Vocational Centre Batch Planning
A district vocational centre is planning short courses in solar technician training and food processing. Each solar batch needs 30 m2 of lab space, 2 instructors and INR 40,000 of equipment. Each food-processing batch needs 20 m2, 1 instructor and INR 60,000. The centre has 240 m2, 12 instructors and INR 480,000. Demand and policy require at least 3 solar batches and at least 2 food-processing batches, and the total number of batches cannot exceed 8. A solar batch enrols 25 students; a food-processing batch enrols 30. The centre must choose whole-number batches. The manager wants the plan that maximises total enrolment, and also wants to know which resource, if any, is left unused and whether adding one more instructor would change the optimum.
1. Write the constraints and find the feasible whole-number plan that maximises total enrolment.
Read the worked answer
Let S = solar batches and F = food-processing batches.
Constraints: S >= 3, F >= 2, S + F <= 8 Space: 30S + 20F <= 240, or 3S + 2F <= 24 Instructors: 2S + F <= 12 Equipment: 40000S + 60000F <= 480000, or 2S + 3F <= 24
Enrolment E = 25S + 30F.
Because food processing gives more students per batch, use the total batch cap fully: F = 8 - S. Then check: Instructors: 2S + (8 - S) = S + 8 <= 12, so S <= 4. Space: 3S + 2(8 - S) = S + 16 <= 24, so S <= 8. Equipment: 2S + 3(8 - S) = 24 - S <= 24, which holds for S >= 0. With S >= 3 and F >= 2, S can be 3 or 4.
S = 3, F = 5: E = 75 + 150 = 225. S = 4, F = 4: E = 100 + 120 = 220.
Maximum enrolment is 225 students with 3 solar and 5 food-processing batches.
2. At the optimum, calculate the unused amount of each resource. Would adding one more instructor change the optimum?
Read the worked answer
At S = 3, F = 5: Space used = 30(3) + 20(5) = 90 + 100 = 190 m2, so 240 - 190 = 50 m2 unused. Instructors used = 2(3) + 5 = 11, so 12 - 11 = 1 instructor unused. Equipment used = 40000(3) + 60000(5) = 120000 + 300000 = 420000 INR, so 480000 - 420000 = 60000 INR unused.
Adding one instructor gives 13 instructors and the constraint 2S + F <= 13. With S + F <= 8, this becomes S + 8 <= 13, so S <= 5. But the optimum still uses only 11 instructors at S = 3, F = 5. The binding limit is the total batch cap of 8, not instructors. Extra instructor capacity alone does not increase enrolment unless the batch cap is also relaxed.
3. If the total batch cap rises to 9 while the other constraints stay the same, find the new optimum and identify which constraints are binding.
Read the worked answer
Use F = 9 - S to use the new cap fully.
Instructors: 2S + (9 - S) = S + 9 <= 13, so S <= 4. Space: 3S + 2(9 - S) = S + 18 <= 24, so S <= 6. Equipment: 2S + 3(9 - S) = 27 - S <= 24, so S >= 3. Also S >= 3 and F = 9 - S >= 2, so S <= 7. Thus S = 3 or 4.
S = 3, F = 6: E = 75 + 180 = 255. S = 4, F = 5: E = 100 + 150 = 250.
The new optimum is 3 solar and 6 food-processing batches, with 255 students. Equipment is exactly used: 2(3) + 3(6) = 24. Instructor capacity has 1 unused: 2(3) + 6 = 12 out of 13. Space used is 3(30) + 6(20) = 210 m2, leaving 30 m2 unused. The binding constraints are the total batch cap and equipment; instructors and space still have slack.