Actividad 3 Comportamiento macroscópico de gases

Informational level: account management

Most closed-loop systems of reusable articles have been traditionally managed without item- level tracking. At best, only account management (aggregate issues and returns in a period of time) information has typically been available. Based only on aggregate issues and returns as input information, some statistical methods have been developed in literature in order to draw some relevant cycle time distribution parameters, such as Bojkow (1991) (Method 1), Goh and Varaprasad (1986) (Method 2a) and Toktay et al. (2000) (Method 2b).

Bojkow (1991) compiles di¤erent methods used for “trippage” calculation. Trippage is a concept closely related to cycle time; it measures the total number of trips made by a reusable article in its life-time (cycle time measures the duration of one trip). Bojkow de…nes the trippage number with (4) ratio:

T rippage = Ia Ia Ra

(4)

where:

Ia: number of articles issued in a given time period (such as the year).

Ia Ra: represents the number of “lost” articles in the same time period.

This ratio provides an order of magnitude of trippage …gures, but su¤ers from gross inaccu- racies which invalidate its use as input in the estimation of reusables ‡eet size. As it does not take into account that articles utilization can span several time periods, the same shortcomings identi…ed for ratio (2) in Table 2 are also present here.

Method 2

A dynamic regression model (Pankratz, 1991) can be formulated to establish the relationship between issues and returns. Returns in period t are a function of sales in past periods:

yt= v0xt+ v1xt 1+ v2xt 2 + ::: + Nt (5)

Where:

–The set of parameters (v0; v1; v2; :::v1) represent the probability that an article issued

on period t, returns to the system either on the same period t, on the next period t + 1, on period t + 2, and in general, i periods afterwards, i = 0; 1; 2; : : : ; provided that the item will ever be returned.

–v₁ represents the probability that an article will never be returned ( v₁ = 1 r, represents the loss rate).

The model (5) can be estimated using historical data (time-series) of articles’issues and re- turns. This estimation consists in calculating the values of the set of parameters (v0; v1; v2; :::v1)

which provides a good approximation to cycle time distribution (including a return rate esti- mate) while (theoretically) circumventing the need of tracking individual items for obtaining such information.

Goh and Varaprasad (1986) (Method 2a) estimate the dynamic regression model in (5) using a transfer function model approach. Model (5) can be rewritten as indicated in (6), where the (potentially) in…nite parameters to be estimated in (5) are expressed by the quotient of two …nite polynomials: yt= v(B) + Nt; yt= w0 w1B w2B2 ::: WsBs 1 1B 2B2 ::: rBr xt b+ Nt; v(B) = w(B)Bb (B) (6)

The Box-Jenkins procedures of transfer function identi…cation, model estimation and diag- nostic checking are carried out to estimate model (6) parameters, using a sample of the past values of the issues fxtg and returns fytg time-series.

In Toktay et al. (2000) (Method 2b), model estimation is done using a distributed lag model approach (Bayesian inference), where a speci…c form, based on theoretical considera- tions, is assumed for lag (vi coe¢ cients). Typical assumptions are geometrically distributed

lags (coe¢ cients that decline exponentially) or Pascal (negative binomial) distributed lags. A distributed lag model has the same form expressed in (5), but in this case, Nt necessarily has

to be gaussian white noise Nt~N (0; ). The disadvantage of this second approach is that a

pre-speci…ed distribution is imposed on the data, while the advantage resides in the relatively parsimonious form of the model: as less parameters are to be estimated, smaller sample sizes are required for estimation.

Method 2 is presumed to provide an approximation to cycle time distribution and return rate while eluding the need of empirically observing reusable articles turnaround behaviour through item-level tracking. However, few attempts have been made to estimate the relationship between returns and past sales using actual data from industry. The method is not reported to be used in practice in none of the industrial case studies we have dealt with. When brought to practice,

the estimates of cycle time distribution and return rate obtained through model (5) are reported to be not always realistic. Estimation problems are reported in Goh and Varaprasad (1986) (daily and weekly counts), Fleischmann et al. (2002), Van Dalen and Van Nunen (2009) and Carrasco-Gallego and Ponce-Cueto (2009b).

We have identi…ed three main shortcomings in this method:

–First, method 2 assume that cycle time distribution is stationary ((v0; v1; v2; :::v1)

parameters are constant values to be determined), while empirical observations show that the turnaround process duration may evolve throughout time. Cycle time aver- age value varies also with the product demand seasonality (Van Dalen et al., 2005, Van Dalen and Van Nunen, 2009; Carrasco-Gallego and Ponce-Cueto, 2009b, Case #1 (Medgas) and Case #2 (LPG)). Then, a non-stationary description of cycle time distribution might be necessary. are not constants but functions of time.

–Second, dynamic regression models used in method 2 rely on the assumption that that causality is unidirectional from the input variable fxtg (sales) to the output

fytg(returns). Feedback from the output to the input is ruled out. If returns timing

and quantity in‡uence somehow sales …gures, the use of dynamic regression models for drawing cycle time and shrinkage rates parameters would be invalidated.

–Third, in practice, returns timing does not depend only on sales timing and the time required for products consumption but also on other operational factors, such as the way transportation is organized between depots. Used articles are usually stocked for some time in order to achieve economies of scale in transportation. Articles do not “freely”return to plant just after use, as suggest the impulse response functions in model (5) (vi coe¢ cients).

Informational level: item-level tracking

Item-level tracking systems provide real empirical data that can be used to obtain information about the average cycle time, a measure of its dispersion and the cycle time distribution shape. In order to explore cycle time behaviour, item-level tracking can be implemented either in the whole ‡eet or just in a sample of articles. The resulting cycle times and other turnaround

process parameters empirically observed in the sample can be inferred (if experiments are well designed) to the reusable articles population (van Dalen and van Nunen, 2009). The technology used to track individual articles can range from human observations registered with pen and paper or in spreadsheets to bar code or RFID tagging. Regardless of the tracking technology used, articles need to be serialized in order to follow cycle times.

Within this informational level, the data obtained through the tracking application can be used by managers for observing the behaviour of the turnaround process and the way it varies over the time.