Problem $3$

a$.(50$ points$)$ Suppose you have been asked to help determine how many units of your product to be produced at each of five manufacturing facilities that your firm owns. The facilities are identified by the letters A through E$.$ You would like to minimize the collective variable $($i$.$e$.,$ per unit$)$ operating costs across all facilities. Those operating costs as well as the production capacities vary somewhat across all five facilities, as shown in the table below.

Facility Facility Capacity

A $2200$

B $2500$

C $2000$

D $2800$

E $3000$

Variable Operating Costs per Unit $$28\mathrm{.}50$

$$27\mathrm{.}90$

$$25\mathrm{.}40$ $$29\mathrm{.}10$ $$28\mathrm{.}60$

There is no restriction on the number of facilities you can use. However, upper management has placed a few restrictions on how the facilities can be used. The percentage of total units produced in B and D combined cannot exceed $60\%$ of the total of all units produced. Also, the total units produced in facility E must be at least half of the total produced by facilities A and C combined.

Total demand for your product is $6000$ units in the upcoming period.

Using linear, integer, or mixed integer programming, determine the number of units to produce at each facility. In addition, use your answer to determine which of your constraints are binding; i$.$e$.,$ for which facilities would you like to acquire additional capacity?

$6$

b$.(50$ points$)$ Now assume that there are fixed as well as variable operating costs at each facility. The fixed costs are incurred if any units at all are produced at the facility, but are the same regardless of how many are produced at the facility. If no units are produced at a facility, the fixed costs are not incurred at that facility. The fixed operating costs vary somewhat across all five facilities, as shown in the table below.

Facility

A B C D E

Fixed Operating Costs

$$59,000$ $$49,000$ $$54,000$ $$52,000$ $$57,000$

Also, management has now decided that at least one $($but not necessarily both$)$ of the following two restrictions must hold: $(1)$ if facility A is not used, then facility E must be used, or $(2)$ if facility D is used, then facility C must also be used.

Using linear, integer, or mixed integer programming, modify your model from part $($a$)$ to determine which facilities to use and the number of units to produce at each facility with these new considerations.