# 11.4: Summary

- Page ID
- 31008

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Manufacturing work cells can be designed using standard calculations. Simulation can be used to validate that the designed cell will operate as intended, in part using a trace of worker actions. The effect of random behavior, such as random times between arrivals and random walking times, can be assessed. WIP levels can be estimated. The use of alternative numbers of workers can be evaluated.

**Problems**

- Compare the cellular manufacturing organization presented in this chapter with the serial line discussed in chapter 7.
- Write down the task sequence for Worker 2 for worker assignment A.
- Model using pseudo - code part of the process for Worker 2 from picking up a part at workstation M2 through arriving at workstation M4, for worker assignment A.
- For the case of two workers and random time between arrivals, estimate the average cycle time for a part to traverse the cell using Little's Law. The cycle time is the time between entering the raw material inventory and entering the FGI. Suppose the average WIP in the cell is 30.8 with a 95% confidence interval for the mean: (22.6, 38.9).
- Does the work cell behave like a CONWIP system? Why or why not?
- An extremely long time between arrivals, say triple the mean, is possible when using the exponential distribution to model this quantity. What is the potential effect of such long times between arrivals on the capacity of the cell?
- Consider the probability distribution of the time worker two takes to complete all tasks once. Assume walking times are normally distributed with same mean and variance as the triangularly distributed times. The mean and variance of a triangular distribution are computed as follows:
Mean: $$\frac{\min +\bmod e+\max }{3}\nonumber$$

Variance: $$\frac{\min ^{2}+\bmod e^{2}+\max ^{2}-\min ^{*} \bmod e-\min ^{*} \max -\bmod e^{*} \max }{18}\nonumber\]- The time for worker two to complete all tasks once is normally distributed. Compute the mean and standard deviation of this distribution. Assume that the minimum is 75% of the mean and the maximum is 125% of the mean as stated on page 11-5. Thus, the distribution is symmetric implying mean = mode.
- What is the probability that the time to complete all tasks once is greater than the takt time?

- Perform a gross capacity analysis for each station in the cell. This means computing the maximum number of parts each station can produce in one work day.
- Print out a trace of the events effecting worker 2 in task assignment A. Does this provide validation evidence?
- Add an additional performance measure to the model: The percent of times a worker traverses an assigned route in more than the takt time. Rerun the model to estimate this performance measure.
- Model the time between arrivals as gamma distributed with mean 55.2 seconds and standard deviation 27.6 seconds. Compare the maximum WIP in the cell to the values in Table 11-4.

**Case ****Problem**^{2}

An injector is produced in two steps: assembly and calibration. This study will focus on the calibration area only. The assembly area can produce a batch of 24 parts in 82 minutes. Each batch is placed on a WIP cart. A batch is only produced if a WIP cart is available. Injectors must be cured for 24 hours after assembly before they can enter the calibration area.

To control work in process, the number of WIP carts is limited to the fewest number need to avoid constraining throughput. Only one WIP cart can be in the calibration area at a time.

The calibration area consists of four workstations that can be labeled W1, W2, W3, and W4. Each workstation processes one injector at a time. A worker is not needed for automated operations and thus is free to due other tasks.

At workstation W1, the worker initiates the injector in 25 seconds. The workstation performs an automated test in 10 seconds. Finally, the worker removes the part in 5 seconds. A manual operation is performed at workstation W2. The operation time is triangularly distributed with minimum 4.0 minutes, mode 5.0 minutes, and maximum 7.8 minutes. At workstation W3, the worker initiates the part in 5 seconds. An automated operation is performed in 4.1 minutes. The worker removes the part in 2 seconds. Workstation W4 is a packing operation performed by the worker in 5 seconds.

The calibration area is served by one worker. Worker walking times between stations are as follows:

Station | W1 | W2 | W3 | W4 |

W1 | 0 | 3 | 7 | 10 |

W2 | 0 | 4 | 7 | |

W3 | 0 | 3 | ||

W4 | 0 |

Determine the number of WIP carts required. Generate a trace of worker tasks to validate the model.

Case Problem Issues

- How should the WIP carts be modeled?
- How should the constraint on the number of WIP carts in the calibration area be modeled?
- How should the injector curing requirement be modeled?

^{2} This application problem is derived from the capstone masters degree project performed by Carrie Grimard.

- Write down the sequence of tasks for the calibration area worker.
- Discuss how the number of WIP carts can effect cycle time.
- Beside throughput, are any other performance measures important? If so, what are they?
- An entity moving through the assembly area represents a WIP cart while an entity moving through the calibration area represents an individual injector. How is the conversion from WIP cart to injector accomplished?
- Specify the experimental strategy for determining the number of WIP carts.
- Discuss how to obtain verification and validation evidence.
- Determine how to model arrivals to the assembly process.
- Determine how to model batch processing times in the assembly area. Note that the batch processing times are the sum of 24 individual processing time. Should this sum be explicitly computed? Can the central limit theorem be applied?
- Compute the expected number of WIP carts needed to maximize throughput.
- What are the initial conditions for the experiment?
Terminating Experiment: The simulation time interval is 5 days (120 hours of work time).