Delay A process where output lags behind its input in some fashion Any belief or perception involves an information delay because we cannot instantaneously update our mental models as new information is received.

Delays always contain stocks

Delays Material delay Information delay Flow is conserved Flow is not conserved Need different modeling structures than material delays. Represent the gradual adjustment of perceptions or beliefs e.g., belief about future orders, given a sudden jump in orders e.g., average production rate of a product

Average length of delay? (Average residence time for a unit) Distribution of the output around the average delay time? Are items processed FIFO, or is there some mixing and reshuffling? Do they all spend the same time in the delay, or are some faster and some slower?

First order material delay Perfect mixing destroys all information about order of entry

S = S0 exp(-t/D)

Higher order delays A delay with n stages, each with 1/n of the total delay time

Pulse response of a third order delay

Response to a pulse input n = ? pipeline delay The higher the order of the delay the less mixing the smaller the variance of the output n = 2 most variance All the four distributions have the same average delay time, but the variances differ.

Pipeline delay For the pipeline delay, the outflow is simply the inflow lagged by the average delay time D.

Pipeline delay Delay time is constant Strictly FIFO Order of exit from the delay is precisely the same as the order of entry Example: Assembly line Outflow(t) = Inflow(t-D)

Little’s Law The equilibrium stock in transit for a delay is always D*I units regardless of the probability distribution of the outflow. S = D * I Stock in transit Input level Delay

Information Delays Delays in measurement or perception of a variable Delays in updating of beliefs and forecasts Flow is NOT conserved (flow is conserved in material delays)

Exponential Smoothing (Adaptive expectations)

Exponential Smoothing (Adaptive expectations) A belief changes when it is in error The larger the error the greater the rate of adjustment in your belief The rate of belief updating is greatest immediately after the change in the actual value of the variable, when the error in the belief is the greatest It’s moving average where the weights wi decline exponentially.

Exponential Smoothing (Adaptive expectations) Challenge: To be responsive without overreacting to noise Survey results (Makridakis et al 1982, 1984, 1995) Exponential smoothing is one of the most commonly used It performs extremely well Approximates judgemental forecasts

Higher order information delays

Let’s first look at the Inventory Let’s first look at the Inventory. The Inventory is regulated by Production less Sales. Sales is constant at 20KWidgets per year. Each employee’s Productivity is 100 Widgets per year, the number each person can make per year. Thus the Production is equal to Productivity * Employment, the number of employees. Production less Sales, which is the change in Inventory per year is Production less Sales = Production – Sales = (Productivity • Employment) – 20K Net Change in Employment is the net effect of all hiring and firing in the firm. The Gap is the discrepancy between Desired Inventory and Inventory. The Time to Close Inventory Gap is a parameter determined by the firm. The Time to Close Inventory Gap and the Gap determine the Production Needed to Close Gap. Production Needed to Close Gap = Gap/Time to Close Inventory Gap

The Production Needed to Close Gap and the Productivity determine the Number of People Needed for Hire. Since we know how many widgets we need to produce in a certain amount of time, we can determine how many people we need to hire by knowing the productivity. Number of People Needed for Hire = Production Needed to Close Gap/Productivity 25K 200 1/4 100 20K 1/2 Net Change in Employment = Number of People Needed for Hire/Hiring Delay The Hiring Delay is the time it takes to hire and train new employees. In this case, it is 3 months or 1/4 years.

Impact of nonlinear adjustment times Perception of job security falls swiftly on news of a layoff and take years to recover

Estimating delays Statistical techniques Firsthand investigation of the process in the field