1. The Energetic company needs to make plans for the energy systems for a new building. The energy needs in the building fall into three categories: (1) electricity, (2) heating water, and (3) heating space in the building. The daily requirements for these three categories (all measured in the same units) are 20 electricity units, 10 heating water units, and 30 space heating units. The three possible sources of energy to meet these needs are electricity, natural gas, and a solar heating unit that can be installed on the roof. The size of the roof limits the largest possible solar heater to 30 units, but there is no limit to the electricity and natural gas available. Electricity needs can be met only by purchasing electricity (at a cost of 50 KWD per unit). Both other energy needs can be met by any source or combination of sources. The unit costs are given in the following table.

Electricity Natural Gas Solar Heater

Water

heating 90 KWD 60 KWD 30 KWD

Space

heating 80 KWD 50 KWD 40 KWD

The objective is to minimize the total cost of meeting the energy needs.

a. Define the decision variables. (3 points)

b. Formulate and solve a linear program to minimize the energy costs of meeting the energy needs. (12 points))

c. Which heating requirement (water or space) would you consider reducing to lower cost? Explain. (5 points)

2. Amal is the director of the Computer Center for a local school. She now needs to schedule the staffing of the center. It is open from 8 A.M. until midnight. Amal has monitored the usage of the center at various times of the day, and determined that the following number of computer consultants are required.

Time of day Minimum number of consultants required

8:00 12:00 3

12:00-16:00 5

16:00-20:00 8

20:00-24:00 5

Two types of consultants can be hired: full-time and part-time. The full-time consultants work for 8 consecutive hours in any of the following shifts: morning (8 A.M.4 P.M.), afternoon (noon8 P.M.), and evening (4 P.M.midnight). Full-time consultants are paid 4 KWD per hour. Part-time consultants can be hired to work any of the four shifts listed in the above table. Part-time consultants are paid 2 KWD per hour. An additional requirement is that during every time period, there must be at least 2 full-time consultants on duty for every part-time consultant on duty. Amal would like to determine how many full-time and how many part-time workers should work each shift to meet the above requirements at the minimum possible cost.

a. Formulate and solve a linear program to help Amal prepare her staffing plan at the lowest possible cost. (15 points)

b. If Amal can lower the minimum number of consultants in any of the four 4-hour shifts, which one should she choose? Explain. (5 points)