Slow going at the pretzel factory

A chocolate covered, peanut butter filled, pretzel factory has three worker pods, and that each pod produces one important part of the snack: pod 1 prepares the chocolate, pod 2 the peanut butter, and pod 3 the pretzel shell.

The net production coming from each pod is balanced so that any given pod produces \(\lambda = N_i s_i\) snacks worth of their part per unit time, where \(N_i\) is the number of workers in pod \(i\), and \(s_i(0)\) is the speed at which a worker in pod \(i\) can produce things.

When everything is working smoothly the factory can output 100 chocolate covered, peanut butter filled pretzels per unit time. One night the workers in pod 1 have a party late into the night and mistakenly dump a thickening agent into the chocolate supply. This makes the chocolate harder to handle, so that the rate at which workers can prepare chocolate drops to \(s_i(T) < s_i(0)\). However, the speed of production in the other pods remains constant. The drop in \(s_1\) causes the overall rate of pretzel production to fall to \(\lambda_T = 60\) per unit time.

What is the new speed of chocolate production per worker, \(s_1(T)\) (in snack equivalents per unit time)?

Assumptions and details

  • The \(N_T = \sum_i N_i = 500\) factory workers always distribute themselves across the pods such that the rate of snack production is maximal, given their current working conditions.
  • When everything is working smoothly, a worker in pod 1 can produce chocolate at the rate \(s_i(0) = 10\) snack equivalents per unit time.
  • The company has a fixed amount of money each day \(M\) to pay workers, and each worker gets paid fixed daily wage \(p\).

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