Probabilistic model of divided conveyer belt to increase input performance of sorter feed systems

Several preparations and precautions have been taken to ensure realistic and ergonomically optimal input operations in the experimental validation procedure. The methods–time measurement (MTM) study provided a basis for designing the workplaces and determining the expected execution time for the various coding and input procedures (see Table 1). To carry out a comprehensive ergonomic design of a workplace, the psychological points of view were considered in addition to the anatomical and physiological factors. The basic conditions of the workplace and the working environment (light, noise, vibration and shock, climate, etc.), as well as the manual loading capacity, were adapted as much as possible to human needs. Care was taken that constantly recurring sequences of motion were supported by the specific kind of the construction for individual workplaces. To guarantee the most natural postures and motion sequences in the workplace design, the body mass of the input staff was taken into consideration. The aim was to ensure that the results of the attempt could not be falsified by insufficient ergonomic boundary conditions [14].

First, a series of experiments was carried out for all coding procedures and input procedures, to determine learning and fatigue curves. The tests for validation of the mutual impediment did not start until it was ensured that the input operator had reached maximum efficiency. External influence, which falsifies the test result, was therefore eliminated as much as possible.

Numerical and experimental evidence for the decline of input performance due to mutual impediment is summarized in Figs. 6 and 7. The possible input performance obtained from simulation with AutoMod decreases. This is in agreement with [11]. Note that it is yet difficult to compare numbers as the simulated systems were rather different. However, experiments with the divided conveyer belt have shown that reality is even worse. As the operator causes chaotic general cargo input (no rhythm), there are not enough “ideal” gaps, i.e. not enough vacant space with sufficient width is generated. As a result, the entire “real” input performance dropped, in fact, to zero at place 4 when conveyer 4 worked with 0.7 m/s and at place 3 when conveyer 4 worked with 0.5 m/s (See Fig. 6).

An even more pronounced decline of input performance at position 2 is shown in Fig. 7. Here, two input positions were observed after they entered a “steady state”. Then, the operator at position 1 gradually increased the input frequency. It is clearly apparent that the short-time increase in input performance at position 1 reduced the input performance at position 2 by almost 50 % due to lack of usable gaps. As a result, the entire input performance was reduced. Again, a simulation carried out simultaneously confirmed the experimental results. This clearly indicates that the origin of mutual impediment is linked to fast and chaotic cargo input.

Even though the initially chosen approach for removing the phenomenon with a divided conveyer belt feed (patent DE 100 51 932 A) seemed promising (cf. [12]), it turned out that as long as the input was spatially restricted a slight reduction of the overall performance still occurred. As a result of this insight, the variability of the input moved to the centre of attention. The probabilistic model described in the next section serves, in addition to practical experiment, as a tool to investigate the interplay between the divided conveyer belt as such and the variability of input.

Results of experiments on the divided conveyer belt prototype system and simulation of a standard conveyer belt feeding system are listed in Table 2. The advantage of variable input over fixed input position (this corresponds to the case where \(b_{\text{E}} = l_{ \hbox{min} }\) in Sect. 4) can clearly be seen. Input is denoted variable, or spatially variable, if the operator is flexible in his choice of where to place the general cargo unit within the input area. Obviously, input can only be variable if the width of the input area is sufficiently much larger than the length of a general cargo unit.

To compute the efficiency (the value 1 corresponds to no idle time), the MTM value \(t_{\text{MTM}}\) for the handling time was taken as the optimal handling time. The reduced value for \(t_{\text{MTM}}\) had to be applied, because a modified experimental design of the workplace simplified the processing procedure. While for the standard conveyer belt system the idle time steadily increases, it decreases steadily for the divided conveyer belt with fixed input and almost vanishes completely if input is variable. The very high idle time at input position 2 is significant for a “fixed” input position. The reason for this is based on the origin of the vacant space, which is relatively narrow at this point in time. This makes further general cargo input difficult. The approach to realizing a bigger speed step at this point must be rejected on account of the subsequent speed steps. Although this table shows some relict of idle time at position 2—there is still a minimal amount of mutual impediment—the system performance does almost entirely depend on the handling time only.

In the following investigation, the basic structure of the model shall mostly be restricted to the case of two conveyer belts, each having their own input area. With the first conveyer belt working at speed \(v_{1}\), the second conveyer belt is accelerated compared with the first by a factor \(a\). Imposed by technical conditions, the speed \(v_{2}\), of the second conveyer shall be limited by \(v_{{{\text{ma}}x}}\), i.e.

Three different lengths:are restricted and related to each other by

While, in principle, \(l_{\text{GC}}\) and \(l_{ \hbox{min} }\) may be different for each individual general cargo unit and thus represent random variables, both shall be assumed constant for simplicity. The handling times at station 1 and 2, denoted \(T_{1}\) and \(T_{2}\), respectively, are treated as random variables following, for example, an equal or suitable normal distribution. An illustration of the basic structure is shown in Fig. 4.

For concretion and comparison, three specific models that differ only in the width of the input area shall be considered explicitly:

Note that the increased input areas in models 2 and 3 permit spatially variable input.

In a real-world application, the minimum necessary handling width \(l_{ \hbox{min} }\) is always larger than \(l_{\text{GC}}\) by a small amount. To avoid unnecessary subtleties, \(l_{ \hbox{min} }\) shall be chosen as the central length of importance in our discussion. The minimum vacant space generated on the transition from one conveyer to the next is

Thus, for \(a \ge 2\), the new generated vacant space is always sufficiently large to accept another general cargo unit.

Generally, the allocation of vacant space between two general cargo units at the end of conveyer 1 is described by the random variable

The condition \(T_{1} \le l_{ \hbox{min} } /{\text{v}}_{1}\) signals self-impediment (see Fig. 2b) that occurs if a self-loaded general cargo unit still blocks the input area at the end of handling the following general cargo unit. If the speed of conveyer 1 is too low or, put in another way, the operator at station 1 acts to fast, the preceding general cargo unit has not had enough time to leave the input area. Assuming that the current general cargo unit then will be placed immediately after the preceding one, there will be effectively no vacant space left between the two units.

Let \(P_{1} \left( {T_{1} } \right)\) be the probability density function of handling times at station 1. Then, the probability \(P_{\text{si}} \left( {T_{1} \le l_{ \hbox{min} } /{\text{v}}_{1} } \right)\) that self-impediment occurs is formally given by

Assume that a given handling time \(T_{1}\) has indeed been smaller than \(l_{ \hbox{min} } /{\text{v}}_{1}\). Then, the operator must wait for a time \(\left( {l_{ \hbox{min} } /{\text{v}}_{1} - T_{1} } \right)\), before the new general cargo unit can be loaded. On average, the waiting or idle time due to self-impediment will be

The value of the integral in the above expression depends on the precise form of the distribution function \(P_{1} \left( {T_{1} } \right)\). By choosing a suitable speed of the first conveyer, it can be ensured that no self-impediment will occur, i.e. \(P_{\text{si}} = 0\) and \(\bar{w}_{\text{si}} = 0\). The expression (1) for the allocation of vacant space then simply reduces its first line for all \(T_{1} \ge 0\). Note that, due to the increased speed, there will be no self-impediment at station 2 (and, in fact, all subsequent stations), provided \(P_{1} \left( {T_{1} } \right) \approx P_{2} \left( {T_{2} } \right)\), i.e. the distributions of handling times are sufficiently similar.

With (1), (2) and (3), the expectation value of vacant space at the end of conveyer 1 can be computed as:

In the special case of \(P_{\text{si}} = 1\), when there is always self-impediment, an average idle time of \(\bar{w}_{\text{si}} = l_{ \hbox{min} } /{\text{v}}_{1} - E\left( {T_{1} } \right)\) results from (3) and the above equation gives consistently \(E\left( {G_{1}^{\text{out}} } \right) = 0\).

After the transition from conveyer 1 to conveyer 2, a general cargo unit moves at increased speed \(v_{2} = av_{1}\), while all following general cargo units still move at speed \(v_{1}\). This “boost” generates extra vacant space according towhere the random variable \(G_{2}^{\text{in}}\) denotes the allocation of vacant space between two general cargo units at the beginning of conveyer 2 after the acceleration with factor \(a\). The expectation value of \(G_{2}^{\text{in}}\) is obviously \(E\left( {G_{2}^{\text{in}} } \right) = aE\left( {G_{1}^{\text{out}} } \right) + \left( {a - 1} \right)l_{ \hbox{min} } .\) In the worst case, two subsequent general cargo units from conveyer 1 come closely packed, with no vacant space in between (\(G_{1}^{\text{out}} = 0)\). Then at least the minimum vacant space \(\left( {a - 1} \right)l_{ \hbox{min} }\) is generated, which is always large enough to accept another general cargo unit if \(a \ge 2\).

Technically, there is always an upper limit \(v_{ \hbox{max} }\) to the speed of conveyers. The speed of the first conveyer much be chosen sufficiently small, such that the speed of the subsequent conveyer(s) does not exceed this maximum. But if self-impediment is to be avoided, the speed of the first conveyer cannot be chosen arbitrarily small. Self-impediment can be completely avoided only if the minimum possible handling time at station 1, \(T_{{1,{ \hbox{min} }}}\), is always larger than \(l_{ \hbox{min} } /v_{1}\). This puts another constraint on the speed \(v_{1}\) of conveyer 1. Thus, in summary, if it is the primary goal to always provide sufficient vacant space at station 2 for another general cargo unit to be supplied, and the secondary goal to avoid self-impediment at station 1, the following optimal solution arises:

It is, in fact, straightforward to generalize this discussion to a system of N conveyer belts with individual speeds \(v_{i}\), each being accelerated with respect to its predecessor by a factor \(a_{i}\). Yet, if there are more than two conveyer belts, it is to ensure that the maximum conveyer belt speed technically possible is not exceeded, i.e. \(v_{N} \le v_{ \hbox{max} }\). The random variable \(G_{i}^{\text{in}}\), which describes the allocation of vacant space at the beginning of conveyer \(i\), depends on the output of conveyer \(i - 1\),

The general treatment of \(G_{i}^{\text{out}}\) must take into account the mixing of different random input sequences for \(i \ge 2\). Ideally, this can be described by a convolution of handling time distributions. Due to mutual impediment, however, the exact treatment is a bit more involved and shall be subject to a future publication.

In principle, the factors \(a_{i}\) can take different values at each conveyer, and fine tuning can lead to optimized performance in real applications. Following along the line of the above discussion, however, we argue that the theoretically optimal solution isfor all conveyers \(i = 2 \ldots N\), in a system of N of conveyer belts. This ensures

In the remaining parts of this section, it shall be assumed that no self-impediment occurs at the input stations.

Idle times at station 2 can arise by blocked input areas at the handling moment. The presence of general cargo from station 1 prevents the input of another general cargo unit at station 2. Of particular interest is the maximum idle time (or waiting time) \(w_{ \hbox{max} }\). Impediment, i.e. blocking caused by the general cargo unit from station 1, begins when there is not enough space left within the input area in front of the blocking general cargo unit (Fig. 8a). Putting the origin to the edge of the input area where general cargo units enter, impediment thus begins when the front edge of the general cargo unit is atwhere the ratio \(k_{\text{E}} = b_{\text{E}} /l_{ \hbox{min} } \ge 1\) has been introduced for convenience. Impediment ends when there eventually occurs enough space within the input area behind the blocking general cargo unit (Fig. 8b). This is the case when the front edge of the blocking unit is atwith the ratio \(k_{\text{GC}} = l_{\text{GC}} /l_{ \hbox{min} } < 1\). The maximum idle time arises when the general cargo unit from station 1 has just begun to block. Then, the maximum distance it has to traverse to end impediment is \(\left( {x_{2} - x_{1} } \right)\). But this maximum distance depends on the width of the input area, and the time it takes to traverse it depends on the speed of conveyer belt 2. Under the condition that \(a \ge 2,\) there is always enough space behind a general cargo unit, and we do not need to consider the presence of more than one blocking general cargo unit. Then, explicitly, the maximum idle time becomes

Note that the maximum idle time becomes zero for

For \(k_{\text{E}} > 2 + k_{\text{GC}} \approx 3\), i.e. for input areas of width \(b_{\text{E}} > 2l_{ \hbox{min} } + l_{\text{GC}} \approx 3l_{ \hbox{min} }\), the expression for the maximum idle time becomes negative, which does not make sense technically. But it simply indicates an (unnecessary) surplus of input space, which may be available or not, depending on whether there is an immediately following general cargo unit or not. In particular, for model 3 with \(b_{E} = 3l_{ \hbox{min} }\), i.e. \(k_{\text{E}} = 3,\) there appears no idle time at all. Mutual impediment arising from the presence of general cargo from station 1 at the input area of station 2 is eliminated.

To quantify the benefit from the solution presented in this paper, it is instructive to consider the average idle time of a general setup with \(b_{\text{E}} = k_{\text{E}} l_{ \hbox{min} }\). Generally, the idle time will take some (random) intermediate value between \(0\) and \(w_{ \hbox{max} }\). Ignoring possible synchronization effects between station 1 and station 2, and assuming that every handling moment is equally probable, the probability that input at station 2 is blocked by the presence of general cargo from station 1 is the blockage probabilitywhere \(L \ge al_{ \hbox{min} }\) is the width of an interval at conveyer 2, measured from front edge to front edge of general cargo units. Note that the above expression can only be interpreted as a probability for \(b_{\text{E}} \le 2l_{ \hbox{min} } + l_{\text{GC}}\). From the point of view of station 2, the worst case is when station 1 works at maximum efficiency and all intervals are of minimum length \(L_{ \hbox{min} }\). The worst-case average idle time can be computed fromwhere \(w\left( x \right)\) is the idle time as a function of the position \(x\) of the front edge of the blocking general cargo unit. With impediment beginning at \(x_{1} \left( 4 \right)\)(4) and ending at \(x_{2}\) (5), it can be writtenand \(w\left( x \right) = 0\), otherwise. Then, the worst-case average idle time becomes

Noting that \(L_{ \hbox{min} } = al_{ \hbox{min} }\) and inserting (4) and (5), we finally get the explicit dependence of the worst-case average idle time on the important technical parameters \(a\), \(v_{1}\) and \(b_{E}\), i.e.which holds for \(a \ge 2\) and \(l_{ \hbox{min} } \le b_{\text{E}} \le 2l_{ \hbox{min} } + l_{\text{GC}} \approx 3l_{ \hbox{min} }\). For comparison, the maximum and worst-case average idle times of the concrete models 1, 2 and 3 are listed in Table 3.

The “worst” case at input station 2 is actually the “best” case at input station 1, which then works with maximum efficiency. In this idealized case, the handling time is always \(T_{\text{opt}}\), which may be adjusted to be, for example, 2.40 or 2.85 s, the MTM values given in Table 2. However, the idealized optimal handling time at station 1 is certainly determined by the speed of conveyer 1 and the minimum necessary handling width, \(T_{\text{opt}} = l_{ \hbox{min} } /v_{1}\). The optimal input at station 1 in general cargo (GC) units per hour then is \(D_{1} \left[ {1/{\text{h}}} \right] = 3600/T_{\text{opt}}\). Nonzero idle times decrease the input performance and in the worst case from the point of view of station 2, \(D_{2} \left[ {1/{\text{h}}} \right] = 3600/\left( {T_{\text{opt}} + w_{\text{avg}}^{\text{worst}} } \right)\). The ratio \(D_{2} /D_{1}\) represents the efficiency of station 2 with respect to the optimal input at station 1 (see Fig. 9). It can be written

Note that this efficiency does, in fact, only depend on two technical parameters, the acceleration factor \(a\) and the width of the input area \(k_{\text{E}}\) relative to the minimum necessary handling width. It has to be reminded, however, that this expression holds only for a limited range of \(a\) and \(k_{\text{E}}\). On the one hand, if \(a < 2\), then in the worst-case scenario considered the efficiency becomes zero, while on the other hand if \(k_{\text{E}} > 2 + k_{\text{GC}} \approx 3\), the presence of more than just one general cargo unit within the input area of station 2 needs to be considered, which has been ignored in present discussion.