Problem 1
Circuit boards for a computer system are to be replaced at intervals over a period of 6 months. Ideally, replacement should occur before an actual breakdown in order to maintain an operational system. Frequent replacement incurs capital expenses and costs of labor for installation. But infrequent replacement may lead to increased maintenance costs and unacceptably high rates of system downtime. If we collect data on the costs of purchase, installation, and maintenance, cost of expected downtime, and salvage value of replaced boards, we can arrive at a tabularized summary of these expenses, such as shown in the Table below. Any circuit board becomes a candidate for replacement after 1 month.
|
Equipment Replacement Costs($)
|
|
Circuit Board Replaced
|
|
|
|
Feb
|
Mar
|
Apr
|
May |
June
|
| Circuit |
Jan
|
5.00
|
6.75
|
8.25
|
12.50 |
16.80
|
| Board |
Feb
|
|
5.25
|
6.25
|
9.50 |
11.50
|
| Installed |
Mar
|
|
|
5.25
|
7.25 |
9.00
|
|
|
Apr
|
|
|
|
5.50 |
8.20
|
|
|
May
|
|
|
|
|
5.80
|
a) Represent this problem as a network and include all the information on it.
b) Solve the problem manually using the appropriate network algorithm to find the optimal replacement policy. Make sure you show your solution step by step.
c) Formulate the problem as a mathematical program. Clearly define your decision variables, objective and constraints.
d) Use Excel to implement the model you formulated in (c) and use Excel Solver to find the optimal solution. Include a snapshot of your solution in the report
Problem 2
A data communications network can be described by the diagram below. Every data link from node i to node j has a capacity which is denoted as a label on the data link in the diagram. Data is being generated at node 1 and is to be routed through the network (not necessarily passing through all other nodes) to node 6 where the data will be used. This is a unidirectional network, which means data does not flow in the opposite direction of the arrows. The amount of data generated at node 1 is exactly the amount of data consumed at node 6. No data is generated or used at intermediate nodes, so all data that enters an intermediate node must leave it, and vice versa.
(a) Solve this problem manually to find the maximum feasible amount of data that can flow through this network? Show your step-by-step solution and present a summary of your final solution
(b) Which links comprise the bottleneck in this network? Explain your answer
(c) Formulate this problem as a mathematical program to find the optimal solution. Clearly define your decision variables, objective and constraints.
(d) Using MS-Excel Solver to find a solution to the model you implemented in part (c). Include a snapshot of your solution in the report
Problem 3
An advertisement agency is trying to determine a TV advertising schedule for a client. The client has three goals (listed here in descending order of importance). It wants its ads to be seen by:
- Goal 1: at least 65 million high-income men (HIM)
- Goal 2: at least 72 million high-income women (HIW)
- Goal 3: at least 70 million low-income people (LIP)
The agency can purchase several types of TV ads: ads shown on live sports shows, on game shows, on news shows, on sitcoms, on dramas, and on soap operas. At most $700,000 total can be spent on ads. The advertising costs and potential audiences (in millions of viewers) of a 1-minute ad of each type are shown in the following table. As a matter of policy, the client requires that at least two ads be placed on each of the following shows: sports shows, news shows, and drama shows. Also, it requires that no more than ten ads be placed on any single type of show. The agency wants to find the advertising plan that best meets its client's goals.
|
Ad Type
|
HIM
|
HIW
|
LIP
|
Cost
|
|
Sports Show
|
7
|
8
|
4
|
$120,000
|
|
Game Show
|
3
|
6
|
5
|
$40,000
|
|
News
|
6
|
3
|
5
|
$50,000
|
|
Sitcom
|
4
|
7
|
5
|
$40,000
|
|
Drama
|
6
|
6
|
8
|
$60,000
|
|
Soap Opera
|
3
|
5
|
4
|
$40,000
|
a) Formulate a Goal Programming model for this problem to determine the number of ads of the different types that best meets the client's goals. Clearly define your decision variables, constraints and objective(s)
b) Solve this problem using Excel Solver. Make sure you label your sheets appropriately and use short comments to describe what you did on each sheet. Include organized snapshots of your Excel solution in the report.
Problem 4
A mower manufacturer produces two types of riding lawn mowers. One is a low-cost mower sold primarily to residential home owners; the other is an industrial model sold to lawn service companies. The company is interested in establishing a pricing policy for the two mowers that will maximize the profit for the product line. A study of the relationships between sales prices and quantities sold of the two mowers has validated the following price-quantity relationships.
q1 = 950 - 1.5p1 + 0.7p2
q2 = 2500 + 0.3p1 - 0.5p2
Where
q1 = number of residential mowers sold
q2 = number of industrial mowers sold
p1 = selling price of the residential mower in dollars
p2 = selling price of the industrial mower in dollars
The accounting department developed cost information on the fixed and variable cost of producing the two mowers. The fixed cost of production for the residential mower is $10,000 and the variable cost is $1,500 per mower. The fixed cost of production for the industrial mower is $30,000 and the variable cost is $4,000 per mower.
a) The company traditionally priced the lawn mowers at $2,000 and $6,000 for the residential and industrial mowers, respectively. Calculate the number of mowers that will be sold, and the profit with this pricing policy?
b) Formulate a mathematical model to solve this problem optimally. Clearly define your decision variables, constraints and objective function
c) Implement the model you formulated in (b) using Excel and find the optimal prices, profit and number of units sold using Excel Solver. Include a snapshot of your solution in the report. Compare the optimal answer you obtained to that in (a)
Problem 5
A cotton grower in south Georgia produces cotton on farms in Statesboro and Brooklet, ships it to cotton gins in Claxton and Millen, where it is processed, and then sends it to distribution centers in Savannah, Perry, and Valdosta, where it is sold to customers for $60 per ton. Any surplus cotton is sold to a government warehouse in Hinesville for $25 per ton. The cost of growing and harvesting a ton of cotton at the farms in Statesboro and Brooklet is $20 and $22, respectively. There are presently 700 and 500 tons of cotton available in Statesboro and Brooklet, respectively. The cost of transporting the cotton from the farms to the gins and the government warehouse is shown in the following table:
|
|
Claxton |
Mon |
SnareMk |
|
Statesboro
|
$4.00 |
$3.00 |
$4.50
|
|
Brooklet
|
$3.5 |
$3.00 |
$3.50
|
The gin in Claxton has the capacity to process 700 tons of cotton at a cost of $10 per ton. The gin in Millen can process 600 tons at a cost of $11 per ton. Each gin must use at least one half of its available capacity. The cost of shipping a ton of cotton from each gin to each distribution center is summarized in the following table:
|
|
Savannah |
Pony
|
Valdosta |
|
Claxton
|
$10 |
$16
|
$15 |
|
Millen
|
$12 |
$18
|
$17 |
Assume that the demand for cotton in Savannah, Perry, and Valdosta is 400, 300, and 450 tons, respectively.
a) Draw a network flow model to represent this problem.
b) Formulate a mathematical model to solve this problem. Clearly define your decision variables, constraints and objective function
c) Implement the model you formulated in (b) using Excel and find the optimal solution using Excel Solver. Include a snapshot of your solution in the report.