Capacity and Throughput

Capacity and Throughput

Capacity and Throughput: Takt, Capacity, Kingman's Equation, Little's Law, OEE and Line Balancing

What the tool does

The tool takes a production line or service process as a sequence of steps and answers the questions a capacity analyst asks first: how fast must each unit leave (takt), how much can each step make (capacity), which step limits the line (bottleneck), how long a unit spends in the line and how much work is in process (flow time and WIP), how well one machine is used (OEE), how the tasks of an assembly line divide into stations for the takt (line balancing), how large a process batch should be when setups compete with waiting (batch sizing), and how much capacity above the mean demand is worth holding when the demand is uncertain (capacity cushion). Every method is a closed form or a small exact search; nothing is read from a proprietary table. Times are entered in minutes and rates reported per hour and per period.

Takt time

Takt time is the available production time divided by the customer demand (DoD CPI/LSS Guidebook 2008, section 2.12 and glossary): takt = (T_av − T_dn)/D_p with the planned downtime (breaks, meetings, planned maintenance) removed from the scheduled time. It is the pace the whole value stream has to keep, and the guidebook's work-balance criterion (section 6.4) is that the pace of each step is set by it. A step whose effective time per machine, t_e/m, exceeds the takt cannot keep up on its own.

Effective capacity of a step

Each step has a natural process time per unit t₀ on one machine, a setup time t_s per process batch of k units, m parallel machines, an availability A (the fraction of scheduled time the machines are up) and a yield Y (the fraction of the units processed that go on). The time per unit on one machine is t_proc = t₀ + t_s/k, the effective time including outages is t_e = t_proc/A, and the capacity of the step in its own good output is m A Y/t_proc units per minute. Because scrap at a later step must be replaced by starting more units earlier, the capacity of step i in good units at the end of the line is m A Π_{j ≥ i} Y_j/t_proc; the smallest of these is the line capacity and the step that has it is the bottleneck. The throughput TH is the smaller of the demand rate and the line capacity; the start rate at step i is r_i = TH/Π_{j ≥ i} Y_j and its utilization u_i = r_i t_proc/(m A). The theory-of-constraints view (Goldratt 1984, cited by the DoD guidebook in section 2.13) is that the bottleneck alone sets the throughput: the page highlights it and gives its utilization and cushion.

Flow time: Kingman's equation chained through the line

Each step is treated as a G/G/m queue fed by the departures of the step before it. Kingman (1961) showed that in heavy traffic the mean wait in a single-server queue is approximately (c_a² + c_e²)/2 · u/(1 − u) · t_e, with c_a and c_e the coefficients of variation of the inter-arrival and the effective process time; the tool uses the m-machine form of Sakasegawa (1977), CT_q = [(c_a² + c_e²)/2] · [u^{√(2(m+1)) − 1}/(m (1 − u))] · t_e, which reduces to Kingman's at m = 1 and is exact for an M/M/1 queue. The departures of a step have squared CV c_d² = 1 + (1 − u²)(c_a² − 1) + (u²/√m)(c_e² − 1) (the stationary-interval approximation of Whitt's Queueing Network Analyzer, which Hopp and Spearman call the linking equation); scrapping at yield Y thins the stream to c² → Y c_d² + 1 − Y (a geometric sum of i.i.d. intervals), and the result is the arrival CV of the next step. The entered CV is that of the effective time per unit, so it should include the effect of outages and setups.

A unit at a step with a process batch of k also waits for its batch to form: with units arriving one at a time at rate r the mean wait is (k − 1)/(2r). A transfer batch b_t moved between steps adds (b_t − 1)/2 departure intervals after every step but the last. The flow time of a step is the sum of the wait to batch, the queue wait, the effective process time and the transfer wait; the flow time of the line is the sum over the steps. The exact M/M/m wait at the same rates (Erlang C) is printed beside the Kingman value: the two agree at m = 1 with c_a = c_e = 1, and the comparison shows how much of the wait is due to the variability actually entered.

Accuracy. The chain is an approximation. In the tool's test an independent discrete-event simulation of the default line (one million units, gamma-distributed times) reproduces the chain exactly when every time is exponential (the M/M/1 tandem, exact by Jackson's theorem, and an M/M/2 station) and differs from it by −0.3 % to +33 % at the individual steps and by +7.6 % on the sum of the queue waits and process times when the CVs are 0.75 and one step has two machines; the test records these as the bound of the approximation. Treat the flow-time figures as estimates of that order, good for comparing designs rather than for a guarantee.

Little's law and the line curves

Little (1961) proved that in any stable queueing system the mean number present equals the arrival rate times the mean time present, WIP = TH · CT; the tool applies it at every step with the step's start rate and on the whole line with the release rate. The curves of throughput and cycle time against the WIP are the closed forms Hopp and Spearman describe as the best, worst and practical-worst cases of a line with bottleneck rate r_b (the smallest m/t_e, in started units) and raw process time T₀ (the sum of the t_e): the critical WIP is W₀ = r_b T₀; the best case has CT = T₀ and TH = W/T₀ up to W₀, then TH = r_b and CT = W/r_b; the worst case runs one unit through at a time, CT = W T₀ and TH = 1/T₀; the practical worst case, in which every unit is equally likely to be at any step and the times are exponential, has CT = T₀ + (W − 1)/r_b and TH = W r_b/(W + W₀ − 1). A real line falls between the best and practical-worst curves; the page marks the Kingman estimate on both charts.

Overall equipment effectiveness

OEE (Nakajima 1988) is availability × performance × quality: availability = run time / planned production time, performance = ideal cycle time × total count / run time, quality = good count / total count, so that OEE = good count × ideal cycle time / planned production time, the fraction of the planned time that was fully productive. The six big losses are the stop time (breakdowns; setup and adjustment), the speed loss (idling and minor stops; reduced speed) and the rejects (start-up rejects; process defects), each expressed in minutes of planned time so that they and the productive time add up to the planned time; the split within each pair comes from the optional inputs (setup time, minor-stop time, start-up rejects). TEEP is the OEE applied to calendar time, OEE × planned time / calendar time; the time-element names follow SEMI E10. A performance above 1 means the entered ideal cycle time is slower than the machine actually ran and should be checked.

Line balancing

Given the tasks of an assembly, their times and their immediate predecessors, the tasks are grouped into stations so that no station's work exceeds the cycle time C (the takt unless another value is entered) and every task is placed with or after its predecessors. The theoretical minimum is ⌈Σt/C⌉ stations. The ranked positional weight heuristic (Helgeson and Birnie 1961) gives each task the weight of its own time plus the times of every task that follows it, takes the tasks in order of decreasing weight and places the first eligible task that fits the open station, opening a new station when none fits; the largest-candidate rule does the same in order of decreasing task time. The exact minimum is found by a depth-first branch and bound over the station sets (tasks within a station in increasing index, bound = stations opened + ⌈(remaining work − room in the open station)/C⌉), searched for lists of up to 12 tasks. For every assignment the page gives the station loads, the balance delay (nC − Σt)/(nC), the line efficiency Σt/(nC) and the smoothness index √Σ(max load − load_k)², and draws the precedence diagram coloured by station. The exact result also shows how far the cycle time could be cut with the same stations (the largest station load).

Batch sizing

At a step with a setup per batch, the time per unit t₀ + t_s/k and hence the utilization fall as the batch grows, while every unit waits longer for its batch to form. The smallest batch that holds the utilization to a target u_t follows from r (t₀ + t_s/k)/(m A) ≤ u_t, k ≥ t_s/(u_t m A/r − t₀), when the right-hand denominator is positive (otherwise no batch reaches the target). The batch that minimizes the time at the step is found by evaluating the wait to batch plus the Kingman queue wait plus the effective time at every k from 1 to the scan limit and taking the least; the chart shows the two opposing terms. The scan is for the chosen step (the bottleneck by default) at its own start rate and arrival variability with one-piece transfer.

Capacity cushion

When the demand in a period is uncertain, holding capacity above the mean demand trades the cost of idle capacity against the cost of unmet demand. With a normal demand of mean D_p and standard deviation σ_D, a shortage cost c_u per unit short and an excess cost c_o per unit of capacity unused, the expected cost is minimized at the newsvendor quantity (Arrow, Harris and Marschak 1951): the capacity C with Φ(z) = c_u/(c_u + c_o), C = D_p + z σ_D. The cushion is (C − D_p)/D_p; the expected shortage is σ_D L(z) with the standard normal loss function L(z) = φ(z) − z(1 − Φ(z)), the expected unused capacity C − D_p + σ_D L(z), and the fill rate 1 − σ_D L(z)/D_p. The normal distribution is evaluated by Marsaglia's (2004) series and the Mills-ratio continued fraction.

Reading the results

The capacity table names the bottleneck and shows the utilization and the takt check of every step; a demand above the line capacity is reported as capacity-limited and the queueing section is then given only for the steps that remain below full utilization. The flow-time table separates the wait to batch, the queue wait and the processing at every step, so the reader sees whether cycle time is driven by batching (lean's usual finding), by variability (the queue column, compared with the exponential value beside it) or by the work itself. The line curves place the estimate between the best and practical-worst cases. The OEE section shows where the planned time of one machine went. The balancing tables compare the two heuristics with the exact minimum, and the batch and cushion sections give the decisions a planner takes from the same data.

Limits

The queueing chain assumes steady state, unlimited buffers, first-come service and renewal arrivals; a WIP cap is only flagged, not modelled (blocking would lower the throughput). The entered CV is that of the effective time per unit; the inflation of variability by long, rare outages (the preemptive-outage formula) is not computed from a repair time and should be reflected in the entered CV. Yields are scrap, not rework. The line curves describe a serial line of single-stream steps in started units. Line balancing handles a single-model line with deterministic task times and no zoning or incompatibility constraints; the exact search is limited to 12 tasks. The newsvendor cushion is a one-period decision with a normal demand and linear costs. The EPA Lean and Environment Toolkit's Figure 3 value stream (milling, welding, painting, assembly: cycle times 2 / 4 / 7 / 2 min, uptime 74 / 67 / 48 / 93 %, lead time 68 days, value-added time 15 min) is reproduced in the test script as a published data set for the capacity and Little's-law arithmetic.