1) The school board for the Bellevue School District has made the decision to purchase 1,350 additional Macintosh computers for computer laboratories in all its schools. Based on past experience, the school board also has directed that these computers should be purchased from some combination of three companies Educomp, Macwin, and McElectronics. In all three cases, the companies charge a discounted variable cost per computer and a fixed delivery and installation cost for these large sales to school districts. The table below shows these charges as well as the capacity (the maximum number of computers that can be sold from the limited inventory) for each of the companies.
|
|
Educomp
|
Macwin
|
McElectronics
|
|
Capacity
|
700
|
700
|
1,000
|
|
Fixed cost
|
$45,000
|
$35,000
|
$50,000
|
|
Variable cost
|
$750
|
$775
|
$700
|
The school board wants to determine the minimum-cost plan for meeting its computer needs
a. Formulate a BIT model in algebraic form for this problem.
b. Formulate and solve this model on a spreadsheet.
c. Now suppose that Macwin has not submitted its final bid yet, so the per computer cost is not known with certainty. Generate a parameter analysis report with RSPE to show the optimal order quantities and total cost of the optimal solution when the cost per computer for Macwin is $680, $690, $700, $710. . . $790, or $800.
PROBLEM 1:
Speedy Delivery provides two-day delivery service of large parcels across the United States. Each morning at each collection center, the parcels that have arrived overnight are loaded onto several trucks for delivery throughout the area. Since the competitive battlefield in this business is speed of delivery, the parcels are divided among the trucks according to their geographical destinations to minimize the average time needed to make the deliveries.
On this particular morning, the dispatcher for the Blue River Valley Collection Center, Sharon Lofton, is hard at work. Her three drivers will be arriving in less than an hour to make the day's deliveries. There are nine parcels to be delivered, all at locations many miles apart. As usual, Sharon has loaded these locations into her computer. She is using her company's special software package, a decision support system called Dispatcher. The first thing Dispatcher does is use these locations to generate a considerable number of attractive possible routes for the individual delivery trucks. These routes are shown in the table below (where the numbers in each column indicate the order of deliveries), along with the estimated time required to traverse the route.
Dispatcher is an interactive system that shows these routes to Sharon for her approval or modification. (For example, the computer may not know that flooding has made a particular route infeasible.) After Sharon approves these routes as attractive possibilities with reasonable time estimates, Dispatcher next formulates and solves a BIP (binary integer programming) model for selecting three routes that minimize their total time while including each delivery location on exactly one route.
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Delivery
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Attractive Possible Routes
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Location
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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A
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1
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1
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1
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B
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2
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1
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2
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2
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2
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C
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3
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3
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3
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3
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D
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2
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1
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1
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E
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2
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2
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3
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F
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1
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2
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G
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3
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1
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2
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3
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H
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1
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3
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1
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I
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3
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4
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|
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2
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|
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Time (hours)
|
6
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4
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7
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5
|
4
|
6
|
5
|
3
|
7
|
6
|
a) Formulate the above problem mathematically as a binary integer programming model. Define you decision variables clearly, define the notation you use, using your decision variables, formulate the objective function and constraints. Finally, combine everything to get the final model.
b) Formulate and solve this problem on a spreadsheet model using Excel.
PROBLEM 2:
An increasing number of Americans are moving to a warmer climate when they retire. To take advantage of this trend, Sunny Skies Unlimited is undertaking a major real-estate development project. The project is to develop a completely new retirement community (to be called Pilgrim Haven) that will cover several square miles. One of the decisions to be made is where to locate the two paramedic stations that have been allocated to the community to respond to medical emergencies. For planning purposes, Pilgrim Haven has been divided into five tracts, with no more than one paramedic station to be located in any given tract. Each station is to respond to all the medical emergencies that occur in the track in which it is located as well as in the other tracts that are assigned to this station. Thus, the decisions to be made consist of (1) the tracts to receive a paramedic station and (2) the assignment of each of the other tracts to one of the paramedic stations. The objective is to minimize the overall average of the response times to medical emergencies. The overall average of the response times is equal to the summation of all of the response times from the located stations to the assigned tracts divided by the summation of the average frequencies.
The following table gives the average response time to a medical emergency in each tract (the rows) if that tract is served by a station in a given tract (the columns). The last column gives the forecasted average number of medical emergencies that will occur in each of the tracts per day.
Formulate the above problem mathematically as a binary integer programming model. Define you decision variables clearly, define the notation you use, using your decision variables, formulate the objective function and constraints. Finally, combine everything to get the final model.
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Fire Station in Tract
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Average Frequency of
Medical Emergencies
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1
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2
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3
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4
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5
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per Day
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|
Response
Times
(min.) to a
medical
emergency
in tract
|
1
2
3
4
5
|
5
12
30
20
15
|
20
4
15
10
25
|
15
20
6
15
12
|
25
15
25
4
10
|
10
25
15
12
5
|
2
1
3
1
3
|
PROBLEM 3:
The school board for the Bellevue School District has made the decision to purchase ,350 additional Macintosh computers for computer laboratories in all its schools. Based on the past experience, the school board has also directed that these computers should be purchased from combinations of three companies
- Educomp, Macwin, and McElectronics. In all three cases, the companies charge a discounted variable cost per computer and fixed delivery and installation cost for these large sales to school districts. The table below shows these charges as well as the capacity (the maximum number of computers that can be sold from the limited inventory) for each of the companies.
|
|
Educomp
|
Macwin
|
Mc Electronics
|
|
Capacity
|
700
|
700
|
1000
|
|
Fixed Cost
|
$45,000
|
$35,000
|
$50,000
|
|
Variable Cost
|
$750
|
$775
|
$700
|
The school board wants to determine the minimum-cost plan for meeting its computer needs.
a) Formulate the above problem mathematically as a binary integer programming model. Define you decision variables clearly, define the notation you use, using your decision variables, formulate the objective function and constraints. Finally, combine everything to get the final model.
b) Formulate and solve this problem on a spreadsheet model using Excel.
PROBLEM 4: Dr. Konur has recently completed a project for Missouri Department of Transportation, which was for optimizing the track inspection planning on the Missouri railroad network. In this question, you are asked to formulate a simpler version of the track inspection planning problem. In particular, suppose that there 5 rail tracks that you can inspect. Each track has different inspection importance and each track has different inspection times. The table below gives the importance level and inspection time for each track.