1. Introduction to Optimization Models EXCEL Modeling of Simple Linear Problems 1 Archis Ghate Assistant Professor Industrial and Systems Engineering archis@uw.edu http://web.mac.com/archis.ghate

2. Nuts and bolts of optimization models Decision variables Parameters (data) Constraints Performance objective Linear problems = constraints and performance objective are linear functions of decision variables 2

3. Problem 1: 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. 3

4. Diet problem continued What decision variables do we need? ‣ Units of wheat consumed every day X ‣ Units of rye consumed every day Y What type of data do we need? What constraints do we need? What is our objective function? 4

5. Diet problem data WheatRyeDR Carbs/unit5720 Proteins/unit4215 Vitamins/unit213 Cost/unit0.60.35 5

6. minimize 0.6X +0.35Y subject to 5X + 7Y ≥ 20 4X + 2Y ≥ 15 2X + Y ≥ 3 X≥ 0 Y≥ 0 Optimal solution : X = 3.611 Y = 0.278 Cost : 2.2639 6

7. Problem 2 : Cancer treatment with radiation therapy One possible way to treat cancer is radiation therapy ‣ An external beam treatment machine is used to pass radiation through the patient’s body ‣ Damages both cancerous and healthy tissues ‣ Typically multiple beams of different dose strengths are used from different sides and different angles Decide what beam dose strengths to use to achieve sufficient tumor damage but limit damage to healthy tissues 7

9. Radiation therapy data AreaBeam 1Beam 2 Restriction on total average dose Healthy anatomy0.40.5Minimize Critical tissue0.30.1at most 2.7 Tumor region0.5 equal to 6 Center of tumor0.60.4at least 6 Decision variables x 1 and x 2 represent the dose strength for beam 1 and beam 2 respectively Fraction of dose absorbed by area (average) 9

10. minimize 0.4x 1 + 0.5x 2 subject to 0.3x 1 + 0.1x 2 ≤ 2.7 0.5x 1 + 0.5x 2 = 6 0.6x 1 + 0.4x 2 ≥ 6 x 1 ≥ 0 x 2 ≥ 0 dose to healthy anatomy dose to critical tissue dose to tumor dose to tumor center Optimal solution : x 1 =7.5 x 2 = 4.5 Dose to healthy anatomy : 5.25 10

11. Problem 3 : Transportation problem 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 X ij the number of boxes shipped from warehouse i to healthcare center j. 11

12. HC1HC2HC3HC4AVAILABILITY W146451365486775 W2352416690791125 W3995682388685100 REQUIREMENT80657085300 Transportation problem data 12

13. minimize 464X 11 +513X 12 +654X 13 +867X 14 + 352X 21 +416X 22 +690X 23 +791X 24 + 995X 31 +682X 32 +388X 33 +685X 34 subject to X 11 +X 12 +X 13 +X 14 = 75 X 21 +X 22 +X 23 +X 24 = 125 X 31 +X 32 +X 33 +X 34 = 100 X 11 +X 21 +X 31 = 80 X 12 +X 22 +X 32 = 65 X 13 +X 23 +X 33 = 70 X 14 +X 24 +X 34 = 85 X ij ≥ 0, for i=1,2,3 and j=1,2,3,4 boxes available must be shipped warehouse material balance health center material balance demand for boxes must be met total shipping cost 13

14. W1 W2 W3 HC1 HC2 HC3 HC4 20 55 80 45 70 30 75 125 100 80 65 70 85 Solution Cost : 152535 14

15. Problem 4 : Air pollution problem 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 reduction. 15

16. Pollutant Required emission reduction (million pounds per year) Particulates60 Sulfur oxides150 Hydrocarbon s 125 Emission reduction (million pounds per year) if the method is employed at the highest possible level Taller Smokestacks FiltersBetter Fuels PollutantF1F2F1F2F1F2 Particulates12925201713 Sulfur oxides354218315649 Hydrocarbons375328242920 Annual cost of employing a method at the highest possible level (million dollars) MethodF1F2 Taller smokestacks 810 Filters 76 Better fuels 119 16

17. Decision variables - fraction of the highest possible level of a method employed MethodF1F2 Taller smokestacks x1x1 x2x2 Filters x3x3 x4x4 Better fuels x5x5 x6x6 17

18. minimize 8x 1 + 10x 2 + 7x 3 + 6x 4 + 11x 5 + 9x 6 subject to 12x 1 + 9x 2 + 25x 3 + 20x 4 + 17x 5 + 13x 6 ≥ 60 35x 1 + 42x 2 + 18x 3 + 31x 4 + 56x 5 + 49x 6 ≥ 150 37x 1 + 53x 2 + 28x 3 + 24x 4 + 29x 5 + 20x 6 ≥ 125 x j ≤ 1, for j=1,2,3,4,5,6 x j ≥ 0, for j=1,2,3,4,5,6 total cost emission reduction requirements cost Optimal solution : (x 1, x 2, x 3, x 4, x 5, x 6 ) = (1,0.623,0.343,1,0.048,1) Cost : 32.16 million dollars 18

19. Reference “Introduction to Operations Research” by Hillier and Lieberman, 9 th edition, McGraw-Hill