Carton Mix with a Mandatory Print Job
Meera runs a packing unit with a cutting machine and a sealing machine. Each small carton (S) needs 4 minutes of cutting and 3 minutes of sealing; each large carton (L) needs 6 minutes of cutting and 5 minutes of sealing. The cutting machine is available for 600 minutes a day and the sealing machine for 480 minutes. Contribution is ₹18 per S carton and ₹30 per L carton. A new print job, if accepted, reserves 120 minutes of cutting time every day and pays a fixed job fee of ₹350 per day; it uses no sealing time. Meera's current plan is 60 S and 40 L cartons a day. Her supervisor argues that because the current plan already fits within capacity, accepting the print job cannot hurt. Overtime can add either 90 cutting minutes or 60 sealing minutes per day for a flat cost of ₹300, not both.
1. Is the supervisor right that accepting the print job cannot hurt? Determine the best carton mix with the job and compare total daily contribution with and without the job.
Read the worked answer
Current plan: cutting = 60×4 + 40×6 = 480 minutes; sealing = 60×3 + 40×5 = 380 minutes. If the print job is accepted, cutting capacity falls to 600 − 120 = 480 minutes, so the current plan just fits. Feasibility is therefore not the issue.
With the job, cutting is the binding constraint at 480 minutes. Contribution per cutting minute is S = 18/4 = 4.5 and L = 30/6 = 5, so make as many L as possible: 480/6 = 80 L, using 400 sealing minutes (within 480). Contribution = 80×30 + 350 = 2,750.
Without the job, solve 4S + 6L = 600 and 3S + 5L = 480. Multiplying the first by 3 and the second by 4 gives 12S + 18L = 1,800 and 12S + 20L = 1,920, so 2L = 120, L = 60 and S = 60. Contribution = 60×18 + 60×30 = 2,880. All-L is not feasible because 100 L would need 500 sealing minutes.
So accepting the job lowers the best achievable contribution by 130 (2,750 vs 2,880). The supervisor is wrong: the job is feasible but consumes scarce cutting time. It only looks harmless if one compares it with the existing, non-optimal plan of 2,280.
2. If the print job is already contracted and cannot be cancelled, should Meera buy the cutting overtime block at ₹300? Show the calculation.
Read the worked answer
Baseline with the job is 80 L plus the job: contribution 2,750.
Cutting overtime adds 90 cutting minutes, taking cutting capacity from 480 to 570; sealing stays at 480. Again, L earns more contribution per cutting minute, so make 570/6 = 95 L. Sealing used = 95×5 = 475 minutes, which is within 480. Contribution = 95×30 + 350 = 3,200.
Incremental contribution = 3,200 − 2,750 = 450. Cost of the overtime block = 300. Net gain = 150. Yes, buy the cutting overtime block, provided the ₹300 is the full incremental cost and the extra cartons can be sold.
3. What is the most Meera should pay for the sealing overtime block if the print job is contracted? Explain.
Read the worked answer
Nothing, or at most ₹0. With the job, cutting capacity is 480 minutes and sealing capacity is 480 minutes. The best plan is 80 L, which uses 480 cutting minutes but only 400 sealing minutes, leaving 80 sealing minutes idle. Cutting, not sealing, is the binding constraint.
Adding 60 sealing minutes cannot increase output because there is no cutting time left to make another carton. The extra sealing capacity therefore has zero value. Any positive payment for it would reduce contribution.