Recall the problem of formulating a linear program to find the optimal schedule for a set of construction activities that would result in the shortest possible construction time for a small commercial establishment (Exercise 1). A subsequent exercise (Exercise 2) presented this problem from the perspective of a manager concerned about minimizing total cost of production. Ex plain how you would generate the tradeoff surface between these two objectives
Exercise 1:
You are responsible for scheduling the activities required for construction of a small commercial establishment. These activities, the expected duration of each, and an indication of which other activities must precede each are shown in the tab le below:
Job | Description | Duration | Must follow lob (s) |
1 | Land leveling and excavation | 9 | — |
2 | Pour retaining walls and foundation | 7 | 1 |
3 | Install basement support structure | 5 | 2 |
4 | Install floor joists | 3 | 2 |
5 | Construct exterior walls | 6 | 3, 4 |
6 | Install walls and flooring | 7 | 4 |
7 | Install ceiling/roof superstructure | 4 | 5, 6 |
8 | Install electrical/mechanical/plumbing | 9 | 6 |
9 | Rough finish interior (wallboard. etc.) | 7 | 6 |
10 | Install roofing material | 5 | 7 |
11 | Finish interior | 8 | 9 |
12 | Landscaping | 11 | 1 |
For example, construction of the exterior walls of the facility is estimated to take six crew days., and may not begin until the basement support structure has been completed—Job 3—and the floor joists have been installed—Job 4.
Formulate a linear program that will determine the starting time for each individual activity such that the facility may be in operation as soon as possible.
Exercise 2:
An alternate approach to solving the job scheduling problem presented in Problem 3.17 is one that seeks to minimize total project costs. This assumes that the costs for each job are also known in advance. Furthermore, it may be possible to accelerate some or all of the construction activities by allocating more resources to those jobs.
Consider the costs indicated for each of the construction activities presented in the table below. The description of each job is the same as in Problem 3.17 with the base costs for each job indicated. An alternate completion time and cost is also included in the table. For example, the installation of the ceiling superstructure, which normally takes four days at a total cost of $2,100, can be completed in two days if you are willing to increase that cost to $3,600.
Job | Base Cost | Base Duration | Crash Cost | Crash Duration |
1 | $2,500 | 9 | $3,100 | 6 |
2 | $2,700 | 7 | $3,200 | 5 |
3 | $1,800 | 5 | $2,600 | 3 |
4 | $1,400 | 3 | $1,700 | 2 |
5 | $4,200 | 6 | $6,300 | 3 |
6 | $3,200 | 7 | $5,600 | 4 |
7 | $2,100 | 4 | $3,600 | 2 |
8 | $4,200 | 9 | $8,300 | 5 |
9 | $2,300 | 7 | $4,600 | 4 |
10 | $4,200 | 5 | $4,900 | 4 |
11 | $3,800 | 8 | $5,200 | 5 |
12 | $4,000 | 11 | $6,400 | 7 |
Your contract also has a penalty clause that if the project is not finished in 20 days, you must pay a penalty computed using the following formula:
Penalty = 6.38 (number of days over 20)2
Formulate a. linear program that determines the scheduled starting time for each job so as to minimize total project cost, including penalty that might be owed.
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