2 1 ¼

  20 S

  2 5/2 -1/4

  1

  40 Z j 60 120 30 2400 C j -Z j

• -30 Objective increases by 30 if 1 additional hour of electricians time is available.

  

Steps to Form the Dual

Steps to Form the Dual To form the Dual:

   If the primal is max., the dual is min., and vice versa.

   The right-hand-side values of the primal constraints become the objective coefficients of the dual.

  

 The primal objective function coefficients become

the right-hand-side of the dual constraints.

   The transpose of the primal constraint coefficients become the dual constraint coefficients.

   Constraint inequality signs are reversed.

  Primal & Dual Primal & Dual      

       

       

  

  X X

  X X

  X X : to Subject : Max Primal: Primal:

  Dual Dual      

       

       

   U U U U U U

  : to Subject : Min

  40 M M-40

  $7 $5 U ₁ S ₁ Z _j C _j - Z _j

  20 A ₁ A ₂ M M 1/2 -1 1/2

  40

  

80

20 -20

$0

  

80

60 $0 $0 1 1/4 -1/4

  

X

₁ X ₂ S ₁ S ₂

  10 $2,400

  30

  C _j Solution Mix Quantity

  Comparison of the Primal and Comparison of the Primal and Dual Optimal Tableaus Dual Optimal Tableaus u al ’s O pt im al S ol u ti on

  30 -10 -30 P ri m al ’s O pt im al S ol u tio n

  

1/2

1 1/4 5/2 -1/4 1 60 120

  

X

₁ X ₂ S ₁ S ₂ $50 $120 $0 $0

  40 $2,400

  20

  X ₂ S ₂ Z _j C _j - Z _j

  $7 $5

  C _j Solution Mix Quantity

• -5/2 1 -1/2