A factory produces m (i = 1, 2, ..., m) products, each of which requires processing on n (j = 1, 2, ..., n) workstations. Let a_ij be the amount of processing time that one unit of the i^th product requires on the j^th workstation. Let the revenue from selling one unit of the i^th product be rⁱ and hi be the holding cost per unit per time period for the i^th product. The planning horizon consists of T (t = 1, 2,..., T) time periods. The minimum demand that must be satisfied in time period t is dit, and the capacity of the j^th workstation in time period t is c_jt. Consider the aggregate planning formulation below, with decision variables Sit (amount of product i sold in time period t), X_it (amount of product i manufactured in time period t) and I_it (amount of product i held in inventory at the end of time period t).

$\mathrm{m}\mathrm{a}\mathrm{x}\sum _{t=1}^T\sum _{i=1}^m\left(r_i{S}_{it}-h_i{I}_{it}\right)$

Subject to

S_it ≥ d_it ∀ i, t

< capacity constraint >

< inventory balance constraint >

X_it, S_it, I_it ≥ 0; I_i0 = 0

The capacity constraints and inventory balance constraints for this formulation respectively are

The correct capacity constraints and inventory balance constraints are:

${\displaystyle \sum _i^m{a}_{ij}{X}_{it}\le c_{jt}\forall j,t}$ and $I_{it}=I_{i,t-1}+X_{it}-S_{it}\forall i,t$

Step-by-Step Derivation and Explanation:

1. Capacity Constraints:
We need to ensure that the total processing time required by all manufactured products on a given workstation in a given time period does not exceed the capacity of that workstation in that time period.
Let:
- i denote the product index (from 1 to m).
- j denote the workstation index (from 1 to n).
- t denote the time period index (from 1 to T).
- a_ij be the processing time required for one unit of product i on workstation j.
- X_it be the amount of product i manufactured in time period t.
- c_jt be the total available capacity of workstation j in period t.

The total time spent on workstation j in period t is the sum of the time required to manufacture each of the m products. This is given by the sum:

$\sum _{i=1}^m{a}_{ij}{X}_{it}$

Since this total time cannot exceed the workstation's capacity c_jt, the capacity constraint for workstation j in period t is:

$\sum _{i=1}^m{a}_{ij}{X}_{it}\le c_{jt}$

Because this limit applies to every workstation j and every period t, the constraint is index-bound by $\forall j,t$ (not $\forall i,t$).

2. Inventory Balance Constraints:
The inventory balance constraint tracks the flow of each product i across successive time periods. For any product i in period t:
- The inventory remaining at the end of the period, I_it, must equal the inventory inherited from the end of the previous period, I_{i, t-1}, plus any new units manufactured in the current period, X_it, minus any units sold in the current period, S_it.

Thus, the balance equation is:

$I_{it}=I_{i,t-1}+X_{it}-S_{it}$

This balance must hold for every individual product i and every time period t, giving the domain $\forall i,t$.