1 Ardavan Asef-Vaziri June-2013LP-Formulation Additional Problems

2 Ardavan Asef-Vaziri June-2013LP-Formulation An appliance manufacturer produces two models of microwave ovens: H and W. Both models require fabrication and assembly work: each H uses four hours fabrication and two hours of assembly, and each W uses two hours fabricatio n and six hours of assembly. There are 600 fabrication hours this week and 450 hours of assembly. Each H contributes $40 to profit, and each W contributes $30 to profit. a)Formulate the problem as a Linear Programming problem. b)Solve it using excel. c)What are the final values? d)What is the optimal value of the objective function? Problem a1

3 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x H : volume of microwave oven type H x W : volume of microwave oven type W Objective Function Max Z = 40 x H +30 x W Constraints Resources 4 x H +2 x W  x H +6 x W  450 Nonnegativity x H  0, x W  0 Problem Formulation

4 Ardavan Asef-Vaziri June-2013LP-Formulation A small candy shop is preparing for the holyday season. The owner must decide how many bags of deluxe mix how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound raisins and 1/3 pounds peanuts, and the standard mix has 1/2 pound raisins and 1/2 pounds peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with. Peanuts cost $0.60 per pounds and raisins cost $1.50 per pound. The deluxe mix will sell for 2.90 per pound and the standard mix will sell for 2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold. a)Formulate the problem as a Linear Programming problem. b)Solve it using excel. c)What are the final values? d)What is the optimal value of the objective function? Problem a2

5 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x 1 : volume of deluxe mix x 2 : volume of standard mix Objective Function Max Z = [ (1/3)-1.5(2/3)] x 1 + [ (1/2)-1.5(1/2)] x 2 Max Z = 1.7x x 2 Constraints Resources (2/3) x 1 +(1/2) x 2  90 (1/3) x 1 +(1/2) x 2  60 Nonnegativity x 1  0, x 2  0 Problem Formulation

6 Ardavan Asef-Vaziri June-2013LP-Formulation Resource Usage per Unit Produced ResourceProduct A Product BAmount of resource available Q212 R122 S334 Profit/Unit$3000$2000 The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them. Problem a3 a)Formulate the problem as a Linear Programming problem. b)Solve it using excel. c)What are the final values? d)What is the optimal value of the objective function?

7 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x A : volume of product A x B : volume of product B Objective Function Max Z = 3000 x A x B Constraints Resources 2 x A +1 x B  2 1 x A +2 x B  2 3 x A +3 x B  4 Nonnegativity x A  0, x B  0 Problem Formulation

8 Ardavan Asef-Vaziri June-2013LP-Formulation The Quality Furniture Corporation produces benches and tables. The firm has two main resources Resources labor and redwood for use in the furniture. During the next production period 1200 labor hours are available under a union agreement. A stock of 5000 pounds of quality redwood is also available. Problem a4. Product mix problem : Narrative representation

9 Ardavan Asef-Vaziri June-2013LP-Formulation Consumption and profit Each bench that Quality Furniture produces requires 4 labor hours and 10 pounds of redwood Each picnic table takes 7 labor hours and 35 pounds of redwood. Total available 1200, 5000 Completed benches yield a profit of $9 each, and tables a profit of $20 each. Formulate the problem to maximize the total profit. Problem a4. Product mix problem : Narrative representation

10 Ardavan Asef-Vaziri June-2013LP-Formulation x 1 = number of benches to produce x 2 = number of tables to produce Maximize Profit = ($9) x 1 +($20) x 2 subject to Labor: 4 x x 2  1200 hours Wood:10 x x 2  5000 pounds and x 1  0, x 2  0. We will now solve this LP model using the Excel Solver. Problem a4. Product Mix : Formulation

11 Ardavan Asef-Vaziri June-2013LP-Formulation Problem a4. Product Mix : Excel solution

12 Ardavan Asef-Vaziri June-2013LP-Formulation Ralph Edmund has decided to go on a steady diet of only streak and potatoes s (plus some liquids and vitamins supplements). He wants to make sure that he eats the right quantities of the two foods to satisfy some key nutritional requirements. He has obtained the following nutritional and cost information. Ralph wishes to determine the number of daily servings (may be fractional of steak and potatoes that will meet these requirements at a minimum cost. Grams of Ingredient per Serving IngredientSteakPotatoesDaily Requirements (grams) Carbohydrates515≥ 50 Protein205≥ 40 Fat152≤ 60 Cost per serving$4$2 Formulate the problem as an LP model. Solve it using excel. What are the final values? What is the optimal value of the objective function? Problem

13 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x 1 : serving of steak x 2 : serving of potato Objective Function Min Z = 4 x 1 +2x 2 Constraints Resources 5 x x 2 ≥ x 1 +5 x 2 ≥ x 1 +2 x 2 ≤ 60 Nonnegativity x 1  0, x 2  0 Problem. Formulation

14 Ardavan Asef-Vaziri June-2013LP-Formulation Problem. Excel Solution

15 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : narrative This is a good example to show that the statement of a problem could be complicated. But as soon as we define the correct decision variables, things become very clear Two sources of pollution: Open furnace and Blast furnace Three types of pollutants : Particulate matter, Sulfur oxides, and hydrocarbons. ( Pollutant1, Pollutant2, Pollutant3). Required reduction in these 3 pollutants are 60, 150, 125 million pounds per year. ( These are RHS) Three pollution reduction techniques : taller smokestacks, Filters, Better fuels. ( these are indeed our activities). We may implement a portion of full capacity of each technique. If we implement full capacity of each technique on each source, their impact on reduction of each type of pollutant is as follows

16 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : narrative Pollutant Taller Filter Better fuel smokestacks B.F.O.F B.F.O.F. B.F.O.F. Particulate Sulfur Hydrocarb The cost of implementing full capacity of each pollutant reduction technique on each source of pollution is as follows Pollutant Taller Filter Better fuel smokestacks B.F.O.F B.F.O.F. B.F.O.F. Cost

17 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : Decision Variables How many techniques?? How many sources of pollution?? How many constraints do we have in this problem??? How many variables do we have Technique i source j

18 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : Decision Variables x 11 = Proportion of technique 1 implemented of source 1 x 12 = Proportion of technique 1 implemented of source 2 x 21 = Proportion of technique 2 implemented of source 1. x 22 = Proportion of technique 2 implemented of source 2 x 31 = Proportion of technique 3 implemented of source 1 x 32 = Proportion of technique 3 implemented of source 2.

19 Ardavan Asef-Vaziri June-2013LP-Formulation Controlling air pollution : Formulation Min Z= 12x 11 +9x x x x x 32 Particulate; 12x 11 +9x x x x x 32  60 Sulfur; 35x x x x x x 32  150 Hydrocarbon; 37x x x x x x 32  125 x11, x12, x21, x22, x31, x32 ???? Pollutant Taller Filter Better fuel smokestacks B.F.O.F B.F.O.F. B.F.O.F. Particulate Sulfur Hydrocarb

20 Ardavan Asef-Vaziri June-2013LP-Formulation An airline reservations office is open to take reservations by telephone 24 hours per day, Monday through Friday. The number of reservation officers needed for each time period is: The union requires all employees to work 8 consecutive hours. Therefore, we have shifts of 12am-8am, 4am-12pm, 8am-4pm, 12pm-8pm, 4pm-12am, 8pm-4am. Hire the minimum number of reservation agents needed to cover all requirements. Personnel scheduling problem PeriodRequirement 12am-4am11 4am-8am15 8am-12pm31 12pm-4pm17 4pm-8pm25 8pm-12am19

21 Ardavan Asef-Vaziri June-2013LP-Formulation The union contract requires all employees to work 8 consecutive hours. We have shifts of 12am-8am, 4am-12pm, 8am-4pm, 12pm-8pm, 4pm-12am, 8pm- 4am. Hire the minimum number of reservation agents needed to cover all requirements. If there were not restrictions of 8 hrs sifts, then we could hire as required, for example 11 workers for 4 hors and 15 workers for 4 hours. Personnel scheduling problem : Narrative representation

22 Ardavan Asef-Vaziri June-2013LP-Formulation Personnel scheduling problem 12 am to 4 am 4 am to 8 am 8 am to 12 pm 12 pm to 4 pm 4 pm to 8 pm 8 pm to 12 am

23 Ardavan Asef-Vaziri June-2013LP-Formulation x 1 = Number of officers in 12 am to 8 am shift x 2 = Number of officers in 4 am to 12 pm shift x 3 = Number of officers in 8 am to 4 pm shift x 4 = Number of officers in 12 pm to 8 pm shift x 5 = Number of officers in 4 pm to 12 am shift x 6 = Number of officers in 8 pm to 4 am shift Personnel scheduling problem : Decision variables

24 Ardavan Asef-Vaziri June-2013LP-Formulation Min Z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 12 am - 4 am : x 1 +x 6  11 4 am - 8 am : x 1 +x 2  15 8 am - 12 pm : +x 2 + x 3  pm - 4 pm : +x 3 + x 4  17 4 pm - 8 pm : +x 4 + x 5  25 8 pm - 12 am : +x 5 + x 6  19 x 1, x 2, x 3, x 4, x 5, x 6  0. Personnel problem : constraints and objective function

25 Ardavan Asef-Vaziri June-2013LP-Formulation Personnel scheduling problem : excel solution