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

1Ardavan Asef-Vaziri June-2013LP-Formulation

Additional ProblemsAdditional Problems


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 fabrication 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


Decision Variables xH : volume of microwave oven type H xW : volume of microwave oven type W

Objective Function Max Z = 40 xH +30 xW

ConstraintsResources4 xH +2 xW 600 2 xH +6 xW 450

NonnegativityxH 0, xW 0

Problem 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


Decision Variables x1 : volume of deluxe mix x2 : volume of standard mix

Objective Function Max Z = [2.9-0.60(1/3)-1.5(2/3)] x1 + [2.55-0.60(1/2)-1.5(1/2)] x2 Max Z = 1.7x1 + 1.5 x2 ConstraintsResources(2/3) x1 +(1/2) x2 90 (1/3) x1 +(1/2) x2 60

Nonnegativityx1 0, x2 0

Problem Formulation


Resource Usage per Unit ProducedResource Product

AProduct B Amount of

resource available

Q 2 1 2R 1 2 2S 3 3 4

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?


Decision Variables xA : volume of product A xB : volume of product B

Objective Function Max Z = 3000 xA +2000 xB

ConstraintsResources2 xA +1 xB 21 xA +2 xB 23 xA +3 xB 4

NonnegativityxA 0, xB 0

Problem 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 period1200 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


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


x1 = number of benches to producex2 = number of tables to produce

Maximize Profit = ($9) x1 +($20) x2

subject to Labor: 4 x1 + 7 x2 1200 hoursWood: 10 x1 + 35 x2 5000 poundsand x1 0, x2 0.

We will now solve this LP model using the Excel Solver.

Problem a4. Product Mix : Formulation


Problem a4. Product Mix : Excel solution

Bench Table Required AvailableLabor 4 7 1200 1200Wood 10 35 5000 5000Contribution Margin 9 20 3185.714Changing Cells 100 114.2857

Microsoft Excel Worksheet


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

Ingredient Steak Potatoes Daily Requirements (grams)Carbohydrates 5 15 ≥ 50

Protein 20 5 ≥ 40Fat 15 2 ≤ 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


Decision Variables x1 : serving of steak x2 : serving of potato

Objective Function Min Z = 4 x1 +2x2

ConstraintsResources5 x1 +15 x2 ≥ 5020 x1 +5 x2 ≥ 4015 x1 +2 x2 ≤ 60

Nonnegativityx1 0, x2 0

Problem . Formulation


Problem . Excel Solution

Steak Potatoe LHS RHSCarbohydrates 5 15 50 ≥ 50Protein 20 5 40 ≥ 40Fat 15 2 24.9 ≤ 60Cost per serving $4 $2 10.9

1.27 2.91

Microsoft Excel Worksheet


Controlling air pollution : narrativeThis 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


Controlling air pollution : narrative

Pollutant Taller Filter Better fuel

smokestacksB.F. O.F B.F. O.F. B.F. O.F.

Particulate 12 9 25 20 17 13Sulfur 35 42 18 31 56 49Hydrocarb. 37 53 28 24 29 20

The cost of implementing full capacity of each pollutant reduction technique on each source of pollution is as follows



Cost 12 9 25 20 17 13


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 haveTechnique i source j


Controlling air pollution : Decision Variables

x11 = Proportion of technique 1 implemented of source 1x12 = Proportion of technique 1 implemented of source 2

x21 = Proportion of technique 2 implemented of source 1.x22 = Proportion of technique 2 implemented of source 2

x31 = Proportion of technique 3 implemented of source 1x32 = Proportion of technique 3 implemented of source 2.


Controlling air pollution : Formulation

Min Z= 12x11+9x12+ 25x21+20x22+ 17x31+13x32 Particulate; 12x11+9x12+ 25x21+20x22+ 17x31+13x32 60Sulfur; 35x11+42x12+ 18x21+31x22+ 56x31+49x32 150Hydrocarbon; 37x11+53x12+ 28x21+24x22+ 29x31+20x32 125 x11, x12, x21, x22, x31, x32 ????



Particulate 12 9 25 20 17 13Sulfur 35 42 18 31 56 49Hydrocarb. 37 53 28 24 29 20


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

Period Requirement12am-4am 114am-8am 158am-12pm 3112pm-4pm 174pm-8pm 258pm-12am 19


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


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

1 2 3 4 5 6

11

15

31

17

25

19

1

1

11

1

1

1

1

1

1

1

1


x1 = Number of officers in 12 am to 8 am shift x2 = Number of officers in 4 am to 12 pm shift x3 = Number of officers in 8 am to 4 pm shift x4 = Number of officers in 12 pm to 8 pm shift x5 = Number of officers in 4 pm to 12 am shift x6 = Number of officers in 8 pm to 4 am shift

Personnel scheduling problem : Decision variables


Min Z = x1+ x2+ x3+ x4+ x5+ x6

12 am - 4 am : x1 +x6 114 am - 8 am : x1 +x2 15

8 am - 12 pm : +x2 + x3 31

12 pm - 4 pm : +x3 + x4 17

4 pm - 8 pm : +x4 + x5 25

8 pm - 12 am : +x5 + x6 19

x1 , x2, x3, x4, x5, x6 0.

Personnel problem : constraints and objective function


Personnel scheduling problem : excel solution

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

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