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I have the linear problem as it follows. I have 3 different types of devices. Type A, Type B, Type C. At any given moment, there is exactly one type of each device installed. So one device A, one device B and one device C.

Each type of device has its own probability to get damaged, so I want to have spare devices if these get damaged, with some constraints.

Given the constraints, the objective function is to find the optimal number of devices of each type so that the probability for them to get damaged all at the same time is minimized. So we are looking for probability that A, B and C will get damaged at the same time, and we want to minimize it by having spare devices for each type.

Probability that type A will get damaged: 0.3 
Probability that type B will get damaged: 0.2 
Probability that type C will get damaged: 0.4 

Let x = number of devices A
Let y = number of devices B
Let z = number of devices C

Those are the constraints:

10x + 15y + 8z <= 150
30x + 40y + 20z <= 400
15000x + 17000y + 13000z <= 200000

If I am not mistaking, the objective function should look like this.

min P = (prob_damaged_A * x) * (prob_damaged_B * y) * (prob_damaged_C * y)
min P = (0.3x) * (0.2y) * (0.8z)

Is it OK if I transform that with natural logarithm into this?

    min P' = log[0.3x * 0.2y * 0.8z]
    min P' = x*log(0.3) + y*log(0.2) + z*log(0.8)
    min P' = -1.20397x - 1.60944y - 0.91629z 

I am not sure since all the signs have changed from positive to negatives, and the coefficients that were largest are now the smallest. For instance, z had its coefficient 0.4 which was the greatest, and now it is the smallest by absolute value: -0.916. Will the values of x, y, z be correct?

Also, since I am looking for the number of devices of each type, i cant have fractions or irrational numbers as solution. Is there a way I can make a constraint that x, y, z must be integers?

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I suspect this should be: $$ \min P = P_{A}^x \times P_{B}^y \times P_{C}^z $$ Or $$ \min \log(P) = x \log(P_A) + y \log(P_B) + z \log(P_C) $$ The integer restrictions on $x,y,z$ can be added when you use a mixed-integer programming solver. E.g.:

enter image description here

The solution looks like:

enter image description here

This is such a small model that any MIP solver will do. (Note: the option optcr indicates a tolerance: we want to find the global optimum).

Note: this is sometimes called the Reliability–Redundancy Allocation Problem (RRAP). Here we have all components in parallel (the systems fails if all components fail). More interesting configurations with some serial and some parallel components make more exciting optimization problems.

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  • $\begingroup$ Hi Erwin ! Are you familiar with the M-method? $\endgroup$ – Evinda Jan 8 '16 at 10:39
  • $\begingroup$ Yes. Actually there are two of them. One is for textbook simplex methods (this one is largely to bother students: practical simplex methods don't use this mechanism). The other is in formulating certain constructs in mixed-integer models. $\endgroup$ – Erwin Kalvelagen Jan 8 '16 at 11:09
  • $\begingroup$ Could you maybe take a look at my question? cs.stackexchange.com/questions/51607/… $\endgroup$ – Evinda Jan 8 '16 at 12:03
  • $\begingroup$ Just follow the steps in your book. They are usually spelled out in excruciating detail. $\endgroup$ – Erwin Kalvelagen Jan 8 '16 at 12:18

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