Suppose a production manager is responsible for scheduling the monthly production levels of a certain product for a planning horizon of twelve months. For planning purposes, the manager was given the following information:

The total demand for the product in month j is dj , for j = 1, 2, : : :, 12. These could either be targeted values or be based on forecasts.

The cost of producing each unit of the product in month j is cj (dollars), for j = 1, 2, …, 12.

The inventory holding cost per unit for month j is hj (dollars), for j = 1, 2, …, 12. These are incurred at the end of each month.

The production capacity for month j is mj , for j = 1, 2, …, 12.

The manager’s task is to generate a production schedule that minimizes the total production and inventory holding costs over this twelve-month planning horizon.

To facilitate the formulation of a linear program, the manager decides to make the following simplifying assumptions:

There is no initial inventory at the beginning of the first month.

Units scheduled for production in month j are immediately available for delivery at the beginning of that month. This means in effect that the production rate is infinite.

Shortage of the product is not allowed at the end of any month.

Formulate an LP to solve the manager’s problem.