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Introduction to Operations Research

Linear Programming. Introduction to Operations Research. Linear Programming provides methods for allocating limited resources among competing activities in an optimal way. Any problem whose model fits the format for the linear programming model is a linear programming problem.

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Introduction to Operations Research

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  1. Linear Programming Introduction to Operations Research

  2. Linear Programming provides methods for allocating limited resources among competing activities in an optimal way. • Any problem whose model fits the format for the linear programming model is a linear programming problem. • Wyndor Glass Co. example • Two variables – Graphical method • Maximize profit Linear Programming Review

  3. Mary diagnosed with cancer of the bladder → needs radiation therapy • Radiation therapy • Involves using an external beam to pass radiation through the patient’s body • Damages both cancerous and healthy tissue • Goal of therapy design is to select the number, direction and intensity of beams to generate best possible dose distribution • Doctors have already selected the number (2) and direction of the beams to be used • Goal: Optimize intensity (measured in kilorads) referred to as the dose Radiation Therapy Example

  4. Radiation Therapy

  5. Graph the equations to determine relationships Minimize Z = 0.4x1 + 0.5x2 Subject to: 0.3x1 + 0.1x2 ≤ 2.7 0.5x1 + 0.5x2 = 6 0.6x1 + 0.4x2 ≥ 18 x1 ≥ 0, x2 ≥ 0 Radiation Therapy

  6. In order to ensure optimal health (and thus accurate test results), a lab technician needs to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protein. But the rabbits should be fed no more than five ounces of food a day. • Rather than order rabbit food that is custom-blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. • Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs $0.20 per ounce. • Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of $0.30 per ounce. • What is the optimal blend? Mixture Problem

  7. Mixture Problem maximum weight of the food is five ounces: X + Y ≤ 5 Minimize the cost: Z = 0.2X + 0.3Y

  8. Graph the equations to determine relationships Minimize Z = 0.2x + 0.3y Subject to: fat: 8x + 12y ≥ 24 carbs: 12x + 12y ≥ 36 protein: 2x + 1y ≥ 4 weight: x + y ≤ 5 x ≥ 0, y ≥ 0 Mixture Problem

  9. When you test the corners at: (0, 4), (0, 5), (3, 0), (5, 0), and (1, 2) you get a minimum cost of sixty cents per daily serving, using three ounces of Food X only. Only need to buy Food X Mixture Problem

  10. You have $12,000 to invest, and three different funds from which to choose. Municipal bond: 7% return CDs: 8% return High-risk acct: 12% return (expected) • To minimize risk, you decide not to invest any more than $2,000 in the high-risk account. • For tax reasons, you need to invest at least three times as much in the municipal bonds as in the bank CDs. • Assuming the year-end yields are as expected, what are the optimal investment amounts? Investment example

  11. Bonds (in thousands): x CDs (in thousands): y High Risk: z • Um... now what? I have three variables for a two-dimensional linear plot • Use the "how much is left" concept • Since $12,000 is invested, then the high risk account can be represented as • z = 12 – x – y Investment example

  12. Constraints: Amounts are non-negative: x ≥ 0 y ≥ 0 z ≥ 0  12 – x – y ≥ 0  y ≤ –x + 12 High risk has upper limit z ≤ 2  12 – x – y ≤ 2  y ≤ –x + 10 Taxes: 3y ≤ x  y ≤ 1/3 x Investment example Objective to maximize the return: Z = 0.07x + 0.08y + 0.12z  Z = 1.44 - 0.05x – 0.04y

  13. When you test the corner points at (9, 3), (12, 0), (10, 0), and (7.5, 2.5), you should get an optimal return of $965 when you invest $7,500 in municipal bonds, $2,500 in CDs, and the remaining $2,000 in the high-risk account. Investment example

  14. Manufacturing Example Machine data Product data

  15. Product Structure

  16. LP Formulation Objective Function xQ 45 xP + 60 max £ 10 xQ 20 xP + 1800 Structural s.t. £ constraints 12 xP + 28 xQ 1440 £ 15 xP + 6 xQ 2040 £ 10 xP + 15 xQ 2400 demand xP 100, xQ 40 xP ≥ 0, xQ ≥ 0 nonnegativity Are we done? Are the LP assumptions valid for this problem? * * = 81.82 = 16.36 x Optimal solution x Q P

  17. Feasible Region

  18. Optimal Solution

  19. Optimal objective value is $4664 but when we subtract the weekly operating expenses of $3000 we obtain a weekly profit of $1664. • Machines A & B are being used at maximum level and are bottlenecks. • There is slack production capacity in Machines C & D. Discussion of ResulTS

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