The set of six equations in four variables

Problem 1 (20pts):
A furniture maker produces TV stands and sells them for 100AED each. The maker's regular
capacity is 10 stands per week, and each stand costs 60AED. But he may hire additional labor to
increase his capacity to 25 stands per week. However, since additional capacity is expensive, any
stand produced beyond the first 10, costs 75AED. For example, 12 stands cost 10 · 60 + 2 · 75 =
750 AED. The furniture maker has a weekly working capital of 1200AED, so his total cost cannot
exceed this amount. The maker has also committed to delivering 4 stands to a customer each
week, but other customers are usually interested in buying his product.
Give a formulation of a linear program allowing to maximize the manufacturer's weekly profit
(do not solve).
Problem 2 (20pts):
The set of six equations in four variables (1)-(6) does not have a unique solution (indeed, most
six equations with four variables don’t).
For each equation i, and values of variables x = (x1, x2, x3, x4), let ei be the absolute difference
(error) between the left-hand side and the right-hand side. For example, for i = 2 and x = (−5, 3, 1, 4),
the error is
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University of Sharjah Spring 2021-22
e2 = |(1)(−5) + 6(3) − (1)(1) − 5(4) − 16| = | − 24| = 24.
Write a linear programming instance that will minimize the total absolute error:
e1 + e2 + e3 + e4 + e5 + e6.
Solve this instance with GAMS. What are the values of x that achieve this error?
Problem 3 (30 pts): CCTV-Cam
CCTV-Cam is a security service company. It employs skilled technicians to conduct maintenance
activities. A demand forecast study reveals the number of hours of skilled maintenance time
that CCTV-Cam requires during the next five months. They are given as follows:
Month 1 (March): 5,000 hours
Month 2 (April): 6,000 hours
Month 3 (May): 7,000 hours
Month 4 (June): 8,500 hours
Month 5 (July): 10,000 hours
At the beginning of March, 50 skilled technicians work for CCTV-Cam. Each skilled technician
can work up to 150 hours per month. To meet future demands, new technicians must be trained.
It takes one month to train a new technician, in order to become a skilled technician. During the
month of training, a trainee must be supervised for 50 hours by a skilled technician.
Each skilled technician is paid $2,000 a month (even if he or she does not work the full 150
hours). During the month of training, a trainee is paid $1,000 a month.
At the end of each month, 5% of CCTV-Cam’s skilled technicians quit to join Aster Computers.
CCTV-Cam must determine the number of technicians who should be trained during each month
t (t = 1, 2, 3, 4, 5) to satisfy demand at minimum cost.
Formulate a Linear Program for this problem.