---
# System prepended metadata

title: The Problem Of Programme Allocation
tags: ['Kantorovich, L.V.', The Best Use of Economic Resources]

---

# The Problem Of Programme Allocation
###### tags: `Kantorovich, L.V.`
###### tags: `The Best Use of Economic Resources`

* $n$ different products (kind of operations)
* $m$ production unit (factories, machines,...)
* $i=1,2,...,m$ / $j=1,2,...,n$
* $a_{ij}$ units of $j$th product are produced by $i$th per unit of time
* $h_{ij}$ the fraction of the working time of the $i$th  spent on the production of the $j$th product


## Problem A
$\{a_{ij}\} (i=1,...,m; j=1,...,n)$, $k_j>0(j=1,...,n)$
with   $\underset {1 \leqq i \leqq m}{max} a_{ij}>0 (j=1,...,n)$

Find the plan $\pi = \{h_{ij}\} (i=1,...,m; j=1,...,n)$ satisfying the following conditions:
1. $h_{ij} \geqq 0$
2. $\sum_{j=1}^n h_{ij}\quad \leqq 1 (i=1,...,m)$
3. The quantity $\mu (\pi)=\underset {1 \leqq j \leqq n}{min} \frac {x^n_j} {k_j}$, 
   where 