Gallup Enterprises out of Gallup, New Mexico, is in the business of producing and then selling sand paintings to tourists. Gallup Enterprises is planning operations for the next year which is composed of four seasons (periods). The sand painters are not enthusiastic about working in the winter and also are somewhat reluctant to work in the summer. This is reflected in the production capacity of Gallup Enterprises for each of the seasons, and it also influences the cost of producing a sand painting from sea to season. In particular, Gallup Enterprises has the capacity to produce 10 paintings in the winter at a cost of $40 each, 30 paintings in the spring a cost of $20 each, 20 paintings in the summer at a cost of $30 each, and 30 paintings in the fall at a cost of $20 each. The demand for sand paint by tourists is quite seasonal, being 5 in the winter, 15 in the spring, 35 in the summer, and 25 in the fall. Regardless of the time of purchase Gallup Enterprises sells the paintings for $80 each.

 

Gallup Enterprises' problem is how many paintings to produce each season (starting with the winter) so that costs can be held to a minimum, while making sure that the demand for the paintings can be satisfied. Alternately, the objective may be stated in terms of maximizing profits from the sale of the paintings. Gallup Enterprises would obviously like to produce paintings in the spring and fall since production costs are at their low during these two seasons. If paintings are produced in one season for sale in a later season, they must be stored in inventory during the interim. It costs $3 to store a painting from one season to the next. This inventory storage must be considered in any plan. In the context of the details outfit above.

How many sand paintings should Gallup Enterprises produce in each season in order to maximize profits? Assume that Gallup Enterprises is be in this business for only one year.