We can quite easily compute the optimal amount of WIP for each production. At the same time the WIP in use notifies the performance of our production when compared to the optimal, worst-case and practical worst-case.

## Terms

Before going any deeper we must look over the terms used in this article.

**TH**

Throughput (TH) means the production that comes out of the line. Usually products/hour.

**WIP**

Work in Process illustrates the parts in production line.

**r _{b}**

Bottleneck speed (r

_{b}) is in parts/time -unit and it is the workstation that has the worst throughput in the long run. It is the mean value after all outages.

**CT**

Production time, cycle time or throughput time illustrates the time that product spends in the production line. It is the time that parts are in WIP condition. From the time they are taken into the production all the way until they are ready.

**Utilization**

Utilization is the work load on workstation. It is calculated from the formula *incoming parts per hour / capacity per hour*.

**Raw process time, T _{0}**

Raw process time (T

_{0}) is the sum of all workstations mean process time. This is only the process time and does not include queuing time.

**Critical WIP, W _{0}**

Critical WIP (W

_{0}) illustrates the amount of WIP that is needed to have the maximal throughput in stabilized environment with no variability. W

_{0}= r

_{b }* T

_{0}.

Depending on calculation period we must include different variables. Example when calculating T_{0} and r_{b} values in long term scheduling we must consider all outages and failures. In short term scheduling we notice only most frequent outages and repair times.

## Bottlenecks in production line

Bottleneck is usually the workstation which is most utilized. In simple production line this is the station which is the slowest. In more complicated line where there is multiple workstations doing same operation or there is loss, bottleneck is not necessarily the slowest workstation. Quicker workstation with bigger incoming rate can be more utilized and is the bottleneck. That’s why the bottleneck should be determined by utilization.

Think about simple production line with two workstations. 1. station needs 1 minute and second 2 minutes to do its job. r_{b} for the first station is 1 and for the second staion 0,5 (0,5 products per hour).

So second station is the bottleneck? If only x percent of the production from the first station is accepted and goes to the second station the utilization for the first station is r/1 = r. 1 – x percent is rejected after the first station. Incoming rate for the second station is x percent so utilization for the second station is xr/0,5 = 2xr.

If acceptance rate is under 50 % then the utilization for the first station is higher. Now when we add the incoming rate r and acceptance rate x stays under 50 % then the first station is overloaded before the second station.

#### Example line

Lets take an example of production line with 4 workstation and 2 hours process time each.

Our line is perfect line where is no loss and it works 24 hours every day.

Rate for each station is r = 0,5 products per hour or one product per 2 hours. Because there is no loss all the stations work with same utilization and bottleneck rate r_{b} = 0,5.

Line is called balanced line because all the stations have same capacity. Raw process time T_{0} is simple the sum of process times on each station.

T_{0} = 8 h.

And critical WIP W_{0} = r_{b}* T_{0} = 0,5 * 8 = 4 units.

This means that at all times we have 1 product on each workstation. We can increase the WIP but it does not increase the throughput because every station can do only 0,5 units per hour. Added WIP is added to the queue in front of the first station and they increases the cycle time CT.

#### Example line 2

For the second example we have unbalanced line where process times are different and each operation has different amount of machines.

Capacity calculations for unbalanced line are little more complex because of the multiple stations for each operation. Example in third operation capacity for the one station is 1/10 products per hour and capacity for the whole third operation is 6 * 1/10 = 0,6 products per hour. Simpler calculation would be the machines divided by process time.

TH for the whole production line can still be computed from the bottleneck rate or from the slowest machine. In this case it is the 2. operation and bottleneck rate r_{b} = 0,4 products per hour.

Notice that bottleneck is not the operation where are the slowest machines (phase 3) nor the phase where is least machines (phase 1). Raw process time T_{0} is still the sum of process times.

T_{0} = 2 + 5 + 10 + 3 = 20 hours.

Critical WIP is also computed the same way.

W_{0} = r_{b}* T_{0} = 0,4 * 20 = 8 products.

In unbalanced line W_{0} is typically less than the amount of machines/workstations. It illustrates the WIP level where maximum production is achieved. However critical WIP applies only in production line with no variability and is therefore almost never the optimal situation.

## Little’s law

WIP = TH * CT

This Little’s law illustrates the amount of WIP and is applicable in all situations.

Formula is applicable also for the single workstation and from the second example we can compute this:

WIP = TH * CT = 0,4 producst/hour * 2 hours = 0,8 products

Because there is only 1 machine for the first operation its utilization is 80 %. Correspondinly in third operation we can input 4 products when the mean utilization is 66,7 % (4 products / 6 machines). we can compute this also from the bottleneck rate and production capacity. In third operation 0,4 / 0,6 = 0,667.

#### Best Case

From the Little’s law we can lead formula for the best production time possible. In this case *w* is for the WIP-level in use.

If w ≤ W_{0} then CT_{best} = T_{0}, otherwise CT_{best} = w/r_{b}

And maximum production TH_{best} = w/T_{0}, if w ≤ W_{0}, otherwise TH_{best} = r_{b}.

#### Worst Case

And the worst possible case is:

CT_{worst} = w * T_{0}

TH_{worst} = 1/T_{0}

Worst case can happen if production from the workstation is moved in batches. In best case scenarion products are immediately moved from workstation to another and there is no interim storage.

#### Practical worst case

In reality the worst case is seldom the case. On the other hand best case is almost impossible to achieve. That’s why we will now compute the practical worse case scenario that can be used as an reference to compare the reality in your production.

CT_{PWC} = T_{0} + [ (w-1) / r_{b} ]

TH_{PWC} = [ w / ( W_{0} + w -1)] * r_{b}

PWC formulas forms an reasonable mean value that indicates many implementations in reality.

## Bottlenecks

Bottleneck rate r_{b} is crucial because it defines the TH for the whole production.

If production performs better than PWC we can find it to be quit good. Otherwise there will be something to improve. From the formulas we can find what should be improved. Example when increasing the bottleneck rate our cycle time will decrease. But this is not always financially reasonable if we would have to buy second machine etc.

Best place to achieve results is always the bottleneck rate and bottleneck station. If this is not realistic we can achieve results also by improving the speed of other stations.

#### Workers as an bottleneck?

Formulas mentioned above are applicable when we have enough workforce and machines are the bottleneck. Example every machine has one employee. In some businesses we can have situation where we have plenty of machines but only few workers. Example machines are cheap (computers) compared to the bottleneck they might create. We have only limited amount of workers and they switch machines depending on the on going operation. Workers will be the bottleneck if we have enough machines.

In this case we can compute our production from the formula TH_{max} = n/T_{0}, where n is the amount of workers and T_{0} is still the raw process time.

Formula is not applicable if workers can do multiple operations in the same time. Example monitor multiple CNC machines. Formula is applicable only if one worker does one job at a time.

TH in the formula implies the best production possible. It seldom is reality because of the different queuing times etc. Example some operation are allowed or skilled to do only part of the workers.

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