# 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.

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