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Production (profit maximization) problem

A company can produce four different types of devices identified as A, B, C and D. The minimum amounts of devices are Amin, Bmin, Cmin and Dmin. The maximum amounts are defined as Amax, Bmax, Cmax and Dmax. The prices of the devices are Ac, Bc, Cc and Dc:

Device	Min amount	Max amount	Price, $
A	650	1000	4.5
B	1700	2200	5.5
C	1100	1400	6.5
D	880	1300	7.0

There are three machines available to produce the devices: M1, M2 and M3. Each machine can operate from Tmin to Tmax minutes:

Machine \ Time	Tmin, min	Tmax, min
M1	55000	65000
M2	60000	69000
M3	62000	68000

The time required to produce each devices on each machine (T) are given in the table (in minutes):

Machine \ Device	A	B	C	D
M1	10	12	8	18
M2	14	11	10	17
M3	13	10	11	20

So, time to produce device A on machine M1 is denote as T_A_M1 and so on. The amounts of devices are required to be found that maximize the profit. The decision variables are amount of devices of each type: Ax, Bx, Cx and Dx.

The objective function is sum of multiplications of devices amounts and their costs:
F = Ax * Ac + Bx * Bc + Cx * Cc + Dx * Dc

There are several constraints: 
T_A_M1 * Ax + T_B_M1 * Bx + T_C_M1 * Cx + T_D_M1 * Dx >= Tmin for M1 T_A_M1 * Ax + T_B_M1 * Bx + T_C_M1 * Cx + T_D_M1 * Dx <= Tmax for M1 and etc.

The problem has been solved successfully with the following plan:

Device	Amount
A	650
B	2200
C	1400
D	1107

The Total Profit is $31877.5

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