Читать книгу Control Theory Applications for Dynamic Production Systems - Neil A. Duffie - Страница 18

For the production work system illustrated in Figure 2.1 it is desired to obtain a continuous-time model that predicts work in progress (WIP) ww(t) hours as a function of the rate of work input to the work system ri(t) hours/day, the nominal production capacity rp(t) hours/day, WIP disturbances wd(t) hours and capacity disturbances rd(t) hours/day. WIP disturbances can be positive or negative due to rush orders and order cancellations, while capacity disturbances usually are negative because of equipment failures or worker absences. Units of hours of work are chosen rather than orders or items, and units of time are days.

Figure 2.1 Continuous variables in a continuous-time model of a production work system.

The rate of work output by the production system ro(t) hours/day is

The WIP is

where ww(0) hours is the initial WIP. Integration often is an element of models of production systems and their components. The corresponding differential equation is

The dynamic behavior represented by this continuous-time model is illustrated in Figure 2.2 for a case where there is a capacity disturbance rd(t) of –10 hours/day that starts at time t = 0 and lasts until t = 3 days. The initial WIP is ww(t) = 30 hours for t ≤ 0 days. The rate of work input is the same as the nominal production capacity, ri(t) = rp(t) hours/day, and there are no WIP disturbances: wd(t) = 0 hours. The response of WIP to the capacity disturbance is shown in Figure 2.2 and is calculated using Program 2.1 for 0 ≤ t ≤ 3 days using the known solution²

Figure 2.2 Response of WIP to 3-day capacity disturbance.

This model represents the production work system using the concept of work flows rather than representing the processing of individual orders. Numerous aspects of real work system operation are not represented such as setup times, operator skills and experience, reduction in actual capacity due to idle times when the work in progress is low, and physical limits on variables. Also, WIP cannot be negative and often cannot be greater than some maximum due to buffer size. Capacity cannot be negative and cannot be greater than some maximum that is determined by physical characteristics such as the number of workers, number of shifts, available equipment, and available product components or raw materials.