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GATE 2021 ME (ME2) – Question 39

Production Planning and Control · Aggregate production planning, scheduling and material requirement planning · 2 marks · Multiple choice

A factory produces $m$ ($i=1,2,\ldots,m$) products, each of which requires processing on $n$ ($j=1,2,\ldots,n$) 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 $T$ ($t=1,2,\ldots,T$) 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 $t$).

$\max\sum_{t=1}^T\sum_{i=1}^m(r_iS_{it}-h_iI_{it})$ subject to <capacity constraint>, <inventory balance constraint>, $S_{it}\ge d_{it}$, $S_{it},X_{it},I_{it}\ge0$; $I_{i0}=0$.

The capacity constraints and inventory balance constraints for this formulation are

  1. $\sum_i a_{ij}X_{it}\le c_{jt}\ (j,t);\ I_{it}=I_{i,t-1}+X_{it}-S_{it}$
  2. $\sum_i a_{ij}X_{it}\le c_{jt}\ (i,t);\ I_{it}=I_{i,t-1}+X_{it}-d_{it}$
  3. $\sum_i a_{ij}X_{it}\le d_{it};\ I_{it}=I_{i,t-1}+X_{it}-S_{it}$
  4. $\sum_i a_{ij}X_{it}\le d_{it};\ I_{it}=I_{i,t-1}+S_{it}-X_{it}$

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Show answer and explanation

Correct answer: (A) $\sum_i a_{ij}X_{it}\le c_{jt}\ (j,t);\ I_{it}=I_{i,t-1}+X_{it}-S_{it}$

Explanation

Capacity is consumed by all products processed at each workstation, giving $\sum_i a_{ij}X_{it}\le c_{jt}$ for every j,t.

Inventory obeys conservation: ending stock=opening stock+production−actual sales. Therefore $I_{it}=I_{i,t-1}+X_{it}-S_{it}$. Both correct constraints appear in **A**; minimum demand need not equal actual sales.