Lot Sizing When the Textbook EOQ Ignores a Setup Constraint
A plant makes a bracket with annual demand 24,000 units, stable at 100/day over 240 working days. Setup cost is estimated at $180, holding cost $4/unit/year, and unit cost $12. The current lot size is 1,200 units (about 12 days of demand). A single machine produces the bracket at 300/day. Setup takes 90 minutes, and the machine is the only one capable of a downstream drilling operation that runs 6 hours/day. The planner proposes raising lot size to 2,400 to 'cut setups in half.' A recent changeover caused a 2-hour overrun and a stockout.
1. Compute the EOQ and the economic production quantity, and compare with the proposed 2,400.
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EOQ: sqrt(2DS/H) = sqrt(2×24,000×180/4) = sqrt(2,160,000) ≈ 1,470 units.
EPQ (production runs): EPQ = sqrt(2DS/H × p/(p−d)), where p=300/day, d=100/day. p/(p−d) = 300/200 = 1.5. So EPQ = 1,470 × sqrt(1.5) ≈ 1,800 units.
The proposed 2,400 is 33% above EPQ, raising holding cost with only a modest setup reduction. The current 1,200 is below EOQ, so setups are more frequent than optimal - but the real constraint is machine time, not the cost tradeoff.
2. Check whether the machine has capacity for the setups at each lot size, and recommend a lot size.
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Machine run time per unit = 1/300 day. Annual run time = 24,000/300 = 80 days. Available machine time: the machine also supports drilling 6 h/day, so usable production time is 240 × (6/24) = 60 days - already less than the 80 days of run time required. This is a capacity infeasibility, not a lot-sizing problem.
Setup time at lot 1,200: 20 setups × 1.5 h = 30 h. At 2,400: 10 × 1.5 = 15 h. Neither fixes a 20-day shortfall.
Recommendation: the lot size question is moot until capacity is resolved. Either add machine capacity, reduce the drilling allocation, or reduce setup time (SMED) to free hours. If capacity were adequate, EPQ ≈ 1,800 is the cost-optimal lot; 2,400 is not justified.
3. The 2-hour changeover overrun caused a stockout. What does this reveal about the lot-size decision?
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It reveals that setup variability, not average setup time, drives stockouts. A 1,200-unit lot gives ~12 days of cover; a 2,400 lot gives ~24 days, which would buffer variability - but at higher holding cost and only if capacity exists. The better fix is to reduce setup time and variability (standardize the changeover, pre-stage tools), which shrinks both the mean and the tail. Lot sizing should be set after setup time is stabilized, using safety stock or buffer to cover remaining variability rather than inflating lot size as a substitute for process control.