A company wants a high level, aggregate production plan for the next 4 months. Projected orders for the companys product are listed in the table. Over the four months period, units may be produced in one month and stored in inventory to meet some later months demand. Because of seasonal factors the cost of production is not constant, as shown in the table. The cost of holding an item in inventory for 1 month is $1 h/unit (consider the holding cost as h), charged for those items in the inventory at the end of the period. The maximum number of units that can be held in inventory is I_max . The initial inventory level at the beginning of the planning horizon is I₀ units; the desired final inventory level at the end of the planning horizon is I_f.

The problem is to determine the optimal amount to produce in each month so that demand is met while minimizing the total cost of production and inventory.

Aggregate Planning Data

Period	1	2	3	4
Demand (units)	D1	D2	D3	D4
Cost of Production ($/unit)	CP1	CP2	CP3	CP4

1. Formulate the problem as an LP model. Suppose that no shortages are allowed, and hence you plan to meet all the demand on time. Note that you must define decision variables you use.

2. Now consider the situation where you are allowed to have backorders in period 2 and period 3, meaning that you can delay the satisfaction of demand in periods 2 and, but all backordered demand should be satisfied latest at the end of the period 4. The following data summarizes the cost of back order per unit per period for each period.

Period	1	2	3	4
Cost of backorder ($/unit/period)	Not allowed	CB2	CB3	Not Allowed

How is your formulation going to change? The problem is to determine the optimal amount to produce in each month so that the total cost of production, inventory and backorders is minimized. You need to define new decision variables, as well as use some of the previous part.

3. When you solve the above problems, part a and part b, let the optimal objective function value you have obtained is OF_a and OF_b respectively. Write an inequality describing their relation.