$7.[$Dual Sourcing$-$ Correct Formulation$]$ In order to provide a correct solution of dual sourcing, let xij

be the amount of shipment made from supplier i to market j$.$ In dual sourcing, every market j has

two suppliers I$1($j$)$ and I$2($j$)$ such that xI$1($j$),$j $+$ xI$2($j$),$j $=$ dj and xij $=0$ for all suppliers except for

I$1($j$)$ and I$2($j$).$ In particular, $($xI$1($j$),$j $=0,$ xI$2($j$),$j $=$ dj$),($xI$1($j$),$j $=$ dj$,$ xI$2($j$),$j $=0)$ or their combination

$($xI$1($j$),$j $=$ dj$/2,$ xI$2($j$),$j $=$ dj$/2)$ are all feasible solutions unlike the case in single sourcing. In other

words, xij$=$ djyij in dual sourcing while the rest of the formulation is given by

m

min

s$.$t$.$

m

$$

i$=1$

$$

n

$$

j$=1$

cijxij

yij $2$ for each market j

i$=1$

n

$$

j$=1$

m

$$

i$=1$

xij $$ ki for each supplier i

xij $$ dj for each market j

$20$

xi$,$j $0,$yi$,$j in $\{0,1\}.$

Note that the formulation is incomplete because xij$$s are not related to yij$$s$.$ Write a constraint or

more to complete the formulation