1. INTRODUCTION TO LINEAR PROGRAMMING
CONTENTS
Introduction to Linear Programming
Applications of Linear Programming
Reference: Chapter 1 in BJS book.

2. Learning objectives
Defining linear programs
Understanding the main steps in modeling problems and assumptions of LPs
Understanding modeling a problem through typical example problems
Understanding the use of some modeling approaches
Graphical approach to solve LPs
Examining different types of LP wrt feasibility and boundedness

3. Linear Programming Problem
Features of Linear programming problem:
Decision Variables
We maximize (or minimize) a linear function of decision variables, called objective function.
The decision variables must satisfy a set of constraints.
Decision variables have sign restrictions.
Example:
Maximize z = 3x1 + 2x2
subject to
2x1 + x2  100
x x2  80
x  40
x1, x2  0

4. A Typical Linear Programming Problem
Linear Programming Formulation:
Minimize c1x1 + c2x2 + c3x3 + …. + cnxn
subject to
a11x1 + a12x2 + a13x3 + …. + a1nxn  b1
a21x1 + a22x2 + a23x3 + …. + a2nxn  b2 :
am1x1 + am2x2 + am3x3 + …. + amnxn  bm
x1, x2, x3 , …., xn  0
or,
Minimize j=1, n cjxj
j=1, n aijxj - xn+i = bi for all i = 1, …, m
xj  0 for all j =1, …, n+m
Linear Objective
Linear Constraints;
- Technology
Uncontrolable
variables/factors
Decision (controlable) variables
- Can take any real value
- Restricted or unrestrited in sign
Standard form representation

5. Matrix Notation
Minimize cx
subject to
Ax = b
x  0
where

6. Linear Programming Modeling and Examples
Stages of an application:
Problem formulation
Finding facts, parameters, relations, controllable factors (decision varb.), uncontrollable factors, objective(s).
Mathematical model
Defining decision variables, writing constrains and objective mathematically, coding the model in a modeling software.
Deriving a solution
Direct solution, decompositon algorithms, search heuristics.

7. Model testing and analysis
Debugging the model, checking the results with problem owner, sensitivity analysis on parameters of the model.
Implementation
Keep monitoring the results, change the formulation and model if needed over time.

8. An Example of a LP
Giapetto’s woodcarving manufactures two types of wooden toys: soldiers and trains. How many of each type to produce?
Constraints:
100 finishing hour per week available
80 carpentry hours per week available
produce no more than 40 soldiers per week
Objective: maximize profit? Minimize cost?

9. An Example of a LP (cont.)
Linear Programming formulation:
Maximize z = 3x1 + 2x2 (Obj. Func.)
subject to
2x1 + x2  100 (Finishing constraint)
x x2  80 (Carpentry constraint)
x  (Bound on soldiers)
x1  (Sign restriction)
x2  (Sign restriction)

10. Assumptions of Linear Programming
Proportionality Assumption
Contribution of a variable is proportional to its value.
Additivity Assumptions
Contributions of variables both in the objective and in the constraints are independent of the values of the other variables. No interactions between the variables.
Divisibility Assumption
Decision variables can take fractional values.
Certainty Assumption
Each parameter is known with certainty.

11. Transportation Problem
The Brazilian coffee company processes coffee beans into coffee at m plants. The production capacity at plant i is ai.
The coffee is shipped every week to n warehouses in major cities for retail, distribution, and exporting. The demand at warehouse j is bj.
The unit shipping cost from plant i to warehouse j is cij.
It is desired to find the production-shipping pattern xij from plant i to warehouse j, i = 1, .. , m, j = 1, …, n, that minimizes the overall shipping cost.
What is Xij? Produced? Shipped?
Why not consider production and shipping together?
Why not maximize profit?

12. Static Workforce Scheduling
Number of full time employees required on different days of the week are given below.
Each employee must work five consecutive days and then receive two days off.
The schedule must meet the requirements by minimizing the total number of full time employees.
Decision variable?

13. Multi-Period Financial Models
Determine investment strategy for the next three years
Money available for investment at time 0 = $100,000
Investment opportunities available : A, B, C, D & E
No more than $75,000 in any invest
Uninvested cash earns 8% interest
Cash flow of these investments:
Decision variables?

14. Cutting Stock Problem
A manufacturer of metal sheets produces rolls of standard fixed width w and of standard length l.
A large order is placed by a customer who needs sheets of width w and varying lengths. He needs bi sheets of length li, i = 1, …, m.
The manufacturer would like to cut standard rolls in such a way as to satisfy the order and to minimize the waste.
Since scrap pieces are useless to the manufacturer, the objective is to minimize the number of rolls needed to satisfy the order.
Decision variable?
Objective ?

15. Multi-Period Workforce Scheduling
Requirement of skilled repair time (in hours) is given below.
At the beginning of the period, 50 skilled technicians are available.
Each technician is paid $2,000 and works up to 160 hrs per month.
Each month 5% of the technicians leave.
A new technician needs one month of training, is paid $1,000 per month, and requires 50 hours of supervision of a trained technician.
Decision variable(s)? Objective?

16. Solution: Transportation Problem
Decision Variables:
xijp: amount of product p shipped from plant i to warehouse j
Formulation:
Minimize z =
subject to
<= ai, i = 1, … , m
 bj, j = 1, … , n
xij  0, i = 1, … , m, j = 1, … , n

17. Solution: Static Workforce Scheduling
LP Formulation:
Min. z = x1+ x2 + x3 + x4 + x5 + x6 + x7
subject to
x x4 + x5 + x6 + x7 ³ 17
x1+ x x5 + x6 + x7 ³ 13
x1+ x2 + x x6 + x7 ³ 15
x1+ x2 + x3 + x x7 ³ 19
x1+ x2 + x3 + x4 + x ³ 14
x2 + x3 + x4 + x5 + x ³ 16
x3 + x4 + x5 + x6 + x ³ 11
x1, x2, x3, x4, x5, x6, x7 ³ 0

18. Solution: Static Workforce Scheduling

19. Solution: Multi period Financial Model
Decision Variables:
A, B, C, D, E : Dollars invested in the investments A, B, C, D, and E
St: Dollars invested in money market fund at time t (t = 0, 1, 2)
Formulation:
Maximize z = B + 1.9D + 1.5E S2
subject to
A + C + D + S = 100,000
0.5A C S = B + S1
A B S = E + S2
A £ 75,000
B £ 75,000
C £ 75,000
D £ 75,000
E £ 75,000
A, B, C, D, E, S0, S1, S2 ³ 0

20. Solution: Multi period Financial Model

21. Solution: Multiperiod Workforce Scheduling
Decision Variables:
xt: number of technicians trained in period t
yt: number of experienced technicians in period t
Formulation:
Minimize z = 1000(x1 + x2 + x3 + x4 + x5) (y1 + y2 + y3 + y4 + y5)
subject to
160y x1 ³ y1 = 50
160y x2 ³ y1 + x1 = y2
160y x3 ³ y2 + x2 = y3
160y x4 ³ y3 + x3 = y4
160y x5 ³ y4 + x4 = y5
xt, yt ³ 0, t = 1, 2, 3, … , 5

22. Cutting Stock Problem
We want certain number of specific length pieces that must be cut out of standard length pieces.
Given a standard sheet of length l, there are many ways of cutting it. Each such way is called a cutting pattern.
The jth cutting pattern is characterized by the column vector aj, where the ith component, namely, aij, is a nonnegative integer denoting the number of sheets of length li in the jth pattern.
Note that the vector aj represents a cutting pattern if and only if i=1,n aijli  l for patter j where each aij is a nonnegative number.
Example; let l=7 feet and we want to have pieces with lengths 2, 3, and 5.
Possible patterns (2x1, 5x1), (2x3), (3x2), (2x2,3x1)
Objective? Decision varb.?

23. Solution:Cutting Stock Problem (contd.)
Formulation: xi: Number of sheets cut using patter j.
There are n possible patterns, m different lengths to produce.
Minimize j=1,n xi
subject to
j=1,n aij xj  bi for each i = 1, …, m
xj  j = 1, …, n
xj integer j = 1, …, n

24. Solution:Cutting Stock Problem (contd.)
Formulation: Let l = 7, and type 1, type 2 and type 3 sizes demanded are 2, 3, and 5 with b1=10, b2=4, b3=3.
possible patterns: j i
Min. z = x1+ x2 + x3 + x4
subject to
3x1+2 x x4  10
x2 + 2x  4
x4  3
xj integer j = 1, …, 4

25. For maximimization problems multiply objective
Modeling issues
Minimize/Maximize absolute value
=>
For maximimization problems multiply objective
with -1 and turn them into minimization problem

26. Modeling issues

27. Modeling issues
X,Y >=0

28. Modeling issues

29. Modeling issues

30. Modeling issues
Fixed charge problem: Cost function with a fixed part

31. Modeling issues

32. Modeling issues
Decision variables

33. Modeling issues

34. Either-or constraints:
Modeling issues
Either-or constraints:

35. Modeling issues
Decision variables

36. Modeling issues
Constraints
What M1 should we use?

37. Modeling issues

38. Modeling issues
If-then constraints

39. Modelling issues

40. Modelling issues

41. Modelling issues
1 out of k constraints must hold
n out of k constraints must hold

42. Modelling issues
Reconsider the example on Gandhi clothing company with the following additional information;
Current clothing supplier (supplier 1) can supply 160 sq-yard per week. Lets say that there are two additional suppliers;
Supplier 2 can supply clothing minimum 100 and maximum 200 sq-yard per week and supplier 3 can supply between 150 and 300 sq-yard per week.
That is supplier 2 and 3 require a minimum purchase in addition to a maximum. Company must work with only one supplier due to logistical reasons. How can we model the same problem with three supplier option?

43. Modelling issues

44. Modelling issues

45. Modelling issues

46. Modelling issues

47. Modelling issues

48. Modelling issues

49. Piecewise linear functions
Modeling issues
Piecewise linear functions
f(x) b3 b1 b4 b2 x
x= z(i)*bi+z(i+1)*b(i+1) where z(i)+z(i+1) = 1 i=0..3
f(x) = z(i)*f(bi)+z(i+1)*f(b(i+1)) where z(i)+z(i+1) = 1 i=0..3
Let y(i) = 1 if b(i) <= x <= b(i+1)
y(i) = 0 otherwise
4 binary variables and
4+1 convex-combination variables (z).

50. Modeling issues
Piecewise linear functions

51. Modelling issues

52. Modelling issues

53. Modelling issues

54. Modelling issues

55. Modelling issues

56. Modelling issues

57. Steps of the geometric solution
Determine and draw the feasible region
Draw the c vector
Draw the iso cost lines (perpendicular to c vector)
Find the last point(s) that iso-cost lines touch the feasible region, when we move towards direction of –c (in minimization). This point (or points) is the optimal solution.

58. Geometric Solution 4. Find the optimal point(s)
3. Draw the iso-cost lines
Draw the feasible
region
2. Draw vector c
minimization; move towards –c
maximization move towards c
as long as (x1,x2) is in feasible region

59. Geometric Solution

60. Geometric Solution

61. Geometric Solution

62. Geometric Solution

63. Unique finite optimal solution

64. Alternative finite optimal solution

65. Unbounded optimal solution
It is always the case that a corner point is among the optimal solution, if there
is a finite optimal solution.

66. Empty feasible region