A company sends products to $12$ different cities. Product shipping is done in two different ways. If the capacity of the truck is full, it is sent to the cities by truck. If the truck is not full, the products are sent by truck to $3$ dealers in the city. Products are expressed in pallets. The capacity of the truck is $12$ pallets and the capacity of the truck is $4$ pallets. The truck is charged at $35$ TL per kilometer. Truck is charged at $550$ TL per trip. $.$ The daily pallet needs of cities are as follows; $1$ pallet for city $1,4$ pallets for city $2,1$ pallet for city $3,2$ pallets for city $4,1$ pallet for city $5,2$ pallets for city $6,1$ pallet for city $7,1$ pallet for city $8,1$ pallet for city $91$ pallet for city, $1$ pallet for city $10,1$ pallet for city $11$ and $2$ pallets for city $12.$ The company's stock includes $2$ pallets for city $1,2$ pallets for city $2,1$ pallet for city $3,1$ pallet for city $4,1$ pallet for city $5,1$ pallet for city $6,1$ pallet for city $7,1$ pallet for city $8,$ There is $1$ pallet for city $9,1$ pallet for city $10,1$ pallet for city $11$ and $1$ pallet for city $12.$ There are certain rules that the factory follows when shipping products. These; The truck cannot move until it is completely full. The truck can visit a maximum of two cities at a time. If it visits a second city, the distance from the first city to the second city is recalculated and charged. The products can wait at the factory for a maximum of three days. If the capacity of the truck is not full at the end of the third day, the pallets are delivered to the dealers by truck. is distributed. Dealers carry the pallets themselves, and we pay a transportation fee for the pallets we send to the dealers according to the rate that a truck normally charges when transporting pallets to the destination city. For example, if a $12-$capacity truck goes to city $1$ and carries $3$ pallets belonging to us$,25\%$ of the total cost is paid. The transportation costs of the provinces to each other are calculated in the table given with the cost matrix. Here, the exit city is $0.$ Based on this data, we will need to decide how to distribute the products to dealers at minimum cost. Should the products be sent by truck or truck, and should the pallets be sent to a single city or two separate cities depending on the production. This is an assignment problem and I want you to write a mathematical model of this problem.

0 city 0 city 1 city 2 city 3 city 4 city 5 city 6 city 7 city 8 city 9 city 10 city 11 city 12 City 0 22575 26985 21630 11130 29680 9555 22120 11655 20580 10640 24185 15855 City 1 22575 49560 44205 33705 52255 32130 44695 34230 43155 33215 46760 38430 City 2 26985 49560 48615 38115 56665 36540 49105 38640 47565 37625 51170 42840 City 3 21630 44205 48615 32760 51310 31185 43750 33285 42210 32270 45815 37485 City 4 11130 33705 38115 32760 40810 20685 33250 22785 31710 21770 35315 26985 City 5 29680 52255 56665 51310 40810 39235 51800 41335 50260 40320 53865 45535 City 6 9555 32130 36540 31185 20685 39235 31675 21210 30135 20195 33740 25410 City 7 22120 44695 49105 43750 33250 51800 31675 33775 42700 32760 46305 37975 City 8 11655 34230 38640 33285 22785 41335 21210 33775 32235 22295 35840 27510 City 9 20580 43155 47565 42210 31710 50260 30135 42700 32235 31220 44765 36435 City 10 10640 33215 37625 32270 21770 40320 20195 32760 22295 31220 34825 26495 City 11 24185 46760 51170 45815 35315 53865 33740 46305 35840 44765 34825 40040 City 12 15855 38430 42840 37485 26985 45535 25410 37975 27510 36435 26495 40040