In order to smooth its production scheduling, a footwear company has decided to use a simple…
In order to smooth its production scheduling, a footwear company has decided to use a simple version of a linear cost model for aggregate planning. The model is:
vi = Unit production cost for product i in each period,
ci = Inventory-carrying cost per unit of product i in each period,
r = Cost per man-hour of regular labor,
o = Cost per man-hour of overtime labor,
dit = Demand for product i in period t,
ki = Man-hours required to produce one unit of product i,
(rm) = Total man-hours of regular labor available in each period,
p = Fraction of labor man-hours available as overtime,
T = Time horizon in periods,
N = Total number of products.
The decision variables are:
Xit = Units of product i to be produced in period t,
Iit = Units of product i to be left over as inventory at the end of period t,
Wt = Man-hours of regular labor used during period (fixed work force),
Ot = Man-hours of overtime labor used during period t.
The company has two major products, boots and shoes, whose production it wants to schedule for the next three periods. It costs $10 to make a pair of boots and $5 to make a pair of shoes. The company estimates that it costs $2 to maintain a pair of boots as inventory through the end of a period and half this amount for shoes. Average wage rates, including benefits, are three dollars an hour with overtime paying double. The company prefers a constant labor force and estimates that regular time will make 2000 man-hours available per period. Workers are willing to increase their work time up to 25% for overtime compensation. The demand for boots and shoes for the three periods is estimated as:
a) Set up the model using 1 man-hour and 1 2 man-hour as the effort required to produce a pair of boots and shoes, respectively.
b) Write the dual problem.
c) Define the physical meaning of the dual objective function and the dual constraints.