I am working on a linear programing problem.

There are 150 maintenance technicians each with their own specialty. Their jobs are Power Plants, Airframes, Electricians, Avionics, and Ordnance. There are 3 shifts, Day, Night, and Mid-shift. There are 18 maintenance rooms available for them to work in. However, 2 rooms can only be used by Powerplants, 2 rooms can only be used by Airframes, and the rest of the 14 rooms can be used by anyone as needed.

The breakdown of the 150 technicians is as follows: 25 Power Plant Technicians, 25 Airframes, 25 Electricians, 30 Avionics, and 45 Ordnance technicians. Power Plants, Airframes, Electricians, and Avionics can only help ordnance technicians when there is available manpower, however Ordnance cannot help the technicians due to qualification issues.

There are maintenance tasks that must be completed by each specialty. Every specialty is allotted 4 and 8 hours to complete task as assigned. Task that does not meet that deadline are completed by the next oncoming shift. Each task requires a minimum of 2 technicians to start.

There are 15-eight-hour task and 40-four-hour task. They are allotted to the specialties as follows:

Airframes has been allotted 2-eight-hour task and 5-four-hour task

Power Plants has been allotted 2- eight-hour task and 5-four-hour task

Electricians has been allotted 10-four-hour task

Avionics has been allotted 1-eight-hour task and 10-four-hour task

Ordnance has been allotted 10-eight-hour task and 10-four-hour task

All 8-hour task must be started on the day shift and is allowed to carry over to the night shift.

All 4-hour task can be started on day shift or night shift and is allowed to be carried over to the mid-shift.

No task is allowed to be started at mid-shift however this shift must have minimum staffing available to cover any spillover from night shift and needs minimum staffing to cover one of every type of task per specialty in the event of emergencies. So, Airframes 1 eight-hour task and 1 four-hour task, power plants1 eight hour task and 1 four hour task, etc.

The goal: Create a linear program that will maximize technicians across all rooms and cover all maintenance task. How many technicians should go on Day shift, Night Shift, and Mid Shift for every maintenance room available?

This is my attempt to solve below:

LetA1 = Airframe Day Shift

A2 = Airframe Night Shift

A3 = Airframe Mid-Shift

P1 = Power Plant Day Shift

P2 = Power Plant Night Shift

P3 = Power Plant Mid-Shift

E1 = Electrician Day Shift

E2 = Electrician Night Shift

E3 = Electrician Mid-Shift

V1 = Avionics Day Shift

V2 = Avionics Night Shift

V3 = Avionics Mid-Shift

B1 = Ordnance Day Shift

B2 = Ordnance Night Shift

B3 = Ordnance Mid-Shift

AB1 = Airframe techs to help Ordnance Techs Day

AB2 = Airframe techs to help Ordnance Techs Night

AB3 = Airframe techs to help Ordnance Techs Mids

PB1 = Power Plant Techs to help Ordnance Techs Day

PB2 = Power Plant Techs to help Ordnance Techs Night

PB3 = Power Plant Techs to help Ordnance Techs Mid

EB1 = Electrician Techs to help Ordnance Techs Day

EB2 = Electrician Techs to help Ordnance Techs Night

EB3 = Electrician Techs to help Ordnance Techs Mid

VB1 = Electrician techs to help Ordnance Techs Day

VB2 = Electrician techs to help Ordnance Techs Night

VB3 = Electrician techs to help Ordnance Techs Mid

X = 8-hour task

Y = 4-hour task

Maximize: A1 + A2 + A3 + P1 + P2 + P3 + E1 + E2 + E3 + V1 + V2 + V3 + B1 + B2 + B3

S.T.

1) A1 + A2 + A3 + P1 + P2 + P3 + E1 + E2 + E3 + V1 + V2 + V3 + B1 + B2 + B3 + AB1 + AB2 + AB3 + PB1 + PB2 + PB3 + EB1 + EB2 + EB3 + VB1 + VB2 + VB3 = 150 {This represents all available technicians across the entire shifts}

2)A1 + A2 + A3 < or = 25 {This represents total Airframe Techs

3) P1 + P2 + P3 < or = 25 {This represents total Power Plant Techs

4) E1 + E2 + E3 < or = 25 {This represents total Electrician Techs

5) V1 + V2 + V3 < or = 30 {This represents total Avionic Techs

6) B1 + B2 + B3 < or = 45 {This represents total Ordnance Techs

7) X + A1 > or = 2 {Represents that task X requires a minimum of 2 airframe techs on days

8) X + A2 > or = 2{Represents that task X requires a minimum of 2 airframe techs on Nights

9) X + A3 > or = 2{Represents that task X requires a minimum of 2 airframe techs on Mid-shift

10) X + P1 > or = 2

11) X + P2 > or = 2

12) X + P3 > or = 2

13) X + V1 > or = 2

14) X + V2 > or = 2

15) X + V3 > or = 2

16) X + B1 > or = 2

17) X + B2 > or = 2

18) X + B3 > or = 2

19) Y + A1 > or = 2

20) Y + A2 > or = 2

21) Y + A3 > or = 2

22) Y + P1 > or = 2

24) Y + P2 > or = 2

25) Y + P3 > or = 2

26) Y + E1 > or = 2

27) Y + E2 > or = 2

28) Y + E3 > or = 2

29) Y + V1 > or = 2

30) Y + V2 > or = 2

31) Y + V3 > or = 2

32) Y + B1 > or = 2

33) Y + B2 > or = 2

34) Y + B3 > or = 2

35) X + A1 + A2 < or = 2 {This represents total 8-hour task for Airframes across 2 shifts

36) X + P1 + P2 < or = 2 {This represents total 8-hour task for Power Plants

37) X + V1 + V2 < or = 1{This represents total 8-hour task for Avionics

38) X + B1 + B2 < or = 10 {This represents total 8-hour task for Ordnance

39) Y + A1 + A2 + A3 < or = 5

40) Y + P1 + P2 + P2 < or = 5

41) Y + E1 + E2 + E3 < or = 10

42) Y + V1 + V2 + V3 < or = 10

43) Y + B1 + B2 + B3 < or = 10

44) X = 15 {This is the total amount of 8-hour task

45) Y = 40 {This is the total amount of 4 hour task

I tried inputting this into solver for excel or QM for Windows. I get no feasible solution.