Formulate the following problem as a mixed integer program (do NOT solve).
A Company manufactures â€œnâ€ different products having unit profits of Pi ( i = 1,2, ...., n). Each product can be manufactured on any of â€œmâ€ available machines. The length of time in minutes it takes to manufacture one unit of product i on machine j is tij. If machine â€œjâ€ is to be used on a particular shift, a setup time in minutes of sj is incurred. During a normal 8 hour shift, R i ( i = 1,2, ...., n), SOME units of product i must be manufactured. There are â€œkâ€ operators (k<m) available to operate machines on a given shift (i.e. no more than k machines can operate on a shift). Because of a shared power source, Machine 2 can be used only if machine 1 is being used. Machines 3 and 4 share the same drive system, and therefore they cannot both operate on the same shift. To operate machines 5 or 6 will require an additional cost of â€˜câ€™ dollars each per shift because of their unique power requirements.
The objective is to determine for a given shift, which machines to operate and how many units of each product to manufacture on each machine in order to maximize profit. Assume no machine downtime during a shift (i.e. each machine is available for 8 hours).
Let:
n = number of different products (discrete integer)
Pi = unit of profit (continuous), â€œiâ€ = 1 to n (discrete integer)
tij = time to produce one unit of product â€œiâ€ on machine â€œjâ€ (continuous, min)
K = workers available per shift (less than # of machines available)
Ri = time available per shift (8 hr assumed) â€œiâ€ = 1 to n (discrete integer)
m = number of available machines (discrete integer)
sj = set up time on machine â€œjâ€ (continuous, min)
Caveats:
Machine 2 & 1 related (dependent): Machine 2 ONLY if Machine 1
Machine 3 & 4 related (dependent): Machine 3 OR Machine 4
Machine 5 & 6 penalty (cost): $c/shift
a. Define your continuous (xij ) and integer variables (yj ):
xij = # of units of product â€œiâ€ (i=1..n) to manufacture on machine â€œjâ€ (j=1â€¦m)
yj = 1 if machine â€œjâ€ used, else = 0 (binary flag)

Formulate the following problem as a mixed integer program (do NOT...