Because dealing with too many carriers can be a cumbersome, Offhaus would like to limit the number of carriers it uses to three. Also, for customer relationship reasons Offhaus wants each city to be assigned to only one carrier $($i$.$e$.,$ no splitting of the demand to a given city across carriers$).$

$($a$)$ Develop and solve a model that will yield the three selected carriers and the city$-$carrier assignments so as to minimize the cost of shipping. Choose the correct optimal solution.

$($i$)$

Carrier $2$: Assigned cities $1,3,4,5,6,$ and $9$

Carrier $5$: Assigned city $3$

Carrier $7$: Assigned city $7,8,$and $10-20$

$($ii$)$

Carrier $2$: Assigned cities $2,3,4,5,6,$ and $9$

Carrier $5$: Assigned cities $7,8,$and $10-20$

Carrier $6$: Assigned city $1$

Total Cost: $$435,404$ Total Cost: $$436,512$

$($iii$)$

Carrier $5$: Assigned cities $1,2,6,7,8$ and $10-20$

Carrier $6$: Assigned cities $3,4,$ and $5$

Carrier $7$: Assigned city $9$

$($iv$)$

Carrier $1$: Assigned cities $11$and $12$

Carrier $3$: Assigned cities $13-20$

Carrier $7$: Assigned cities $1-10$

Total Cost: $$475,542$ Total Cost: $$385,519$

$-$ Select your answer $-$

$($b$)$ Offhaus is not sure if there is the correct number of carriers to select. Run the model you developed in part $($a$)$ for allowable carriers varying from one to seven. Based on results, how many carriers would you recommend?