Question

Formulate the following problem as a mixed integer program (do NOT solve).
A Company manufactures “n” different products having unit profits of Pi ( i = 1,2, ...., n). Each product can be manufactured on any of “m” available machines. The length of time in minutes it takes to manufacture one unit of product i on machine j is tij. If machine “j” 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 “k” 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 ‘c’ 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), “i” = 1 to n (discrete integer)
tij = time to produce one unit of product “i” on machine “j” (continuous, min)
K = workers available per shift (less than # of machines available)
Ri = time available per shift (8 hr assumed) “i” = 1 to n (discrete integer)
m = number of available machines (discrete integer)
sj = set up time on machine “j” (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 “i” (i=1..n) to manufacture on machine “j” (j=1…m)
yj = 1 if machine “j” used, else = 0 (binary flag)

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Choice 1’s objective function is wrong since the additional cost c pertains to machines 5 and 6. This is correctly written in Choice 2 and 3. Beyond the objective function, everything goes well with choice 2 and 3 until the non-negativity of x_ij. It is wrongly formulated in Choice 2, where it should have been x_ij≥0 instead of x_ij≤0. So choice 3 remains correct. Now, let’s check the other constraints set up in choice 3. The machine availability limit is correct since the number of machines to be used must be at most the number of workers. However, the machine usage limit/shift is wrong. R_i is the total time needed to produce item i....

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