$1-(4$pts$)$ An optimization model must decide how to allocate available supplies $s_i$ at sources $i=1,$dots,$m$ to meet requirements $r_j$ at customers $j=1,$dots,$n.$ Using decision at variables

$w_{i,j}$ : amount allocated from source $i$ to customer $j$

formulate each of the following requirements$($constraints$)$ in a single line $($hint: use summation sign, for all$/$each notation where necessary$).$

a$.$ The amount allocated from sources $25$ cannot exceed the supply available at $25.$

b$.$ The amount allocated from each source i cannot exceed the supply available at $i.$

c$.$ The amount allocated to customer $n$ from sources $1$ and $2$ should equal half the requirement at $n.$

d$.$ The amount allocated to each customer $j$ should equal the requirement at $j.$

$2-(6$ pts$,4$ pts for graduate students$)$ Write each of the following as compactly as possible using summation $()$ and "for all" $(AA)$ notation.

a$)$ min$3y_{3,1}+3y_{3,2}+4y_{4,1}+4y_{4,2}$

b$)max{y}_{1,3}+2y_{2,3}+3y_{3,3}+4y_{4,3}$

c$)$ max${}_1{y}_{1,4}+{}_2{y}_{2,4}+$cdots$+{}_p{y}_{p,4}$

d$)$ min${}_1{y}_1+{}_2{y}_2+$cdots$+{}_t{y}_t$

e$)$

$y_{1,1}+y_{1,2}+y_{1,3}+y_{1,4}=S_1$

$y_{2,1}+y_{2,2}+y_{2,3}+y_{2,4}=S_2$

$y_{3,1}+y_{3,2}+y_{3,3}+y_{3,4}=S_3$

f$)$

$a_{1,1}{y}_1+a_{2,1}{y}_2+a_{3,1}{y}_3+a_{4,1}{y}_4=c_1$

$a_{1,2}{y}_1+a_{2,2}{y}_2+a_{3,2}{y}_3+a_{4,2}{y}_4=c_2$

$a_{1,3}{y}_1+a_{2,3}{y}_2+a_{3,3}{y}_3+a_{4,3}{y}_4=c_3$