A new automatic storage and retrieval $($ASAR$)$ area is being added to an existing warehouse on

land already owned by the company. It will have $>=1$ aisles, each with pallet storage cells on

both sides and a stacker crane moving in the middle which can carry a pallet to$/$from any location

in the aisle. Storage must be provided for a total of at least p pallet cells of width w$,$ depth d$,$ and

height h feet.

All racks will be $>=1$ cells high, and k cells from one end of the aisle to the other on each side.

The total building height should not be more than t feet, including a clearance of u feet between

the top of the racks and the ceiling. Aisle width will be $150\%$ of pallet depth to allow for clear

passage of pallets carried by the cranes, and their length should accommodate the k storage cells

plus one extra pallet width to provide for an input$/$output station at the end.

The ASAR area and its input$/$output stations will be enclosed with a roof and walls on three sides,

being open only on the end $($perpendicular to the aisles$)$ where it adjoins the existing warehouse.

Engineers want to find a minimum total cost design for the new facility using unit cost of cranes

$($c$1),$ unit cost of steel racks per pallet storage cell $($c$2),$ together with c$3,$ c$4,$ and c$5$ being the

construction cost per square of foundation$/$floors$,$ ceilings, and sidewalls, respectively.

Formulate an NLP model to choose an optimal design in terms of decision variables n$,$ m$,$ k$,$ and

building exterior dimensions x $($perpendicular to the aisles$)$ y $($parallel to the aisles$),$ and z$($height$).$

Find the optimal solution if the company wants to store $500$ pallets $($w$=4,$ d $=4,$ and h $=4$ feet$).$

The total building height should not be more than $25$ feet, including a clearance of $2$ feet between

the top of the racks and the ceiling. The unit cost of cranes is $$25,000,$ unit cost of steel racks per

pallet storage cell is $$100,$ and the construction cost per square of foundation$/$floors$,$ ceilings,

and sidewalls are $$10$