Introduction to Optimization Models

OR Mini-courseJuly 31, 2009

Archis GhateAssistant Professor

Industrial and Systems EngineeringThe University of Washington, Seattle

What are “Optimization Models”?• One possible definition - mathematical models

designed to help institutions and individuals decide how to

‣ allocate scarce resources

‣ to activities

‣ to make the most of their circumstances.

• More generally, mathematical models designed to help us make “better” decisions.

• Several other possible definitions.

What kind of decisions?• Any kind really... but here are some examples

‣ How many cars to produce? What radiation intensity to use in cancer treatment? How many outpatients to schedule every day? Whether to start a new project? Whether to buy a stock? Which retirement plan to choose? How many medical service centers to open? Where, what and how much supplies to stock in storage? How many nurses to employ? Which doctors to have on call? How to schedule flights? How to route traffic? How to respond to a natural disaster?

• These are difficult decisions that involve an inordinate number of factors.

‣ Optimization models attempt to capture some key components to build a reasonable replica of the real situation.

‣ Provide a systematic, quantitative way to evaluate and select decisions.

FAQs about this talk• Who is this talk for?

‣ People who have never seen an optimization model and have minimal or no mathematics background but

‣ May benefit from using optimization models in their profession.

• What if you are none of the above?

‣ Take a nap.

• How is this talk going to proceed?

‣ We are going to build a few simple toy models and solve some of them with EXCEL.

Nuts and bolts of optimization models

decision variables

activity levels or choices

constraintsresource

limitations or other restrictions

objective function(s)

performance metric(s)

Example 1: a diet problem

• To decide the quantities of different food items to consume every day to meet the daily requirement (DR) of several nutrients at minimum cost.

• What type of data do we need?

‣ food items under consideration.

‣ nutrition information.

‣ cost of food items.

Diet problem : toy example

Wheat Rye DR

Carbs/unit 5 7 20

Proteins/unit 4 2 15

Vitamins/unit 2 1 3

Cost/unit 0.6 0.35

Diet example continued

• What decision variables do we need?

‣ Units of wheat consumed every day XW

‣ Units of rye consumed every day XR

• What constraints do we need?

• What is our objective function?

minimize 0.6XW + 0.35XR

subject to

5XW + 7XR ≥ 204XW + 2XR ≥ 152XW + XR ≥ 3

XW ≥ 0 XR ≥ 0

Solution : XW =3.611 XR = 0.278Cost : 2.2639

Example 2 : Transportation problem

• Adapted from Hillier and Lieberman (2005).

• A non-profit organization manages three warehouses and four healthcare centers. The organization has estimated the requirements for a specific vaccine at each healthcare center in number of boxes of vials. The organization knows the number of boxes of vials available at each warehouse. They want to decide how many boxes of vials to ship from the warehouses to the healthcare centers so as to meet the demand for the vaccine at minimum total shipping cost.

• Decision variables Xij the number of boxes shipped from warehouse i to healthcare center j.

HC1 HC2 HC3 HC4 AVAILABILITY

W1 464 513 654 867 75

W2 352 416 690 791 125

W3 995 682 388 685 100

REQUIREMENT 80 65 70 85 300

W1

W2

W3

HC1

HC2

HC3

HC4

20

55

80

45

7030

75

125

100

80

65

70

85

Solution

Cost : 152535

Example 3 : Cancer treatment with radiation therapy

• Taken from Hillier and Lieberman (2005).

• One possible way to treat cancer is radiation therapy.

‣ An external beam treatment machine called a linear accelerator is used to pass radiation through the patient’s body.

‣ Damages both cancerous and healthy cells.

‣ Typically multiple beams of different strengths are used from different sides and different angles.

• Decide what beam strengths to use to achieve sufficient tumor damage but limit damage to healthy regions.

Radiation therapy dataFraction of energy absorbed by

area

Area Beam 1 Beam 2 Restriction on total average dose

Healthy anatomy 0.4 0.5 Minimize

Critical tissue 0.3 0.1 at most 2.7

Tumor region 0.5 0.5 equal to 6

Center of tumor 0.6 0.4 at least 6

Decision variables x1 and x2 represent the strengths for beam 1 and beam 2 respectively

minimize 0.4x1 + 0.5x2

subject to

0.3x1 + 0.1x2 ≤ 2.7

0.5x1 + 0.5x2 = 6

0.6x1 + 0.4x2 ≥ 6x1 ≥ 0 x2 ≥ 0

dose to healthy anatomy

dose to critical tissue

dose to tumor

dose to tumor center

Optimal solution : x1 =7.5 x2 = 4.5Dose to healthy anatomy : 5.25

Example 4: Air pollution problem

• Adapted from Hillier and Lieberman (2005)

• A steel plant has been ordered to reduce its emission of 3 air pollutants - particulates, sulfur oxides, and hydrocarbons

• The plant uses 2 furnaces

• The plant is considering 3 methods for achieving pollution reductions - taller smokestacks, filters, better fuels

• The 3 methods are expensive, so the plant managers want to decide what combination of the 3 to employ to minimize costs and yet achieve the required emission reductions

PollutantRequired emission reduction (million pounds per year)

Particulates 60

Sulfur oxides 150

Hydrocarbons 125

Emission reduction (million pounds per year) if the method is employed at the highest possible level

Taller Smokestacks

Filters Better Fuels

Pollutant F1 F2 F1 F2 F1 F2

Particulates 12 9 25 20 17 13

Sulfur oxides 35 42 18 31 56 49

Hydrocarbons 37 53 28 24 29 20

Annual cost of employing a method at the highest possible level (million dollars)

Method F1 F2

Taller smokestacks 8 10

Filters 7 6

Better fuels 11 9

Decision variables - fraction of the highest possible level of a method employed

Method F1 F2

Taller smokestacks x1 x2

Filters x3 x4

Better fuels x5 x6

minimize 8x1 + 10x2 + 7x3 + 6x4 + 11x5 + 9x6

subject to

12x1 + 9x2 + 25x3 + 20x4 + 17x5 + 13x6 ≥ 6035x1 + 42x2 + 18x3 + 31x4 + 56x5 + 49x6 ≥ 15037x1 + 53x2 + 28x3 + 24x4 + 29x5 + 20x6 ≥ 125

xj ≤ 1, for j=1,2,3,4,5,6 xj ≥ 0, for j=1,2,3,4,5,6

total cost

emission reduction

requirements

Optimal solution : (x1 , x2 , x3 , x4 , x5 , x6)=(1,0.623,0.343,1,0.048,1)Cost : 32.16 million dollars

Example 5: Production problem

• A company makes three types of laptops: type 1, type 2, and type 3.

• Three types of machines are used to make these laptops. The machines have limited machine time available for production.

Machine Available TimeMachine 1 500Machine 2 350Machine 3 150

• Making one laptop requires the following machine times:

Machine Type 1 Type 2 Type 3Machine 1 9 3 5Machine 2 5 4 0Machine 3 3 0 2

• The production costs of making one laptop are as follows:

Laptop Prod. costType 1 250Type 2 100Type 3 150

• The company does not want to make more than 20 type 3 laptops.

• The price that the company needs to set to sell x laptops of type 1 is 350+100x-1/3.

• The price that the company needs to set to sell y laptops of type 2 is 150+400y-1/4.

• The price that the company needs to set to sell z laptops of type 3 is 200+50z-1/2.

• Given this information, how many laptops of each type should the company make if they want to maximize their profit? (ignore the fact that “number of laptops” has to be an integer).

• Class exercise: develop a mathematical model.

maximize (350+100x-1/3-250)x + (150+400y-1/4-100)y+ (200+50z-1/2-150)z

subject to9x + 3y + 5z ≤ 5005x + 4y + ≤ 3503x + + 2z ≤ 150

z ≤ 20 x ≥ 0 y ≥ 0 z ≥ 0

Answer (approximate) : Type 1= 26, Type 2 = 54, Type 3= 20Profit (approximate) : 6390

Example 6: Chess queens problem

• How many queens can you place on an 8*8 chessboard so that no queen threatens another?

• Where would you place these queens?

• Class exercise: develop an optimization model.

taken from wikipedia

A five step approach to optimization models

• Define/describe the problem and gather data

• Formulate a mathematical model to represent the real problem

• Develop a computer based procedure for deriving solutions to the model

• Test/refine the model, perform sensitivity analyses

• Implement

References

• Hillier, F., and Lieberman, G., Introduction to operations research, McGraw Hill, 8th edition, 2005.