1. Operations Management Linear Programming Module B

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

3. 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 = mc2
Y = X

4. 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.

5. 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.

6. 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.

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

8. Linear Equations and Inequalities
 4x1 + 6x2  9  4x1 x2+ 6x2  9
 3x - 4y + 5z = 8  3x - 4y2 + 5z = 8
 3x/4y =  3x/4y = 8y
same as 3x - 32y = 0
 4x1 + 5x3 =  4x = 8

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

10. Formulating LP Problems
Word Problem
Mathematical Expressions
Solution
Formulation
Computer

11. 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?

12. 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

13. 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?

14. 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

15. 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

16. 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 hours
assembly time hours

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

18. 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:
x1 = Number of Walkmans to produce each month.
x2 = Number of Watch-TVs to produce each month.

19. Formulation Example - Objective
Hours Required to
Produce 1 Unit
Department x1 x2
Available Hours
Walkmans
Watch-TV’s
This Month
Electronic 4 3
225
Assembly 2 1
100
Profit/unit $7 $5

20. Formulation Example - Objective
Hours Required to
Produce 1 Unit
Department x1 x2
Available Hours
Walkmans
Watch-TV’s
This Month
Electronic 4 3
225
Assembly 2 1
100
Profit/unit $7 $5
Objective:
Maximize: 7x1 + 5x2

21. Formulation Example - 1st Constraint
Hours Required to
Produce 1 Unit
Department x1 x2
Available Hours
Walkmans
Watch-TV’s
This Month
Electronic 4 3
225
Assembly 2 1
100
Profit/unit $7 $5
Objective:
Maximize: 7x1 + 5x2
Constraint 1:
4x1 + 3x2  (Electronic Time hrs)

22. Formulation Example - 2nd Constraint
Hours Required to
Produce 1 Unit
Department x1 x2
Available Hours
Walkmans
Watch-TV’s
This Month
Electronic 4 3
225
Assembly 2 1
100
Profit/unit $7 $5
Objective:
Maximize: 7x1 + 5x2
Constraint 1:
4x1 + 3x2  (Electronic Time hrs)
Constraint 2:
2x1 + x2  (Assembly Time hrs)

23. Complete Formulation (4 parts)
x1 = Number of Walkmans to produce each month.
x2 = Number of Watch-TVs to produce each month.
Maximize: 7x1 + 5x2
4x1 + 3x2  225
2x1 + x2  100
x1, x2 0

24. Formulation Example - Max Profit
Suppose you are not given the profit for each product, but are given:
The selling price of a Walkman is $63 and the selling price of a Watch-TV is $43.
Each hour of electronic time costs $10 and each hour of assembly time costs $8.
Profit = Revenue - Cost
Walkman profit = $63 - ($10/hr 4 hr + $8/hr  2 hr) = $7
Watch-TV profit = $43 - ($10/hr 3 hr + $8/hr  1 hr) = $5

25. Formulation Example - Optimal Solution
x1 = 37.5 Walkmans
x2 = 25 Watch-TVs
Profit = $387.5/month
Can you make 37.5??
Can you round to 38??
NO!! 4  25 = 227 (> 225!)

26. 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!

27. Formulation Example Graph
4x1+3x2 225 (electronics)
120
100
2x1+x2 100 (assembly) 80
Number of Watch-TVs (X2) 60 40 20 10 20 30 40 50 60 70 80
Number of Walkmans (X1)

28. Feasible Region 2x1+x2 100 (assembly) 120 100 80
4x1+3x2 225 (electronics)
120
100
2x1+x2 100 (assembly) 80
Number of Watch-TVs (X2) 60 40
Feasible Region 20 10 20 30 40 50 60 70 80
Number of Walkmans (X1)

29. Possible Solution Points
4x1+3x2 225 (electronics) 20 40 60 80
100
120 10 30 50 70
Number of Walkmans (X1)
Number of Watch-TVs (X2)
Feasible Region
2x1+x2 100 (assembly)
Possible Corner Point Solution 80

30. Possible Solutions Profit = 7 x1 + 5 x2 1. x1 = 0, x2 = 0 profit = 020 40 60 80
100
120 10 30 50 70 X1 X2
Feasible Region

31. 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.

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

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

34. Formulation #1
xi = Number of ounces of ingredient i used in snack food.
Minimize: 0.2x x x3
8x x x3  20 (protein)
3x x x3  40 (carbs.)
150x x x3  900 (calories)
x1, x2, x3 0

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

36. Formulation #1 - Additional Constraints
xi = 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 = 150x1
total calories = 150x x x3

37. Formulation #1 - Additional Constraints
xi = 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.

38. Formulation #1 - Additional Constraints
xi = 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.

39. 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 fertilizer includes 20% Nitrogen, 10% Phosphate and 40% Potash.
NuGrow makes three different fertilizers, packaged in 40 lb. bags: , and The fertilizer sells for $8/bag and at least 3000 bags must be produced next month. The fertilizer sells for $4/bag. The 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:

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

41. 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.

42. Formulation #2 - continued
Product
Minimum
req’d (bags)
N Ph Po In
3000
4000
Price
($/bag) 8 4 6
lbs. of ingredient per bag
xi = Number of bags of fertilizer type i to make next month.
i=1: i=2: i=3:

43. 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?

44. 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

45. Formulation #2 - Objective
xi = Number of bags of fertilizer type i to make next month. :
Maximize Profit = Revenue - Cost
Revenue = 8x1 + 4x2 + 6x3
Cost = (cost per bag of type 1) x1
+ (cost per bag of type 2) x2
+ (cost per bag of type 3) x3
Cost per bag is cost of all ingredients in a bag.

46. Formulation #2 - Costs
xi = Number of bags of fertilizer type i to make next month.
Cost for one bag of type 1 ( )
= cost for N  ($300/ton=$0.15/lb)
+ cost for Ph  ($200/ton=$0.10/lb)
+ cost for Po  ($400/ton=$0.20/lb)
+ cost for In  ($100/ton=$0.05/lb)
= $5.4
Similarly:
Cost for one bag of type 2 ( ) = $3.2
Cost for one bag of type 3 ( ) = $4.4

47. Formulation #2 - Objective
xi = Number of bags of fertilizer type i to make next month. :
Maximize Profit = Revenue - Cost
Revenue = 8x x x3
Cost = 5.4x x x3
Maximize 2.6x x x3

48. Formulation #2
xi = Number of bags of fertilizer type i to make next month. :
Maximize: 2.6x x x3
8x x x3  (N)
4x x x3  (Ph)
16x x x3  (Po)
x1, x2, x3 0
x  ( )
x3  ( )

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

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

51. Formulation #3 Paper Stock Strength Color Cost/Ton Availability
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
R $ tons/week
R $ tons/week
R $ tons/week

52. Formulation #3
Paper Stock Strength Color Cost/Ton Availability
R $ tons/week
R $ tons/week
R $ tons/week
Paper Strength Color Price/Ton Max. Demand
P1   $ tons/week
P2   $ tons/week
Produce 2 products (papers) from 3 ingredients (paper stocks)
to maximize profit (= revenue - cost).
Constraints: Availability(3); Demand(2); Strength(2); Color(2)

53. 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?

54. 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.

55. Formulation #3 - Key! x11 x12 x21 x22 x31 x32
Produce 2 products (papers: P1 and P2) from 3 ingredients (paper stocks: R1, R2 and R3).
xij = Number of tons of stock i in paper j; i=1,2,3 j=1,2
P P2 R1 R2 R3
x11 x12
x21 x22
x31 x32

56. Formulation #3
xij = Number of tons of stock i in paper j; i=1,2,3 j=1,2
P P2 R1 R2 R3
x11 x12
Tons of stock R1 used = x11 + x12
Tons of stock R2 used = x21 + x22
Tons of stock R3 used = x31 + x32
x21 x22
x31 x32

57. Formulation #3
xij = Number of tons of stock i in paper j; i=1,2,3 j=1,2 :
Amount of paper P1 produced = x11 + x21 + x31
Amount of paper P2 produced = x12 + x22 + x32
P P2 R1 R2 R3
x11 x12
x21 x22
x31 x32

58. Formulation #3 - Objective
Paper Stock Strength Color Cost/Ton Availability
R $ tons/week
R $ tons/week
R $ tons/week
Maximize Profit = Revenue - Cost
Cost = 150(tons of R1) + 110(tons of R2) + 50(tons of R3)
Cost = 150(x11 + x12) (x21 + x22) (x31 + x32)

59. Formulation #3 - Objective
Paper Strength Color Price/Ton Max. Demand
P1   $ tons/week
P2   $ tons/week
Revenue = 200(tons of P1 made) + 100(tons of P2made)
Revenue = 200(x11 + x21 + x31) (x12 + x22 + x32)

60. Formulation #3 - Objective
Maximize profit = Revenue - Cost
Revenue = 200(x11 + x21 + x31) + 100(x12 + x22 + x32)
Cost = 150(x11 + x12) + 110(x21 + x22) + 50(x31 + x32)
Maximize 50x x x x x x32

61. Formulation #3 - Constraints
Paper Stock Strength Color Cost/Ton Availability
R $ tons/week
R $ tons/week
R $ tons/week
Availability of each ingredient
x x  (R1)
x x  (R2)
x x32  (R3)

62. Formulation #3 - Constraints
Paper Strength Color Price/Ton Max. Demand
P1   $ tons/week
P2   $ tons/week
Demand for each product
x11 + x21 + x  (P1)
x12 + x22 + x32  (P2)

63. Formulation #3 - Constraints
Paper Stock Strength Color Cost/Ton Availability
R $ tons/week
R $ tons/week
R $ tons/week
Paper Strength Color Price/Ton Max. Demand
P1   $ tons/week
P2   $ 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

64. Formulation #3 - Constraints
Paper Stock Strength Color Cost/Ton Availability
R $ tons/week
R $ tons/week
R $ tons/week
Paper Strength Color Price/Ton Max. Demand
P1   $ tons/week
P2   $ tons/week
Strength P1: average strength from ingredients of P1 7

65. Formulation #3
xij = Number of tons of stock i in paper j; i=1,2,3 j=1,2
Maximize 50x x x x x x32
x x  40
x x  60
x x32 100
x x x  70
x x x32 120
(strength P1) x x x  0
(strength P2) x x32  0 (color P1) x x x  0
(color P2) x x x32  0
x11, x12, x13, x21, x22, x23 0