1. 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	# XXXXX	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 4, and the advertising budget is $10000.
a. Develop a model that will maximize the number reached and achieve an exposure quality of at least 1000.
b. Solve the model that you developed in part a.
c. Suppose the advertising budget is raised to $12,000, how would your solution to part b change?



2. Larkin Industries manufactures several lines of decorative and functional metal items. The most recent order has been for 1200 door lock units for an apartment complex developer. The sales and production departments must work together to determine delivery schedules. Each lock unit consists of three components: the knob and face plate, the actual lock itself, and a set of two keys. Although the processes used in the manufacture of the three components vary, there are three areas where the production manager is concerned about the availability of resources. These three areas, their usage by the three components, and their availability are detailed in the table.

Resource	Knob and Plate	Lock	Key (each)	Available
Brass Alloy	12	5	1	15000 units
Machining	18	20	10	36000 minutes
Finishing	15	5	1	12000 minutes

A quick look at the amounts available confirms that Larkin does not have the resources to fill this contract. A subcontractor, who can make an unlimited number of each of the three components, quotes the prices below.

Component	Subcontractor Cost	Larkin Cost
Knob and Plate	10.00	6.00
Lock	9.00	4.00
Keys (set of 2)	1.00	.50

Develop a linear programming model that would tell Larkin how to fill the order for 1200 lock sets at the minimum cost.


3. Super City Discount Department Store is open 24 hours a day. The number of cashiers needed in each four hour period of a day is listed below. Suppose each cashier is paid $12.00 per hour with a positive hourly shift differential of 30% of the hourly wage for workers who are on the job between 10:00 P.M. and 6:00 A.M.

Period	Cashiers Needed
10 p.m. to 2 a.m.	8
2 a.m. to 6 a.m.	4
6 a.m. to 10 a.m.	7
10 a.m. to 2 p.m.	12
2 p.m. to 6 p.m.	10
6 p.m. to 10 p.m.	15

If cashiers work for eight consecutive hours, how many should be scheduled to begin working in each period in order to minimize the number of cashiers needed?



4. Canning Transport is to move goods from three factories to three distribution centers. Information about the move is given below.

Source	Supply	Destination	Demand
A	200	X	50
B	100	Y	125
C	150	Z	125

 

 

Shipping costs are:

 

 

Â	Destination
Source	X	Y	Z
A	3	2	5
B	9	10	--
C	5	6	4
Â	(Source B cannot ship to destination Z)

a. Develop the network representation of this problem.
b. Formulate this problem as a linear program.
c. Solve the problem.







5. RVW (Restored Volkswagens) buys 15 used VW's at each of two car auctions each week held at different locations. It then transports the cars to repair shops it contracts with. When they are restored to RVW's specifications, RVW sells 10 each to three different used car lots. There are various costs associated with the average purchase and transportation prices from each auction to each repair shop. Also there are transportation costs from the repair shops to the used car lots. RVW is concerned with minimizing its total cost given the costs in the table below.

a.

Given the costs below, draw a network representation for this problem.

Â

Â

Repair Shops

Â

Â

Used Car Lots

Â

Â

Â

S1

S2

Â

Â

L1

L2

L3

Â

Â

Auction 1

550

500

Â

S1

250

300

500

Â

Â

Auction 2

600

450

Â

S2

350

650

450

Â

b.

Formulate this problem as a transshipment linear programming model.

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6. Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year. The projects, the number of engineers and the number of support personnel required for each project, and the expected profits for each project are summarized in the following table:

Â	Project
Â	1	2	3	4	5	6
Engineers Required	20	55	47	38	90	63
Support Personnel Required	15	45	50	40	70	70
Profit ($1,000,000s)	1.0	1.8	2.0	1.5	3.6	2.2

 

 

Formulate an integer program that maximizes Tower's profit subject to the following management constraints:

1) Use no more than 175 engineers
2) Use no more than 150 support personnel
3) If either project 6 or project 4 is done, both must be done
4) Project 2 can be done only if project 1 is done
5) If project 5 is done, project 3 must not be done and vice versa
6) No more than three projects are to be done.













7. Your express package courier company is drawing up new zones for the location of drop boxes for customers. The city has been divided into the seven zones shown below. You have targeted six possible locations for drop boxes. The list of which drop boxes could be reached easily from each zone is listed below.

Zone	Can Be Served By Locations:
Downtown Financial	1, 2, 5, 6
Downtown Legal	2, 4, 5
Retail South	1, 2, 4, 6
Retail East	3, 4, 5
Manufacturing North	1, 2, 5
Manufacturing East	3, 4
Corporate West	1, 2, 6

Let xi = 1 if drop box location i is used, 0 otherwise. Develop a model to provide the smallest number of locations yet make sure that each zone is covered by at least two boxes.