L4_10 - 1/27 ESI 6314 Deterministic Methods in Operations...

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Unformatted text preview: 1/27 ESI 6314 Deterministic Methods in Operations Research Lecture Notes 4 University of Florida Department of Industrial and Systems Engineering / REEF 2/27 A diet problem Four foods, available for consumption, their cost, nutrition content and daily ingestion requirements are given: Type of food cost calories chocolate sugar fat (cents) (ounces) (ounces) (ounces) Brownie 50 400 3 2 2 Chocolate ice cream (1 scoop) 20 200 2 2 4 Cola (1 bottle) 30 150 4 1 Pineapple cheesecake (1 piece) 80 500 4 5 Required ingestion 500 6 10 8 We want to design a diet of minimum total cost. 3/27 A diet problem The decision variables: x 1 = number of brownies eaten daily x 2 = number of scoops of chocolate ice cream eaten daily x 3 = number of bottles of cola drunk daily x 4 = number of pieces of pineapple cheesecake eaten daily The objective (minimize total cost of diet): min z = 50 x 1 + 20 x 2 + 30 x 3 + 80 x 4 Constraints: 1 Daily calorie intake must be at least 500 calories. 2 Daily chocolate intake must be at least 6 oz. 3 Daily sugar intake must be at least 10 oz. 4 Daily fat intake must be at least 8 oz. 4/27 A diet problem 1 Daily calorie intake must be at least 500 calories: 400 x 1 + 200 x 2 + 150 x 3 + 500 x 4 ≥ 500 2 Daily chocolate intake must be at least 6 oz. 3 x 1 + 2 x 2 ≥ 6 3 Daily sugar intake must be at least 10 oz. 2 x 1 + 2 x 2 + 4 x 3 + 4 x 4 ≥ 10 4 Daily fat intake must be at least 8 oz. 2 x 1 + 4 x 2 + x 3 + 5 x 4 ≥ 8 5/27 A diet problem min z = 50 x 1 + 20 x 2 + 30 x 3 + 80 x 4 s.t. 400 x 1 + 200 x 2 + 150 x 3 + 500 x 4 ≥ 500 (calorie) 3 x 1 + 2 x 2 ≥ 6 (chocolate) 2 x 1 + 2 x 2 + 4 x 3 + 4 x 4 ≥ 10 (sugar) 2 x 1 + 4 x 2 + x 3 + 5 x 4 ≥ 8 (fat) x i ≥ 0 (i=1,2,3,4) (sign) The optimal solution: x 1 = x 4 = 0 , x 2 = 3 , x 3 = 1 , z = 90 . 6/27 Workforce scheduling Number of full-time employees required on different days of the week is given by the following table: 1. Monday 17 2. Tuesday 13 3. Wednesday 15 4. Thursday 19 5. Friday 14 6. Saturday 16 7. Sunday 11 Each employee must work five consecutive days and then receive two days off [union rule]. Design a schedule that meets the requirements by minimizing the total number of full-time employees. 7/27 Workforce scheduling Proposed formulation: x i- number of employees working on day i , i = 1 , . . . , 7. The corresponding LP: min z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 s . t . x 1 ≥ 17 x 2 ≥ 13 x 3 ≥ 15 x 4 ≥ 19 x 5 ≥ 14 x 6 ≥ 16 x 7 ≥ 11 x i ≥ 0 ( i = 1 , 2 , . . . , 7) 8/27 Workforce scheduling min z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 s . t . x 1 ≥ 17 x 2 ≥ 13 x 3 ≥ 15 x 4 ≥ 19 x 5 ≥ 14 x 6 ≥ 16 x 7 ≥ 11 x i ≥ 0 ( i = 1 , 2 , . . . , 7) This formulation is incorrect!...
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This note was uploaded on 12/14/2010 for the course ESI 6314 taught by Professor Vladimirlboginski during the Fall '09 term at University of Florida.

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L4_10 - 1/27 ESI 6314 Deterministic Methods in Operations...

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