Question

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 \( r_i \) and \( h_i \) 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 \( d_{it} \), and the capacity of the \( j^{th} \) workstation in time period \( t \) is \( c_{jt} \). Consider the aggregate planning formulation below, with decision variables \( S_{it} \) (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 \( t \)). \[ \text{max} \sum_{t=1}^{T} \sum_{i=1}^{m} (r_i S_{it} - h_i I_{it}) \] subject to \[ S_{it} \ge d_{it} \forall i, t \] <capacity constraint> 
<inventory balance constraint>
\[X_{it}, S_{it}, I_{it} \ge 0; \, I_{i0} = 0 \] The capacity constraints and inventory balance constraints for this formulation are 
 

• A. \( \sum_{i} a_{ij} X_{it} \le c_{jt} \forall j, t \text{ and } I_{it} = I_{i,t-1} + X_{it} - S_{it} \forall i, t \)
• B. \( \sum_{i} a_{ij} X_{it} \le c_{jt} \forall i, t \text{ and } I_{it} = I_{i,t-1} + X_{it} - d_{it} \forall i, t \)
• C. \( \sum_{i} a_{ij} X_{it} \le d_{it} \forall i, t \text{ and } I_{it} = I_{i,t-1} + X_{it} - S_{it} \forall i, t \)
• D. \( \sum_{i} a_{ij} X_{it} \le d_{it} \forall i, t \text{ and } I_{it} = I_{i,t-1} + S_{it} - X_{it} \forall i, t \)

Hint:

In production planning models, ensure that the capacity constraints and inventory balance equations are properly represented to manage resources effectively.

Answer 1

The Correct Option is A

The given problem is an optimization problem where the objective is to maximize the total revenue minus the holding costs, subject to constraints on production and inventory balance. Capacity constraints The capacity of each workstation is represented by: \[ \sum_{i} a_{ij} X_{it} \le c_{jt} \forall j, t \] This ensures that the total processing time for all products \( i \) on workstation \( j \) in time period \( t \) does not exceed the available capacity \( c_{jt} \). Inventory balance constraint The inventory balance equation is given by: \[ I_{it} = I_{i,t-1} + X_{it} - S_{it} \forall i, t \] This equation ensures that the inventory at the end of each period \( t \) is calculated by adding the produced units in period \( t \) to the inventory from the previous period and subtracting the sold units in period \( t \). Thus, the correct capacity and inventory balance constraints are as stated in option (A).
Final Answer: (A)