SYSEN 5200 Homework 4
1. The igure represents an oil pipeline network. The nodes represent pumping and/or receiving stations. The lengths in miles of the diferent segments of the network are shown on the respective arcs. The bi-directional segments allow lows in both directions. The supplies at stations 1 and 3 are respectively 60 and 50 barrels per day. The demands at the stations 2 and 4 are respectively 85 and 25 barrels per day. Assume that the transportation cost per shipped barrel is proportional to the distance and we are interested in minimizing the total transportation cost.
Figure 1: Problem 1
(a) (15 points) Optimize the transportation schedule (i.e. write the problem that consists of an objective function and constraints).
(b) (5 points) Is this a linear or non-linear optimization problem? Please briely explain.
2. A tech startup was founded in 2021. The following table shows an estimate of active numbers ofcon- tractors the company needs for any given year (which includes new hires as well as those stayingon due to a longer-term contract) over the next 5 years:
The company can choose to sign diferent length contracts with each contractor it hires and wants tokeep more contractors than needed during some years of the planning horizon. The startup estimatesthat the cost of recruiting and maintaining the contractors is a function of their length of stay withthe company. The following table summarizes these estimates:
The startup is interested in the recruiting schedule to minimize the total cost.
(Hint: Think about the number of contractors the startup needs to hire for diferent horizons – 1, 2,3, 4, 5 years – in each year of the planning horizon.)
(a) (15 points) Write the problem as an optimization problem. You can ignore the fact that the numbers of engineers cannot take fractional values.
(b) (5 points) Is this a linear or non-linear optimization problem? Please briely explain.
(c) (20 points) Determine the optimal solution by using Excel’s solver or Python’s packages. If you use Excel, be sure to submit the Excel sheet and solver window. If you use Python, be sure to submit your fully commented code and output.
3. A company would like to choose the site of a new distribution center that will service four sales centers located in Arlington (A), Bainbridge (B), Charlotte (C) and Dryden (D). The table below gives the location of these sales centers in the plane, where the units are in miles. The table also gives the number of daily deliveries to each sales center from the distribution center. Suppose that truck delivery costs are s1 per mile.
(a) (15 points) Where should the distribution center be located so as to minimize the daily trans- portation cost, assuming that the trucks travel in a straight line? Formulate the problem as an optimization problem.
(b) (5 points) Is this a linear or non-linear optimization problem? Please briely explain.
(c) (20 points) Use Excel’s solver / Python to determine the optimal solution: what are the optimal location for the distribution center and minimum transportation cost? Be sure to submit a screenshot of the Excel sheet and solver window or your code that you used to solve the problem.