Consider the following LP formulation to model $Pm|prmp|C_{\text{m}\text{a}\text{x}}$ :

minimize,$C_{\text{m}\text{a}\text{x}}$

subject to

${\displaystyle \underset{i=1}{\overset{m}{}}}{x}_{ij}=p_j,j=1,$dots,$n$

${\displaystyle \underset{i=1}{\overset{m}{}}}{x}_{ij}{C}_{\text{m}\text{a}\text{x}},j=1,$dots,$n$

${\displaystyle \underset{j=1}{\overset{n}{}}}{x}_{ij}{C}_{\text{m}\text{a}\text{x}},i=1,$dots,$m$

$x_{ij}0,i=1,$dots,$m,j=1,$dots,$n.$

$C_{\text{m}\text{a}\text{x}}0,$

where:

Parameter:

$p_j$ : processing time of job $)=(1,2,$dots,$n$

Decision variables:

$x_{ij}$ : the total time job $j$ spends on machine $)=1,2,$dots,$n$;$i=(1,2,$dots,$m$

$C_{\text{m}\text{a}\text{x}}$: makespan

Consider $P2|$prmp $|C_{\text{m}\text{a}\text{x}}$ problem with two identical parallel machines $)=(2$ and three jobs $)=(3$ with $p_1,p_2$ and $p_3.$ Select a different random integer processing time for each job from the interval $1,20.$ While selecting processing times, make sure they do not follow a certain patten such as $1-2-3$ or $2-4-6$ etc. If the same data set is selected by different teams, the question may not be graded.