Consider the following linear programming problem

Max        8X + 7Y

                        s.t.         15X + 5Y  <  75

                                      10X + 6Y  <  60

                                          X +   Y  <  8

                                              X, Y  ≥  0         

a.         Set up and solve using Management Scientist, Excel Solver, or an online LP solver.           

b.        What are the values of X and Y at the optimal solution?

c.        What is the optimal value of the objective function?

 An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below.

Medium

Cost Per Ad

# Reached

Exposure Quality

TV

500

10000

30

Radio

200

3000

40

Newspaper

400

5000

25

The number of TV ads cannot exceed the number of radio ads by more than 10% (for example, if there are 100 radio ads, then there cannot be more 110 TV ads). The number of radio ads cannot exceed the number of newspaper ads by more than 13.

If in addition the advertising budget is $10000, develop the model that will maximize the number reached and achieve an exposure quality of at least 1000 (assume that fractional numbers of ads are ok):

  1. a.    Provide the objective function and set of constraints. define the decision variables

 

  1. b.   Find the optimal solution using Management Scientist, Lindo, Excel Solver, or online interactive LP software.
  2. c.    List the values of the objective function and the decision variables in the optimal solution you’ve found.         

Tots Toys makes a plastic tricycle that is composed of three major components:  a handlebar-front wheel-pedal assembly, a seat and frame unit, and rear wheels.  The company has orders for 12,000 of these trikes.  

As indicated in Table I below, the company obviously does not have the resources available to manufacture everything needed for the completion of 12000 tricycles, so it has arranged to purchase additional components, as necessary.   

Regarding the outsourcing option, the company can purchase components from one of two suppliers. Supplier A charges a dollar less per component than Supplier B, as indicated in the second table. However, there are limits to the number of components that are available from Supplier A: 

  • up to 5,000 handlebar-front wheel-pedal assemblies
  • up to 5,000 seat and frame units
  • up to 10,000 rear wheels 

Develop a linear programming model to tell the company how many of each component should be manufactured and how many should be purchased from each supplier in order to provide 12000 fully completed tricycles at the minimum cost. (Don’t forget that there are TWO rear wheels per trike) 

a. provide the objective function and set of constraints. define the decision variables

b. find the optimal solution using Management Scientist, Lindo, Excel Solver, or online interactive LP software.

c. list the values of the objective function and the decision variables in the optimal solution you’ve found.

(Guidelines: (1) there are three components, each of which can either be manufactured or purchased from one of two sources – this tells you how many decision variables there are. (2) There are three resources that are utilized when components are produced (not when they are purchased), which determines the set of resource constraints. (3) we need constraints to ensure that we have adequate supplies of each of the three components, and as indicated, each component can be purchased from one of two sources or manufactured. Bear in mind that it may be cost effective to manufacture different percentages of each component. (4) There are limits to how many components of each type that we can purchase from supplier A.

 

 

 

Table I: resource limitations for in-house manufacturing

In-house manufacturing:

  Requirements

 

 

Component

Plastic

Time

Space

Front

3

10

2

Seat/Frame

4

6

2

Each rear wheel

.5

2

.1

     Available

50000

160000

30000

 

 

Table II: Parts Production and Purchase Costs:

 

 

 

 

 

 

Cost to Manufacture

Cost to Purchase from Supplier A

Cost to Purchase from Supplier B

Front

8

12

13

Seat/Frame

6

9

10

Rear Wheel

1

3

4

 

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