1. IndE 310 Linear Programming
UW Industrial Engineering
Instructor: Prof. Zelda Zabinsky

2. Operations Research “The Science of Better”:
The discipline of applying advanced analytical methods to help make better decisions
Operations – problems that concern how to conduct and coordinate the operations within an organization
Research – use of the scientific method to address these problems
Interdisciplinary – brings together:
mathematics
statistics
industrial engineering
management (“management science”)
systems engineering...
Highly application oriented

4. Applications of Operations Research
Business
Manufacturing
Project management
Scheduling
Facility layout and location
Many more
Service
Transportation and logistics
Marketing
Queueing
Economics & Finance
Auctions
Portfolio selection
Capital investment
Many more
Health
Scheduling
Queueing
Biotechnology
And more
Military

5. Examples of O.R. Applications
China
Select and schedule projects for future energy needs
New Haven Health Department
Design an effective needle exchange program to combat HIV
Continental Airlines
Optimize the reassignment of crews to flights
IBM
Reengineer supply chain
Proctor and Gamble
Redesign production and distribution system
Many more

6. (and make recommendations)
Operations Research
Structuring the real-life situation into a mathematical model, abstracting the essential elements so that a solution relevant to the decision maker's objectives can be sought
This involves looking at the problem in the context of the entire system
(Understand and)
Model
(and verify)
Exploring the structure of such solutions and developing systematic procedures for obtaining them
Developing a solution, including the mathematical theory, if necessary, that yields an optimal value of the system measure of desirability (or possibly comparing alternative courses of action by evaluating their measure of desirability)
(Strategize and)
Solve
(and make recommendations)

7. Operations Research Modeling Toolset
Queueing Theory
Markov Chains
PERT/ CPM
Network Programming
Dynamic Programming
Simulation
Markov Decision Processes
Inventory Theory
Linear Programming
Stochastic Programming
Forecasting
Integer Programming
Decision Analysis
Nonlinear Programming
Game Theory

8. Operations Research Modeling Toolset
311
Queueing Theory
310
Markov Chains
PERT/ CPM
Network Programming
Dynamic Programming
Simulation
Markov Decision Processes
Inventory Theory
Linear Programming
312
Stochastic Programming
Forecasting
Integer Programming
Decision Analysis
Nonlinear Programming
Game Theory
312
312

9. IndE 310 Linear Programming Transportation and Assignment Problems
Modeling
Solving
Simplex method
Foundations of the simplex method
Duality and sensitivity
Transportation and Assignment Problems
Network Problems
Shortest path
Minimum spanning tree
Minimum cost, maximum flow
PERT/CPM

10. Interesting Links
“Operation Everything”, The Boston Globe, June 27, operationeverything html
Links available on the course web

11. O.R. Modeling Approach

12. O.R. Modeling Approach Define the problem and gather data
Formulate a (mathematical) model to represent the problem
Develop a computer-based procedure for deriving solutions
Test model and refine it
Prepare for the ongoing application of the model
Implement

13. Defining the problem and gathering data
First order of business:
Study the relevant system
Develop a well-defined statement of the problem
Right answer for the wrong problem?
Defining objectives
What data are needed?
Collect them (easily said)

14. Formulating a (mathematical) model
We need to create a model that can be used for
Abstracting the essence of the subject
Showing interrelationships
Facilitating analysis
A model: E=mc2
A mathematical model usually consists of:
Decision variables
Parameters
Objective function
Constraints

15. Deriving solutions A common theme in O.R.:
Search for an optimal solution
Develop a procedure for deriving solutions to the mathematical model
e.g. what is the best speed to obtain the highest possible energy?
What kind of a procedure?
Usually based on theoretical foundations
Exact vs. heuristic (slow vs. fast?)
Post-optimality and sensitivity analysis

16. Testing, preparing and implementing
Need to validate the model
“Debugging”
Plausible results?
Prepare to apply as prescribed by management
Operating procedures, supporting systems, managerial reports…
Implementation
Coordination, detailed indoctrination, new courses of action

17. Peeking into Chapter 3 Wyndor Glass Co. Example
Wyndor Glass produces glass windows and doors
They have 3 plants:
Plant 1: makes aluminum frames and hardware
Plant 2: makes wood frames
Plant 3: produces glass and makes assembly
Two products proposed:
Product 1: 8’ glass door with aluminum siding
Product 2: 4’ x 6’ wood framed glass window
Some production capacity in the three plants is available to produce a combination of the two products
Problem is to determine the best product mix to maximize profits

18. Peeking into Chapter 3 Wyndor Glass Co. Data
The O.R. team has gathered the following data:
Number of hours of production time available per week in each plant
Number of hours of production time needed in each plant for each batch of new products
Estimated profit per batch of each new product
Production time per batch (hr)
Production time available per week (hr)
Product
Plant 1 2 4 12 3 18
Profit per batch
$3,000
$5,000

19. Peeking into Chapter 3 Formulate LP Model
Identify the parameters
(activities/values you cannot control)
Identify the decision variables
(activities/values that you can control and need to make a decision on)
Identify the objective function
(function of the decision variables for minimization/maximization)
Identify the constraints
(limitations you cannot control)

20. Peeking into Chapter 3 Graphical Solution Setup
The model:

21. Prototype Example Graphical Solution
The model:

22. LP Modeling
Wyndor Glass Co. is only one of a vast number of applications that LP can be used to address
In general, the most common type of an LP addresses:
The allocation of limited resources to competing activities for maximizing the value of these activities

23. LP Terminology
The allocation of limited resources to competing activities for maximizing the value of these activities
Activities, n
Resources, m
Decision variables – or level of activities, x
Objective function – or value of activities, Z
Constraints
functional
non-negativity

24. LP Terminology Feasible region, feasible solution Infeasible solution
Optimal solution
Extreme-point – or corner-point feasible solutions
Parameters

25. Standard (Canonical) Form of an LP Model
Maximize Z = c1x1 + c2x2 + … + cnxn
subject to a11x1 + a12x2 + … + a1nxn ≤ b1
a21x1 + a22x2 + … + a2nxn ≤ b2 …
am1x1 + am2x2 + … + amnxn ≤ bm
x1 ≥ 0, x2 ≥ 0, …, xn ≥ 0

26. Matrix Form of an LP Model

27. Other Forms that can be Converted into Standard Form
Objective function: Minimize Z=cx (instead of Maximize Z=cx)
Functional constraints: Ax ≥ b or Ax = b (instead of Ax ≤ b)
Non-negativity constraints: x unrestricted in sign (instead of x ≥ 0)

28. Assumptions of Linear Programming
Proportionality
Additivity
Divisibility
Certainty

29. LP Modeling Examples

30. A Transportation Example
A company has 2 plants and 3 warehouses
Supply at plants
100 units in Plant 1, 200 units in Plant 2
Sales potential at warehouses
150 units, 200 units, and 350 units at Warehouses 1, 2 and 3, respectively
Revenue
12 $/unit, 14 $/unit and 15 $/unit at Warehouses 1, 2 and 3, respectively
Cost of manufacturing one unit at plant i and shipping to w/h j:
How many units to ship from each plant to each w/h to maximize profits?
Warehouse 1 2 3
Plant 8 10 12 7 9 11

32. Personnel Scheduling (p.56)
Union Airways is adding more flights and needs to hire additional customer service agents
Each agent works an eight-hour shift
The five possible shifts are
Shift 1: 6:00 am – 2:00 pm
Shift 2: 8:00 am – 4:00 pm
Shift 3: Noon – 8:00 pm
Shift 4: 4:00 pm – Midnight
Shift 5: 10:00 pm – 6:00 am

33. Minimum number of agents needed
Personnel Scheduling
Minimum number of agents needed per two-hour time periods:
Time period
Time periods covered
Minimum number of agents needed
Shift 1 2 3 4 5
6:00 am – 8:00 am  48
8:00 am – 10:00 am 79
10:00 am – noon 65
Noon – 2:00 pm 87
2:00 pm – 4:00 pm 64
4:00 pm – 6:00 pm 73
6:00 pm – 8:00 pm 82
8:00 pm – 10:00 pm 43
10:00 pm – midnight 52
Midnight – 6:00 am 15
Daily cost per agent
$170
$160
$175
$180
$195

35. Reclaiming Solid Wastes (p.52)
A recycling center takes four types of material
Material 1: Newsprint
Material 2: White paper
Material 3: Mixed paper
Material 4: Cardboard
Three products are reclaimed
Grades A, B, C
The SAVE-IT company wants to determine the amount of each grade to produce and the mix of materials in each grade to maximize profit

36. Reclaiming Solid Wastes
Grade
Specification
Amalgamation Cost per lb ($)
Selling Price per lb ($) A
Mat’l 1: Not more than 30% of total Mat’l 2: Not less than 40% of total Mat’l 3: Not more than 50% of total Mat’l 4: Exactly 20% of total
3.00
8.50 B
Mat’l 1: Not more than 50% of total Mat’l 2: Not less than 10% of total Mat’l 4: Exactly 10% of total
2.50
7.00 C
Mat’l 1: Not more than 70% of total
2.00
5.50
Products
Solid waste materials
Mat’l
lbs per week available
Treatment cost per lb ($) 1
3,000
3.00 2
2,000
6.00 3
4,000
4.00 4
1,000
5.00
Use at least half of each material collected
Cannot use more than $30,000 per week for treatment of mat’ls