Linear Programming
Module B
B-1

Outline
 What is Linear Programming (LP)?
 Characteristics of LP.
 Formulating LP Problems.
 Graphical Solution to an LP Problem.
 Formulation Examples.
 Computer Solution.
 Sensitivity Analysis.
B-2

Optimization Models
 Mathematical models designed to have optimal (best) solutions.
 Linear and integer programming.
 Nonlinear programming.
 Mathematical model is a set of equations and inequalities that describe a system.
 E = mc 2
 Y = 5.4 + 2.6 X
B-3

What is Linear Programming (LP)?
 Mathematical technique to solve optimization models with linear objectives and constraints.
 NOT computer programming!
 Allocates scarce resources to achieve an objective.
 Pioneered by George Dantzig in World War II.
B-4

Examples of Successful LP
Applications
 Scheduling school buses to minimize total distance traveled.
 Allocating police patrols to high crime areas to minimize response time.
 Scheduling tellers at banks to minimize total cost of labor.
B-5

Examples of Successful LP
Applications - continued
 Blending raw materials in feed mills to maximize profit while producing animal feed.
 Selecting the product mix in a factory to make best use of available machine- and labor-hours available while maximizing profit.
 Allocating space for tenants in a shopping mall to maximize revenues to the leasing company.
B-6

Characteristics of an LP Problem
1 Deterministic (no probabilities).
2 Single Objective: maximize or minimize some quantity (the objective function).
3 Continuous decision variables (unknowns to be determined).
4 Constraints limit ability to achieve objective.
5 Objectives and constraints must be expressed as linear equations or inequalities.
B-7

Linear Equations and Inequalities
 4x
1
+ 6x
2
 9  4x
1 x
2
+ 6x
2
 9
 3x - 4y + 5z = 8  3x - 4y 2 + 5z = 8
 3x/4y = 8y  3x/4y = 8 same as 3x - 32y = 0
 4x
1
+ 5x
3
= 8
 4x
1 x
B-8

Formulating LP Problems
Word Problem
Formulation
Mathematical Expressions
Computer
Solution
B-9

Formulating LP Problems
1. Define decision variables.
2. Formulate objective.
3. Formulate constraints.
4. Nonnegativity (all variables  0).
B-10

Formulation Example
You wish to produce two products: (1) Walkman and
(2) Watch-TV. Each Walkman takes 4 hours of electronic work and 2 hours of assembly time. Each
Watch-TV takes 3 hours of electronic work and 1 hour of assembly time. There are 225 hours of electronic work time and 100 hours of assembly time available each month. The profit on each
Walkman is $7; the profit on each Watch-TV is $5.
Formulate a linear programming problem to determine how many of each product should be produced to maximize profit?
B-11

Formulation Example
You wish to produce two products…. Each Walkman takes 4 hours of electronic work and 2 hours of assembly time.… How many of each product should be produced to maximize profit?
Producing 2 products from 2 materials.
Objective: Maximize profit
B-12

Formulation Example
You wish to produce two products: (1) Walkman and
(2) Watch-TV. Each Walkman takes 4 hours of electronic work and 2 hours of assembly time. Each
Watch-TV takes 3 hours of electronic work and 1 hour of assembly time.
There are 225 hours of electronic work time and 100 hours of assembly time available each month.
The profit on each
Walkman is $7; the profit on each Watch-TV is $5.
Formulate a linear programming problem to determine how many of each product should be produced to maximize profit?
B-13

Formulation Example - Objective
.… The profit on each
Walkman is $7; the profit on each Watch-TV is $5.
Maximize profit: $7 per Walkman
$5 per Watch-TV
B-14

Formulation Example -
Requirements
... Each Walkman takes 4 hours of electronic work and 2 hours of assembly time. Each
Watch-TV takes 3 hours of electronic work and 1 hour of assembly time.
...
Requirements:
Walkman 4 hrs elec. time 2 hrs assembly time
Watch-TV 3 hrs elec. time 1 hr assembly time
B-15

Formulation Example - Resources
... There are 225 hours of electronic work time and 100 hours of assembly time available each month. …
Available resources: electronic work time 225 hours assembly time 100 hours
B-16

Formulation Example - Table
Department
Electronic
Assembly
Profit/unit
Hours Required to
Produce 1 Unit
Walkmans Watch-TV’s
4 3
2 1
$7 $5
Available Hours
This Month
225
100
B-17

Formulation Example - Decision
Variables
 What are we deciding? What do we control?
 Number of products to make?
 Amount of each resource to use?
 Amount of each resource in each product?
 Let:

 x
1 x
2
= Number of Walkmans to produce each month.
= Number of Watch-TVs to produce each month.
B-18

Formulation Example - Objective
Department
Electronic
Assembly
Profit/unit
Hours Required to
Produce 1 Unit x
1 x
2
Walkmans Watch-TV’s
4 3
2 1
$7 $5
Available Hours
This Month
225
100
B-19

Formulation Example - Objective
Department
Electronic
Assembly
Hours Required to
Produce 1 Unit x
1 x
2
Walkmans Watch-TV’s
4 3
2 1
Profit/unit $7 $5
Objective : Maximize: 7x
1
+ 5x
2
Available Hours
This Month
225
100
B-20

Formulation Example - 1st
Constraint
Department
Electronic
Assembly
Hours Required to
Produce 1 Unit x
1 x
2
Walkmans Watch-TV’s
4 3
2 1
Available Hours
This Month
225
100
Profit/unit $7 $5
Objective : Maximize: 7x
1
+ 5x
2
Constraint 1: 4x
1
+ 3x
2
 225 (Electronic Time hrs)
B-21

Formulation Example - 2nd
Constraint
Department
Electronic
Assembly
Hours Required to
Produce 1 Unit x
1 x
2
Walkmans Watch-TV’s
4 3
2 1
Available Hours
This Month
225
100
Profit/unit $7 $5
Objective : Maximize: 7x
1
+ 5x
2
Constraint 1:
Constraint 2:
4x
1
2x
1
+ 3x
2
+ x
2
 225 (Electronic Time hrs)
 100 (Assembly Time hrs)
B-22

Complete Formulation (4 parts) x
1 x
2
= Number of Walkmans to produce each month.
= Number of Watch-TVs to produce each month.
Maximize: 7x
1
+ 5x
2
4x
1
2x
1
+ 3x
2
+ x
2
 225
 100 x
1
, x
2
 0
B-23

Formulation Example - Max Profit
 Suppose you are not given the profit for each product, but are given:
 The selling price of a Walkman is $60 and the selling price of a Watch-TV is $40.
 Each hour of electronic time costs $10 and each hour of assembly time costs $8.
 Profit = Revenue Cost
Walkman profit = $60 ($10/hr  4 hr + $8/hr  2 hr) = $4
Watch-TV profit = $40 ($10/hr  3 hr + $8/hr  1 hr) = $2
B-24

Formulation Example - Optimal
Solution x
1 x
2
= 37.5 Walkmans produced each month.
= 25 Watch-TVs produced each month.
Profit = $387.5/month
 Can you make 37.5??
 Can you round to 38??
NO!! That requires 227 hrs of electronic time.
4  38 + 3  25 = 227 (> 225!)
B-25

Graphical Solution Method - Only with 2 Variables!
 Draw graph with vertical & horizontal axes
(1st quadrant only).
 Plot constraints as lines, then as planes.
 Find feasible region.
 Find optimal solution.
 It will be at a corner point of feasible region!
B-26

100
80
60
40
20
0
0
Formulation Example Graph
4x
1
+3x
2
 225 (electronics)
2x
1
+x
2
 100 (assembly)
20 40 60 80
Number of Walkmans (X
1
B-27
)

Feasible Region
100
80
4x
1
+3x
2
 225 (electronics)
2x
1
+x
2
 100 (assembly)
60
40
20
0
0
Feasible
Region
20 40 60 80
Number of Walkmans (X
1
B-28
)

Possible Solution Points
100
80
60
40
20
0
0
Feasible
Region
20 40
X
1
60 80
B-29
4x
1
+3x
2
 225 (electronics)
2x
1
+x
2
 100 (assembly)
Corner Point Solutions

100
80
Opitmal Solution
1. x
1
2. x
1
Profit = 7 x
1
+ 5 x
2
= 0, x
= 0, x
2
2
= 0 profit = 0
= 75 profit = 375
3. x
1
4. x
1
= 50, x
2
= 37.5, x
2
= 0 profit = 350
= 25 profit = 387.5
60
40
20
0
0
Feasible
Region
20 40
X
1
60 80
B-30

Formulation #1
A company wants to develop a high energy snack food for athletes. It should provide at least 20 grams of protein, 40 grams of carbohydrates and 900 calories. The snack food is to be made from three ingredients, denoted A, B and C. Each ounce of ingredient A costs $0.20 and provides 8 grams of protein, 3 grams of carbohydrates and 150 calories. Each ounce of ingredient B costs $0.10 and provides 2 grams of protein, 7 grams of carbohydrates and 80 calories. Each ounce of ingredient C costs $0.15 and provides 5 grams of protein, 6 grams of carbohydrates and 100 calories.
Formulate an LP to determine how much of each ingredient should be used to minimize the cost of the snack food.
B-31

Formulation #1
How many products?
How many ingredients?
How many attributes of products/ingredients?
B-32

Formulation #1
How many products?
1
How many ingredients?
3
How many attributes of products/ingredients?
3
Do we know how much of each ingredient (or resource) is in each product?
B-33

Formulation #1
Ingredient cost
A $0.2/oz
B
C
$0.1/oz
$0.15/oz
Snack food protein
8
2
5
 20 carbo.
3
7
6
 40 calories
150
80
100
 900
B-34

Formulation #1
Ingredient cost
A $0.2/oz
B
C
$0.1/oz
$0.15/oz
Snack food protein
8
2
5
 20 carbo.
3
7
6
 40 calories
150
80
100
 900
Variables: : x i
= Number of ounces of ingredient i used in snack food.
i = 1 is A; i = 2 is B; i = 3 is C
B-35

Formulation #1 x i
= Number of ounces of ingredient i used in snack food.
Minimize: 0.2x
1
+ 0.1x
2
+ 0.15x
3
8x
3x
150x
1
1
1
+ 2x
2
+ 7x
2
+ 80x
2
+ 5x
3
+ 6x
3
+ 100x
3
 20 (protein)
 40 (carbs.)
 900 (calories) x
1
, x
2
, x
3
 0
B-36

Formulation #1 - Additional
Constraints x i
= Number of ounces of ingredient i used in snack food.
1. At most 20% of the calories can come from ingredient A.
B-37

Formulation #1 - Additional
Constraints x i
= Number of ounces of ingredient i used in snack food.
1. At most 20% of the calories can come from ingredient A.
calories from A = 150x
1 total calories = 150x
1
+ 80x
2
+ 100x
3
150x
1
150x
1
 80x
2
 100x
3 or
120x
1
 16x
2
 20x
3
 0
 0.2
B-38

Formulation #1 - Additional
Constraints x i
= Number of ounces of ingredient i used in snack food.
2. The snack food must include at least 1 ounce of A and
2 ounces of B.
3. The snack food must include twice as much A as B.
B-39

Formulation #1 - Additional
Constraints x i
= Number of ounces of ingredient i used in snack food.
2. The snack food must include at least 1 ounce of A and
2 ounces of B.
x
1
 1 x
2
 2
3. The snack food must include twice as much A as B.
x
1
 2x
2
B-40

Formulation #1 - Additional
Constraints x i
= Number of ounces of ingredient i used in snack food.
4. The snack food must include twice as much A as B and C.
5. The snack food must include twice as much A and B as C.
B-41

Formulation #1 - Additional
Constraints x i
= Number of ounces of ingredient i used in snack food.
4. The snack food must include twice as much A as B and C.
x
1
= 2x
2 x
1
= 2x
3 or x
1
= 2(x
2
+ x
3
)
5. The snack food must include twice as much A and B as C.
x
1
= 2x
3 x
2
= 2x
3 or x
1
+ x
2
= 2x
3
B-42

Formulation #2
2. Plant fertilizers consist of three active ingredients, Nitrogen,
Phosphate and Potash, along with inert ingredients.
Fertilizers are defined by three numbers representing the percentages of Nitrogen, Phosphate, Potash. For example a 20-10-40 fertilizer includes 20% Nitrogen, 10% Phosphate and 40% Potash.
NuGrow makes three different fertilizers, packaged in 40 lb.
bags: 20-10-40, 10-10-10 and 30-30-10. The 20-10-40 fertilizer sells for $8/bag and at least 3000 bags must be produced next month. The 10-10-10 fertilizer sells for
$4/bag. The 30-30-10 fertilizer sells for $6/bag and at least
4000 bags must be produced next month. The cost and availability of the fertilizer ingredients is as follows:
B-43

Formulation #2 - continued
Ingredient
Nitrogen (N)
Phosphate (Ph)
Potash (Po)
Inert (In)
Amount Available
(tons/month)
20
30
40 unlimited
Cost
($/ton)
300
200
400
100
Formulate an LP to determine how many bags of each type of fertilizer NuGrow should make next month to maximize profit.
B-44

Formulation #2
Produce 3 products (fertilizers) from 4 ingredients.
Do we know how much of each ingredient (or resource) is in each product?
If ‘ YES ’, variables are probably amount of each product to produce.
If ‘ NO ’, variables are probably amount of each ingredient (or resource) to use in each product.
B-45

Formulation #2 - continued
Product
20-10-40
10-10-10
30-30-10
Minimum req’d (bags)
3000
4000
Price
($/bag)
8
4
6 lbs. of ingredient per bag
N Ph Po In
8 4 16 12
4 4 4 28
12 12 4 12 x i
= Number of bags of fertilizer type i to make next month.
i=1: 20-10-40 i=2: 10-10-10 i=3: 30-30-10
B-46

Formulation #2 - Constraints
Produce 3 products (fertilizers) from 4 ingredients.
3 variables.
How many constraints?
Usually:
- one (or two) for each ingredient
- one (or two) for each final product
- others?
B-47

Formulation #2 - Constraints
Produce 3 products (fertilizers) from 4 ingredients.
3 variables.
How many constraints?
Usually:
- one for each ingredient ( 3, no constraint for Inert )
- one for each final product ( 2, no constraint for type 2 )
- others? ( no )
3 variables, 5 constraints
B-48

Formulation #2 - Objective x i
= Number of bags of fertilizer type i to make next month.
: Maximize Profit = Revenue Cost
Revenue = 8x
1
+ 4x
2
+ 6x
3
Cost = (cost per bag of type 1) x
+ (cost per bag of type 2) x
1
+ (cost per bag of type 3) x
2
3
Cost per bag is cost of all ingredients in a bag.
B-49

Formulation #2 - Costs x i
= Number of bags of fertilizer type i to make next month.
Cost for one bag of type 1 (20-10-40)
= cost for N 8  0.15
($300/ton=$0.15/lb)
+ cost for Ph 4  0.10
($200/ton=$0.10/lb)
+ cost for Po 16  0.20
($400/ton=$0.20/lb)
+ cost for In 12  0.05
($100/ton=$0.05/lb)
= $5.4
Similarly:
Cost for one bag of type 2 (10-10-10) = $3.2
Cost for one bag of type 3 (30-30-10) = $4.4
B-50

Formulation #2 - Objective x i
= Number of bags of fertilizer type i to make next month.
: Maximize Profit = Revenue Cost
Revenue = 8x
1
+ 4x
2
+ 6x
3
Cost = 5.4x
1
+ 3.2x
2
+ 4.4x
3
Maximize 2.6x
1
+ 0.8x
2
+ 1.6x
3
B-51

Formulation #2 x i
= Number of bags of fertilizer type i to make next month.
Maximize: 2.6x
1
+ 0.8x
2
+ 1.6x
3
8x
4x
16x x
1
1
1
1
+ 4x
+ 4x
2
+ 4x
2
2
+ 12x
3
+ 12x
3
+ 4x
3

 40000 (N)
60000 (Ph)
 80000 (Po) x
3
 3000 (20-10-40)
 4000 (30-30-10) x
1
, x
2
, x
3
 0
B-52

Formulation #2 - Additional
Constraints x i
= Number of bags of fertilizer type i to make next month.
1. NuGrow can produce at most 4000 lbs. of 10-10-10 fertilizer next month.
2. The 20-10-40 fertilizer should be at least 50% of the total production.
B-53

Formulation #2 - Additional
Constraints x i
= Number of bags of fertilizer type i to make next month.
1. NuGrow can produce at most 4000 lbs. of 10-10-10 fertilizer next month.
x
2
 100
2. The 20-10-40 fertilizer should be at least 50% of the total production. x
1
 0.5
(x
1
 x
2
 x
3
)
B-54

Formulation #3
4.
NuTree makes two 2 types of paper (P1 and P2) from three grades of paper stock. Each stock has a different strength, color, cost and
(maximum) availability as shown in the table below. Paper P1 must have a strength rating of at least 7 and a color rating of at least 6. Paper P2 must have a strength rating of at least 6 and a color rating of at least 5. Paper P1 sells for $200/ton and the maximum demand is 70 tons/week. Paper P2 sells for $100/ton and the maximum demand is 120 tons/week. NuTree would like to determine how to produce the two paper types to maximize profit.
Paper Stock Strength Color Cost/Ton Availability
R1
R2
R3
8
6
3
9
7
4
$150 40 tons/week
$110 60 tons/week
$ 50 100 tons/week
B-55

Formulation #3
Paper Stock Strength Color Cost/Ton Availability
R1
R2
R3
8
6
3
9
7
4
$150 40 tons/week
$110 60 tons/week
$ 50 100 tons/week
Paper Strength Color Price/Ton Max. Demand
P1  7  6 $200 70 tons/week
P2 tons/week
 6  5 $100 120
Produce 2 products (papers) from 3 ingredients (paper stocks) to maximize profit (= revenue - cost).
Constraints: Availability(3); Demand(2); Strength(2); Color(2)
B-56

Formulation #3 - Decision
Variables
Produce 2 products (papers) from 3 ingredients (paper stocks).
Do we know how much of each ingredient is in each product?
B-57

Formulation #3 - Decision
Variables
Produce 2 products (papers) from 3 ingredients (paper stocks).
Do we know how much of each ingredient is in each product?
NO!
 6 variables for amount of each ingredient in each final product.
B-58

Formulation #3 - Key!
Produce 2 products (papers: P1 and P2) from 3 ingredients (paper stocks: R1, R2 and R3).
x ij
= Number of tons of stock i in paper j; i=1,2,3 j=1,2
R1
R2
R3
P1 x
11 x
21 x
31 x
12 x
22 x
32
P2
B-59

Formulation #3 x ij
= Number of tons of stock i in paper j; i=1,2,3 j=1,2
R1
R2
R3
P1 x
11 x
21 x
31 x
12 x
22 x
32
P2
Tons of stock R1 used = x
11
+ x
12
Tons of stock R2 used = x
21
+ x
22
Tons of stock R3 used = x
31
+ x
32
B-60

Formulation #3 x ij
= Number of tons of stock i in paper j; i=1,2,3 j=1,2
: Amount of paper P1 produced = x
11
Amount of paper P2 produced = x
12
+ x
21
+ x
31
+ x
22
+ x
32
R1
R2
R3
P1 x
11 x
21 x
31 x
12 x
22 x
32
P2
B-61

Formulation #3 - Objective
Paper Stock Strength Color Cost/Ton Availability
R1 8 9 $150 40 tons/week
R2
R3
6
3
7
4
$110
$ 50
60 tons/week
100 tons/week
Maximize Profit = Revenue Cost
Cost = 150(tons of R1) + 110(tons of R2) + 50(tons of R3)
Cost = 150(x
11
+ x
12
) + 110(x
21
+ x
22
) + 50(x
31
+ x
32
)
B-62

Formulation #3 - Objective
Paper Strength Color Price/Ton Max. Demand
P1  7  6 $200 70 tons/week
P2  6  5 $100 120
Revenue = 200(x
11
+ x
21
+ x
31
) + 100(x
12
+ x
22
+ x
32
)
B-63

Formulation #3 - Objective
Maximize profit = Revenue Cost
Revenue = 200(x
11
+ x
21
+ x
31
) + 100(x
12
+ x
22
+ x
32
)
Cost = 150(x
11
+ x
12
) + 110(x
21
+ x
22
) + 50(x
31
+ x
32
)
Maximize 50x
11
+ 90x
21
+ 150x
31
- 50x
12
- 10x
22
+ 50x
32
B-64

Formulation #3 - Constraints
Paper Stock Strength Color Cost/Ton Availability
R1 8 9 $150 40 tons/week
R2
R3
6
3
7
4
$110 60 tons/week
$ 50 100 tons/week
Availability of each ingredient x
11 x
21 x
31
+ x
12
+ x
22
+ x
32
 40 (R1)
 60 (R2)
 100 (R3)
B-65

Formulation #3 - Constraints
Paper Strength Color Price/Ton Max. Demand
P1  7  6 $200 70 tons/week
P2 tons/week
 6  5 $100 120
Demand for each product x
11
+ x
21
+ x
31 x
12
+ x
22
+ x
32
 70 (P1)
 120 (P2)
B-66

Formulation #3 - Constraints
Paper Stock Strength Color Cost/Ton Availability
R1 8 9 $150 40 tons/week
R2
R3
6
3
7
4
$110 60 tons/week
$ 50 100 tons/week
Paper Strength Color Price/Ton Max. Demand
P1
P2
 7
 6
 6
 5
$200 70 tons/week
$100 120 tons/week
Strength and color are weighted averages, where weights are tons of each ingredient used.
1 ton of R1 + 1 ton of R2 = 2 tons with Strength = 7
2 tons of R1 + 1 ton of R2 = 3 tons with Strength = 7.333
B-67

Formulation #3 - Constraints
Paper Stock Strength Color Cost/Ton Availability
R1 8 9 $150 40 tons/week
R2
R3
6
3
7
4
$110 60 tons/week
$ 50 100 tons/week
Paper Strength Color Price/Ton Max. Demand
P1
P2
 7
 6
 6
 5
$200 70 tons/week
$100 120 tons/week
Strength P1: average strength from ingredients of P1  7
8x x
11
11


6x x
21
21

 x
3x
31
31  7 or x
11
 x
21
 4x
31
 0
B-68

Formulation #3 x ij
= Number of tons of stock i in paper j; i=1,2,3 j=1,2
Maximize 50x
11 x
11
+ 90x
21
+ 150x
31
(strength P1)
(strength P2)
(color P1)
(color P2) x x
11
11
- x x
21
+ x
21
21 x
31
+ x
31
- 4 x
31
- 50x
+ x x
12
12
12
2x
12
- 10x
+ x
+ x
22
22
22
3x
11
+ x
21 x
- 2 x
31
11
, x
12
, x
13
, x
21
4x
12
, x
22
+ 2x
, x
23
12
 0
+ 50x
32
+ x
32
 40
 60
 100
 70
+ x
32
 120
 0
- 3x
32
- x
32
 0
 0
 0
B-69