A binomially distributed production process revisited: a pedagogical approach.

Sutterfield, J.S. ; Nkansah, Paul

INTRODUCTION

The economics of rework/re-manufacturing in production systems is very important since it affects the time and cost to achieve a required production yield. The goal is to achieve the required production yield within the expected time at a minimum cost. A number of research papers have attempted to address various aspects of this issue. For instance, Abdel-Malek and Asadathorn (1996), Inderfurth and Teutner (2003), Flapper, et al (2002), and Flapper and Teutner (2003), studied planning and control problems of rework; Inderfurth, et al (2003) and Lindner, et al (2001) examined lotsizing issues of rework; and Inderfurth, et al (2005), Chase, et al (2006), and Nevins and Whitney (1989) discussed costs associated with rework. In other papers, Nasr, et al (1998) showed that the area of remanufacturing employs a very significant number of workers in the United States and Lund (1998) discussed the fact that remanufacturing accounts for a very significant portion of the business volume in the United States. Also, Inderfurth and van der Laan (2001) analyzed the relationship between lead-time and stochastic inventory control in remanufacturing. Similarly, Tang and Grubbstrom (2005) examined stochastic lead times and deterministic demands in manufacturing/remanufacturing with returns.

Of interest in this paper is the problem of rework in production systems producing relatively large numbers of products in a single lot, using a stable manufacturing system and performing quality sampling at the manufacturing site. Often, in such production systems, quality sampling is performed less frequently and is usually deferred until the time of shipment of a lot, at which time a sample is drawn to assess the quality of the lot. Also, for such systems, when the probability of producing a defective unit is low, it is desirable to know how many production units must be run in order to reasonably expect to have an adequate number of good items (i.e., less defectives) to satisfy a given production quota. In fact, we would want to size the production run such that it has a high probability, say 95% or 99%, of satisfying the required production quota.

We first consider the single-level process used by Nevins and Whitney (1989) in their discussion of the effect of recycling and rework. The process is shown schematically in Figure 1.

[FIGURE 1 OMITTED]

The following variables for this model are defined as follows:

q--required production yield

M--unit material cost

u--number of units which must be processed to achieve "q"

P--processing cost, including testing

w--number of units which must be reworked

R--cost of rework and/or material replacement

y--capability of process P

In the above process, it is assumed that "q" units of material are started through the production process, each having a cost of "M." As each undergoes the production process, it is tested, and incurs a cost of "P" for production and testing. Some of the units processed will be found good, and contribute toward satisfying the required production yield "q." However, a number "w" of these units will be found defective, and will either have to be reworked or replaced. Thus, "w" is comprised of those units that can be successfully reworked, and some number of makeup units to account for those units that cannot be successfully reworked. Then, as this system arrives at steady state, the number of units "u" to pass through the production process is ...

u = q + w

Further, Nevins and Whitney describe the process as having a binomial distribution with the number of good items as the binomial random variable and the probability of a good item as the process capability "y," which is expressed as ...

y = q + 1/q + m + 1,

where "m" is the mean number of units failing to pass the inspection process.

As indicated earlier, the above process assumes a binomial distribution where the production yield is set equal to the mean of the binomial variable. Thus, if "n" is the number of units entering the process and "q" is the required production yield, then q=n*y. From this relationship, the number of units needed to achieve the required production yield is n=q/y. We argue that this number is too conservative. To mitigate this problem, the above model allows rework of defective units. However, it does not incorporate the number of salvageable units in the formula for determining the number of input units. Thus, while reworking defective units might reduce the number of units actually used to satisfy the required production yield, there would still be a problem of oversupply of input units.

In this paper, we revisit the Nevins and Whitney's single-level process of Figure 1 and reexamine the nature of the process, taking into account salvageable units, to determine a more reasonable number of input units that will result in a given production yield.

METHODOLOGY

The question that is to be answered with this analysis is that given a certain Yield "Y," that must be met by a manufacturing process "M," having capability "c," and given a certain number of times through the rework cycle "R," how many units of "N," must be started? Figure 2 below depicts this process, and is exhibited as the basis for developing the pertinent equations for the proposed approach to the production rework cycle.

As "N" units make their first pass through "M," c*N of them contribute toward meeting the quota "Y," and (1-c)*N are found to be defective and are sent through the rework cycle as indicated by "R." Of those units sent through the rework cycle, a fraction "w" can be reworked and "1-w" cannot be. It is evident that if the production process were perfect, viz, if it had a capability "c = 1," that Y would be equal to N and only one pass would be necessary for each unit produced in order to satisfy the production quota "Y."

[FIGURE 2 OMITTED]

The following variables for this model are defined as follows:

N--number of units required to realize a production yield of Y, where N [greater than or equal to] Y

M--Manufacturing and testing process

R--rework cycle

c--capability of process

Y--required production yield

w--fraction of rejected units that can be reworked

Next, the (1-c)*N defective units are processed through "M" a second time. Again, some of them are found to be good and contribute toward meeting the production quota "Y" while others are found still to be defective. The number of good units contributing toward satisfying "Y" can be tabulated as a function of the number of passes through the rework cycle. This is as follows:

Reworkable

Passes through P Good Units Defective Units Units

1 c*N (1 - c)*N (1 - c)*N*w

2 c*(1 - c)*N*w[(1 - c).sup.2]*N*w [(1 - c).sup.2]*N*[w.sup.2]

3 c*[(1 - c).sup.2]*N*[w.sup.2] [(1 - c).sup.3]*N*[w.sup.2] (1 - c)3*N*[w.sup.3]

p c*[(1 - c).sup.p-1]*N*[w.sup.p-1] [(1 - c).sup.p]*N*[w.sup.p-1] [(1-c).sup.p]*N*[w.sup.p]

Now it is evident that the number of good units can be written as the sum of a series with "p" terms. Letting "S" be the sum of the series, we then have

1) S = c*N + c*(1 - c)*N*w + c*[(1 - c).sup.2]*N*[w.sup.2] + .... + c*[(1 - c).sup.p-1]*N*[w.sup.p-1] After factoring out "c*N," the sum in equation 1) becomes ...

S = c*N*[1 + (1 - c)*w + [(1 - c).sup.2]*[w.sup.2] + .... + [(1 - c).sup.p-1]*[w.sup.p-1]]

Now the term inside the brackets is a geometric series, which may be summed by the approach that follows. Multiplying both sides of equation 2) by "(1 - c)*w," equation 3) is obtained as ...

S*(1 - c)*w = c*N*[ (1-c)*w + (1-c)2*[w.sup.2] + (1-c)3*[w.sup.3] + .... + (1-c)p*wp]

Multiplying through equation 3) by minus 1, and adding it to equation 2), equation 4) is obtained as .

S - S*(1-c)*w = c*N*[1 - [(1 - c).sup.p]*[w.sup.p]]

Solving for "S," equation 5) is obtained as ...

5) S = c*N*[1-[(1-c).sup.p]*[w.sup.p]]/1-(1 - c)*w

Now, the sum of the series "S" after "p" passes through the rework cycle must equal the required production Yield "Y." Then solving for "N" in terms of "Y," equation 6) is obtained as

6) N = Y*[1-(1 - c)*w]/c*[1 -[(1 - c).sup.p]*[w.sup.p]]

Thus, in order to arrive at a production quota "Y" after "p" passes through the rework cycle, it is necessary to start "N" units. Again it should be noted that if "M" were perfect (i.e. c=1), "N" would be equal to "Y."

DISCUSSION

In the previous Nevins and Whitney model (Figure 1), the production process is assumed to be binomially distributed and the required production yield is set equal to the mean of the binomial. Thus, using the variables of Figure 2, we have

7) Y=N*C

Hence, the number of startup units "N" to achieve a required production yield "Y" is

8) N = Y/C

We will now compare equation 6) with equation 8). Clearly, both equations are the same if p=1 or w=0. However, for 0<c<1, 0<w<1, and p>1, equation 6) is always smaller than equation 8). This is illustrated in Table 1 below for "c" values of 0.8 and 0.7, w=0.6, and Y=200.

In the above illustration, the two approaches give the same results for p=1. However, for p=2 and c=0.8, 27 fewer startup units are required for the proposed approach. As "p" increases beyond 2, the reduction tapers off to 220, resulting in a total reduction of 30 units. For a less capable process with c=0.7, the reduction in startup units is 44 for p=2 and tapers off to 234 for p=5. Even at p=2, if startup units cost $100 each, the proposed approach will reduce production cost by $2700 when c=0.8 and $4400 when c=0.7; and the reduction in cost will grow linearly with production volume.

CONCLUSION

In this paper, we have derived a new formula for determining the number of startup units to achieve a required production yield without assuming any probability distribution. This approach results in fewer startup units than would be required by the current approach of assuming a binomial distribution and using the mean of the binomial to determine the number of startup units. The reduction in startup units is greatest for a rework cycle of 2 and then tapers off as the rework cycle increases beyond 2. Also, the reduction in startup units grows linearly with production volume resulting in greater savings in production cost, especially for costly startup units.

Although the purpose of this analysis has been primarily pedagogical, an important practical use for which the foregoing model might be employed is as a cost cutting tool for companies having older and less capable equipment since the reduction in startup units also increases with lower process capability. Furthermore, this approach can be used in multistage production systems where a startup unit goes through a series of processes to become a finished product. In this case, one would start with the last process and work backward to the first, treating the estimate of startup units for a succeeding process as the required yield for the previous process. In this way, we can estimate the number of startup units for the first process.

REFERENCES

Abdel-Malek, Layek, Asadathorn, Nutthapol (1996), "An Analytical Approach to Process Planning with Rework Option," International Journal of Production Economics, 46-47, pp. 511-520

Chase, Richard B., Jacobs, F. Robert and Aquilano, Nicholas J. (2006), Operations Management for Competitive Advantage, McGraw-Hill-Irwin, Boston, MA

Flapper, S. D. P., Fransoo, J. C., Broekmeulen, R. A. C. M., and Inderfurth, K. (2002), "Planning and Control of Rework in the Process Industries: A Review," Production Planning & Control, 1, pp. 26-34

Flapper, S. D. P., and Teutner, R. H. (2003), "Logistic Planning of Rework with Deteriorating Work In-process, Control of Rework," Working paper, Department of Technology Management, Eindhoven University of Technology, The Netherlands

Guide, V. Daniel R., Jr., (2001), "Managing Product Returns for Remanufacturing," Journal of Production and Operations Management, 10(2), pp. 142-153

Inderfurth, K., and Teutner, R. H. (2003), "Production Planning and Control of Closed-loop Supply Chains, Cited by Guide, V. D. R., Jr., van Wassenhove, L. N., (editors), Business Perspectives on Closed-loop Supply Chains, Carnegie-Mellon University Press, Oxford, pp. 149-173

Inderfurth, K., Lindner, G., Rahaniotis, N. P. (2003), "Lot Sizing in a Production System with Rework and Product Deterioration," Working paper 1/2003, Faculty of Economics and Management, Otto-von-Guericke-University, Madgeburg, Germany

Inderfurth, K., Kovalyov, Mikhail, Y., Ng, C. T., Werner, Frank (2005), "Cost Minimizing Scheduling of Work and Rework Processes on a Single Facility Under Deterioration of Reworkables," International Journal of Production Economics, 105, pp. 345-357

Lindner, G., Buscher, U., Flapper, S. D. P. (2001), "An Optimal Lot and Batch Size Policy for a Single Item Produced and Remanufactured on One Machine," Working paper 10/2001, Faculty of Economics and Management, Otto-von-Guericke-University, Madgeburg, Germany

Lund, R. (1998), "Remanufacturing: An American Process," Proceedings of the Fifth International Congress for Environmentally Conscious design and Manufacturing, Rochester Institute of Technology, Rochester, NY, pp. 23-30

Nasr, N., Hughson, E. Varel, and Bauer, R. (1998), "State-of-the-art Assessment of Remanufacturing Technology," Proceedings for National Center for Remanufacturing, Rochester Institute of Technology, Rochester, NY

Nevins, J. E., and Whitney, Daniel E. (1989), Concurrent Design of Products and Processes, Mc Graw-Hill, pp. 345-370

J. S. Sutterfield, Florida A&M University

Paul Nkansah, Florida A&M University

Table 1: Comparison of N Using Equation 8) and N Using Equation 6)
for c values of 0.8 and 0.7, w = 0.6, and Y = 200.

            N Using Binomial       N Using Proposed
               Equation 8)       Formula Equation 6)

p           c=0.8      c=0.7      C=0.8      C=0.7
1            250        286        250        286
2            250        286        223        242
3            250        286        220        236
4            250        286        220        235
5            250        286        220        234