Each machine on Widgetco’s assembly line gets out ofwhack an average of once a minute. Laborers are assignedto reset a machine that gets out of whack. The companypays each laborer cs dollars per hour and estimates that eachhour of idle machine time costs the company cm dollars inlost production. Data indicate that the time betweensuccessive breakdowns of a machine and the time to reset amachine are exponential. Widgetco plans to assign eachworker a certain number of machines to watch over andrepair. Let M total number of Widgetco machines, w number of laborers hired by Widgetco, and R Mw machines assigned to each laborer.a Express Widgetco’s hourly cost in terms of R and M.b Show that the optimal value of R does not dependon the value of M.c Use calculus to show that costs are minimized bychoosingR d Suppose cm 78¢ and cs $2.75. Widgetco has 200 machines, and a laborer can reset a machine in an aver-age of 7.8 seconds. How can Widgetco minimize costs? e In parts (a)–(d), we have tacitly assumed that at anypoint in time, the rate at which the machines assigned toa worker break down does not depend on the number ofhis or her assigned machines that are currently workingproperly. Does this assumption seem reasonable?
Each machine on Widgetco’s assembly line gets out of
whack an average of once a minute. Laborers are assigned
to reset a machine that gets out of whack. The company
pays each laborer cs dollars per hour and estimates that each
hour of idle machine time costs the company cm dollars in
lost production. Data indicate that the time between
successive breakdowns of a machine and the time to reset a
machine are exponential. Widgetco plans to assign each
worker a certain number of machines to watch over and
repair. Let M total number of Widgetco machines, w
number of laborers hired by Widgetco, and R M
w
machines assigned to each laborer.
a Express Widgetco’s hourly cost in terms of R and M.
b Show that the optimal value of R does not depend
on the value of M.
c Use calculus to show that costs are minimized by
choosing
R
d Suppose cm 78¢ and cs $2.75. Widgetco has 200
machines, and a laborer can reset a machine in an aver-
age of 7.8 seconds. How can Widgetco minimize costs?
e In parts (a)–(d), we have tacitly assumed that at any
point in time, the rate at which the machines assigned to
a worker break down does not depend on the number of
his or her assigned machines that are currently working
properly. Does this assumption seem reasonable?
a) Widgetco's hourly cost can be expressed as the sum of the cost of labor and the cost of idle machine time:
Hourly cost = labor cost + cost of idle machine time
The cost of labor is the product of the hourly wage and the number of laborers:
labor cost = cs * w
The cost of idle machine time is the product of the cost of idle time per machine and the number of machines that are idle:
cost of idle machine time = cm * (M - R * w)
The number of machines that are idle is equal to the total number of machines minus the number of machines assigned to each laborer times the number of laborers:
number of idle machines = M - R * w
Substituting these expressions into the formula for hourly cost, we get:
Hourly cost = cs * w + cm * (M - R * w)
b) To find the optimal value of R, we need to differentiate the hourly cost with respect to R and set the derivative equal to zero:
d(Hourly cost)/dR = -cm * w = 0
Since cm and w are both positive, this equation implies that the optimal value of R is independent of M.
c) To show that the cost is minimized by choosing R, we need to differentiate the hourly cost with respect to R twice and check whether the second derivative is positive or negative. If it is positive, the function is convex, and the value of R that makes the first derivative equal to zero is a minimum. If it is negative, the function is concave, and the value of R that makes the first derivative equal to zero is a maximum.
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