$[$Question $912]$ Kim's Kitchen makes Kimchi for famous restaurants worldwide. To meet demand, they are considering setting up a facility in each region to achieve lower transportation costs and possibly avoid duties that could be imposed if the product is imported from another region. Therefore, Kim's kitchen is interested in determining the appropriate size of the facility to build in each location.

$($Use $x_{ij}$ amount of item transporting & Li Hi low and high capacity binary variables$)$

$\backslash$table$[[,\backslash$table$[[$Demand Region$],[$Production and Transportation Cost$],[$per One Million Units$]],\backslash$table$[[$Low$],[$Capacity$]],\backslash$table$[[$High$],[$Capacity$]]],[\backslash$table$[[$Supply$],[$Region$]],\backslash$table$[[$North$],[$Corea$]],$Atlantis,$\backslash$table$[[$Jurassic$],[$World$]],\backslash$table$[[$Fixed$],[$Cost$]],$Capacity,$\backslash$table$[[$Fixed$],[$Cost$]],$Capacity$],[\backslash$table$[[$North$],[$Corea$]],20,60,81,100,4,320,10],[$Attlantis$,15,50,32,120,4,350,10],[\backslash$table$[[$Juurassic$],[$World$]],12,45,15,150,4,400,10],[$Demand$,7,8,5,,,,]]$

Find the constraints for the following logical constraints.

$"$If you want to build high$-$capacity facility in north corea, you need to build low$-$capacity facility in north corea first."

a$)L_1=H_1$

b$)L_1{H}_1$

c$)L_1{H}_1$

d$)L_1+H_1=1$

If high$-$capacity facility is built in Attlantis, then low capacity must be built in North Corea as well.

a$)H_2{L}_1$

b$)H_2=L_1$

c$)H_2{L}_1$

d$)H_2{L}_2$

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If low capacity facility is built in Juurassic World, then at least one low or high capacity must be built in North Corea as well.

$($Which means low, high, or low high are all possible.$)$

a$)L_3{H}_3+L_2$

b$)L_3{H}_1+L_1$

c$)L_3=H_1+L_1$

d$)L_3{H}_1-L_1$

Use excel$-$solver and find the optimal solution $($Min$).$ You must add constraints from previous constraints $($#$9,10,11).$

a$)$ in range $01000$

b$)$ in range $10002000$

c$)$ in range $20003000$

d$)$ in range $30004000$

e$)$ above $4000$