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Five-step production line at 400 units per shift: takt, step capacities and the bottleneck, flow time by the Kingman chain against a discrete-event simulation, WIP by Little's law, OEE with the six losses, a nine-task line balance and the newsvendor capacity cushion

Tool	Capacity and Throughput
Source	U.S. Department of Defense (2008). Continuous Process Improvement / Lean Six Sigma Guidebook, Revision 1, section 2.12 (takt time); Little, J. D. C. (1961). "A Proof for the Queuing Formula: L = λW." Operations Research, Vol. 9, No. 3, pp. 383–387; Kingman, J. F. C. (1961). "The Single Server Queue in Heavy Traffic." Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 57, No. 4, pp. 902–904; Helgeson, W. B., and Birnie, D. P. (1961). "Assembly Line Balancing Using the Ranked Positional Weight Technique." Journal of Industrial Engineering, Vol. 12, No. 6, pp. 394–398.
Level	Undergraduate

Problem

A line of five steps (Cut, Machine with two machines, Weld, Paint, Test and pack) with process times 0.6 / 1.4 / 0.6 / 0.4 / 0.5 min, setups 5 / 6 / 5 / 16 / 2 min per batch of 20 / 20 / 20 / 40 / 20, availabilities 0.95 / 0.90 / 0.92 / 0.88 / 0.98, yields 0.99 / 0.98 / 0.99 / 0.97 / 1.0 and CV 0.75 must deliver 400 good units in an 8-hour shift with 30 minutes of planned downtime, released with CV 1.0. Find the takt, each step's capacity and utilization, the bottleneck and line capacity, the flow time and WIP by Kingman's equation chained through the line, the critical WIP and the practical-worst-case point, the OEE of a machine that ran 400 of 450 planned minutes making 360 units (350 good) at a 1.0 min ideal cycle, the minimum stations for a nine-task assembly at the takt, the batch size at the bottleneck and the capacity cushion for a demand standard deviation of 60 with shortage cost 50 and excess cost 10.

Inputs to Enter

Steps (t_0, t_s, k, m, A, Y, CV): Cut 0.6, 5, 20, 1, 0.95, 0.99, 0.75; Machine 1.4, 6, 20, 2, 0.90, 0.98, 0.75; Weld 0.6, 5, 20, 1, 0.92, 0.99, 0.75; Paint 0.4, 16, 40, 1, 0.88, 0.97, 0.75; Test and pack 0.5, 2, 20, 1, 0.98, 1.0, 0.75
Demand 400 per period, available 480 min, planned downtime 30 min, arrival CV 1.0, transfer batch 1
OEE: planned 450 min, run 400 min, ideal cycle 1.0 min, 360 made, 350 good, setup 20 min, minor stops 15 min, start-up rejects 4, calendar 1,440 min
Tasks (min, predecessors): A 0.45; B 0.30 (A); C 0.55 (A); D 0.40 (B, C); E 0.60 (C); F 0.35 (D); G 0.50 (E); H 0.25 (F, G); I 0.65 (H); cycle time = takt
Batch scan at the bottleneck to k = 100 at target utilization 0.85; demand standard deviation 60, shortage cost 50, excess cost 10 per unit

Open the Capacity and Throughput tool (loaded with this example)

Answers

Takt 450/400 = 1.125 min per unit (53.33 per hour); line yield 0.9317, release rate 57.24 per hour
Bottleneck Machine: capacity to the line end 2 × 0.9 / 1.7 × 0.9411 = 59.79 good units per hour, 448.4 per shift (12.1 % above the demand), utilization 0.8921; Cut 0.8536, Weld 0.8552, Paint 0.8331, Test 0.5442
Kingman chain: queue waits 4.08 / 4.62 / 3.43 / 2.64 / 0.42 min (exact M/M/m at the same rates 5.22 / 7.36 / 5.46 / 4.54 / 0.73), wait to batch 9.96 / 10.06 / 10.26 / 21.28 / 10.69 min; flow time 82.67 min, WIP 76.60 units (batching 75 % of the flow time); discrete-event simulation of the queue waits −0.3 / +8.2 / +32.6 / +4.3 / −10.7 % by step, +7.6 % on the total
Critical WIP W_0 = 63.53/h × 5.229 min = 5.536 units; at 76.6 units the practical worst case gives CT 76.6 min against the chain's 82.7 (batching)
OEE = 0.8889 × 0.9000 × 0.9722 = 0.7778 (= 350 × 1.0 / 450); losses 30 / 20 / 15 / 25 / 4 / 6 min; TEEP 0.2431
Line balance at C = 1.125 min: Σt = 4.05, minimum ⌈3.6⌉ = 4; RPW 5 stations (A C | E B | D G | F H | I), largest candidate 4, exact 4 (A C | B D F | E G | H I), balance delay 0.10, smoothness 0.229
Batch at Machine: k ≥ 27.3 for u ≤ 0.85; least time at k = 19 (16.56 min); newsvendor fractile 0.8333, z 0.9674, capacity 458.0 (cushion 14.5 %), expected shortage 5.32, fill rate 98.7 %

Notes

The tool's default inputs are this case; press Submit. The line and the task list are illustrative (stated on the page); test_capacity_throughput.php checks every number against closed forms, the exact M/M/1 and M/M/c queues, an independent numpy Kingman chain, a 1,000,000-unit discrete-event simulation (exact to 0.5 % when every time is exponential), a subset dynamic program for the exact station count on the default and 10 seeded instances, scipy for the newsvendor, and reproduces the EPA Lean and Environment Toolkit (2007) Figure 3 value stream (cycle times 2 / 4 / 7 / 2 min, uptimes 74 / 67 / 48 / 93 %, value-added time 15 min, lead time 68 days, bottleneck Painting at 4.11 per hour).