Other factors such as quality, lead time and flexibility influence the overall performance of a supply chain [2, 3]. Measuring and monitoring these performance variables is a precursor for performance improvement [4, 5]. One aspect of the supply chain that has not been thoroughly studied is the impact on lead time when supply chains become more complex. The number of tiers in a supply chain refers to the number of vendors in series for a single product. If one vendor produces a product from start to finish, this is a one-tier supply chain. If one vendor performs some of the processes, then sends the partially completed item to a second vendor to finish the processes, this is a two-tier supply chain. The designation continues for as many suppliers are in the supply chain for a product [6].

Each vendor in a supply chain consists of a unique organization with unique operating characteristics. Each organization has variation in its lead time due to uncontrolled operational situations [7]. Machine breakdowns, workforce attendance, material inconsistencies, quality errors, process bottlenecks and priority conflicts are common causes for lead-time variation. With each additional tier in a supply chain, there are additional factors that contribute to lead time variation [8].

Each vendor must estimate the lead time to complete the processes they are responsible for performing. Vendors are often evaluated based on their ability to meet their promised lead time [9]. A vendor does not want to tell a customer that they can deliver product within a specified lead time and fail to meet the promise. Consequently, many vendors will promise a lead time that they are confident they can meet.

Although few vendors perform statistical analysis on their lead-time performance, they use some method to estimate their average lead time and "worst case" lead time. With statistical techniques, a worst-case lead time may be estimated using average and standard deviation of lead-time performance. If a vendor is asked to quote a lead time that they can consistently meet, the vendor might quote the upper end of a 98 percent confidence interval for their lead time. From the customer's perspective the vendor performs acceptable 99 percent of the time. Any lead time less than the quoted lead time is viewed as acceptable to the customer, so only the upper tail of the limit would be viewed as unacceptable. If the vendor produces the product in less than the quoted lead time, they will hold the product until it is due. Although this practice may result in additional cost to the vendor, it has no direct affect on the customer.

Methodology: Lead-Time Confidence Interval


Lead time is a variable that can be analyzed using many common statistical techniques [10]. The average, variance, standard deviation and coefficient of variation (CV) can be computed from a sample of lead-time values. The coefficient of variation is a ratio — a combination of the average and the standard deviation [11, 12]. The coefficient of variation is computed as the standard deviation divided by the average. This unitless value indicates the amount of variation in a data set relative to the average. A data set with an average of 10 and standard deviation of 1 will have a CV of 1/10 = 10 percent. Likewise, a data set with an average of 50 and a standard deviation of 5 will also have a CV of 5/50 = 10 percent.

If the processes to produce a product are all performed in a one-tier supply chain, the lead time for that product is a function of the average lead time and standard deviation for that single vendor. If the processes are distributed between two vendors, the average lead time may stay the same. If the work content is split evenly, the first vendor needs one half of the average time to perform their processes and the second vendor needs one half of the average time to perform their processes. There is no delay in average time by using multiple vendors. However, the variation in the lead time is affected by the number of vendors involved. Each vendor has some variation that must be included in its allotted lead time. If each vendor experiences the same amount of lead time variation, then two vendors will have twice the variance in lead time as one vendor. As each vendor in a supply chain adds a time "cushion" for unexpected delays in their processes, the overall expected lead time of the supply chain continues to grow.

Confidence intervals are often used to provide a range of expected values for a variable [13, 14]. A 98 percent confidence interval for the lead time of a product would be approximately the average (A) plus or minus two times the standard deviation (S) or A +/- 2S. That is to say, there is a 98 percent probability that the true average lead time lies within this confidence interval. Since the coefficient of variation contains the ratio of the values of the average and standard deviation, it can be used to estimate the confidence interval.

The formula for CV can be manipulated so the standard deviation is expressed in terms of the CV and the average. The upper portion of the confidence interval can also be manipulated to contain only the CV and the average value for lead time.

With a known CV this allows us to express the confidence interval in terms of only a constant multiplied by the average. This can also be expressed as a percent of the average. This expression provides a simple multiplier to the average lead time to estimate the upper and lower confidence limits. The upper confidence interval limit is the value of interest to the customer, since it indicates the worst case expected lead time.

Figure 1 shows an example of how the upper confidence interval limit for lead time of a process grows as the number of tiers grows.

Figure 1: The upper confidence interval limit grows along with the number of tiers.

Each line in the graph represents a different supply chain with a different coefficient of variation. Coefficients of variation of 10 percent, 20 percent and 40 percent for each tier are graphed. There are many programs and techniques associated with variability reduction in processes [15, 16]. Variability reduction may move a supply chain from a higher CV line to a lower CV line. However, the number of tiers in the supply chain also has a significant effect on lead-time confidence interval limits. As an example, even a supply chain characterized by a low CV = 10 percent with five tiers has a significant variation in lead time. In order to meet the promised lead time consistently, this supply chain would need to quote a lead time 152 percent of its average lead time. A product that requires 10 days on average to complete would need to be quoted at 15 days in order to allow for variations in lead time from each of the five tier vendors.

Summary


As long as variability in lead time is a reality, the number of tiers in a supply chain will have a significant impact on the overall lead time of a product. With each sequential addition of a vendor to a supply chain, the opportunities for delay increase. Each vendor introduces a new set of uncontrolled variables that can create variability in lead time. When a supply chain is designed, this concept needs to be considered and balanced against other decision criteria. A complex supply chain that incorporates many different specialized vendors will have more chances for disruption than one with fewer tiers.

The use of lead-time confidence intervals provides an accurate model of what can be expected based on the number of tiers in a supply chain. This approach acknowledges and quantifies the inherent variability as a supply chain becomes more complex. Supply chain designers should utilize this technique to evaluate the effect of adding tiers to their supply chain. The confidence interval technique is a common statistical analysis tool used in many different applications. It provides a window of expectations for a variable rather than a simple average. Supply chain decisions should be based on all of the important performance variables such as cost, lead time, quality and service. Utilizing the supply chain confidence interval will provide a decision-maker with an analytical estimate for expected lead time that is based on variability and the number of vendors involved.



Cliff Welborn is an assistant professor in the Management and Marketing Department of Middle Tennessee State University. Prior to his academic appointment, he spent 18 years in manufacturing industries in various positions of manufacturing engineering and advanced planning.

References

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