Neil A. Duffie

Control Theory Applications for Dynamic Production Systems


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is the fraction of a worker’s day required for an order, and T days is the period between calculations of the number of workers to assign to the product.

      

      Figure 2.11 Response of desired workforce to fluctuations in demand.

      2.3 Delay

      Delays are common in production systems and sources of delay include data gathering and communication, decision-making and implementation, setup times, processing times, and buffers. For example, decisions may not be made until sometime after relevant information is obtained, and, for logistical reasons, implementation of decisions may not be immediate. Disturbances may not have immediate effects, and these effects may not be detected until they have propagated through a production system. Delays often are detrimental and limit achievable performance; therefore, it is important to include delays in models when they are significant.

      Example 2.7 Continuous-Time Model of Delay in a Production System

      Figure 2.12 Lead time and transportation delays in a two-company production system.

      If the order input rate to Company B is demand ri(t) orders/day and the order input rate to Company A is rA(t) orders/day, the order output rates from Companies B and A, rB(t) and ro(t), respectively, are

r Subscript upper B Baseline left-parenthesis t right-parenthesis equals r Subscript i Baseline left-parenthesis t minus upper L Subscript upper B Baseline right-parenthesis

      Shipping is described by

r Subscript upper A Baseline left-parenthesis t right-parenthesis equals r Subscript upper B Baseline left-parenthesis t minus upper D right-parenthesis

      Combining the delays, the relationship between demand and the completed order output rate of Company A is

r Subscript o Baseline left-parenthesis t right-parenthesis equals r Subscript i Baseline left-parenthesis t minus upper L Subscript upper B Baseline minus upper D minus upper L Subscript upper A Baseline right-parenthesis

      Example 2.8 Discrete-Time Model of Assignment of Production Workers with Delay

      Order input rate ri(kT) orders/day is measured regularly with a period of T days, weekly for example, and the portion of production capacity provided by permanent workers rp(kT) orders/day is adjusted; however, because of logistical issues in hiring and training, there is a delay of dT days in implementing permanent worker adjustment decisions where d is a positive integer. The exponential filter is used to focus adjustments in permanent worker capacity on relatively low frequencies:

r Subscript f Baseline left-parenthesis k upper T right-parenthesis equals alpha r Subscript i Baseline left-parenthesis k upper T right-parenthesis plus left-parenthesis 1 minus alpha right-parenthesis r Subscript f Baseline left-parenthesis left-parenthesis k minus 1 right-parenthesis upper T right-parenthesis

      where 0 < α ≤ 1. A relatively high value of weighting parameter α results models relatively rapid adjustment of permanent worker capacity, whereas a relatively low value of weighting parameter α models significant smoothing and relatively slow adjustment of permanent worker capacity.

      The portion of production capacity provided by permanent workers is

r Subscript p Baseline left-parenthesis left-parenthesis k plus d right-parenthesis upper T right-parenthesis equals r Subscript f Baseline left-parenthesis k upper T right-parenthesis

      where dT days is the delay in implementing permanent worker capacity adjustments. Hence, the portions of fluctuating order input that are addressed by permanent worker capacity rp(kT) orders/day and cross-trained capacity rc(kT) orders/day are

r Subscript p Baseline left-parenthesis left-parenthesis k plus d right-parenthesis upper T right-parenthesis equals alpha r Subscript i Baseline left-parenthesis k upper T right-parenthesis plus left-parenthesis 1 minus alpha right-parenthesis r Subscript p Baseline left-parenthesis left-parenthesis k minus 1 plus d right-parenthesis upper T right-parenthesis r Subscript c Baseline left-parenthesis k upper T right-parenthesis equals r Subscript i Baseline left-parenthesis k upper T right-parenthesis minus r Subscript p Baseline left-parenthesis k upper T right-parenthesis

      2.4 Model Linearization

      A component behaves in a linear manner if input x1 produces output y1, input x2 produces output y2, and input x1