For the given scenario, please provide complete AMPL IDE code $(.$mod, $.$dat, $.$run,$)$

$1$ Problem description: In an intelligent warehouse, stock$-$keeping units$1$

$($SKUs$)$ are initially stored on storage racks in the picking area. The warehouse has a number of workstations where the SKUs are picked from the racks by robot arms and placed into the packages to be shipped. A large number of AGVs, which pick up racks from the picking area and transport them to the workstations, surround the

storage racks. The figure on the cover shows typical racks being carried by AGVs $($the small robots at the bottom$).$ Each rack has $10$ storage grids, each storing only one type of SKU. A typical order consists of certain quantities of various types of SKUs. When an order arrives, it is distributed among the workstations. $($Note that distributing the orders to the workstations is itself an important problem; we initially assume that this is given to us as part of our input$).$ Each workstation contains several berths for the AGVs. After receiving an order, the dispatch centre examines the types and the number of SKUs in that order and sends commands to one or multiple AGVs to pick up and transport racks to the workstation. The candidate racks include the ones in the picking area and the ones already being carried by AGVs.

If a rack has been assigned to a workstation, it needs to be transported by an AGV. In particular, if a rack is currently not moving, an idle AGV will be assigned to pick it up and deliver it to the target workstation. If instead a rack is currently moving and it is not assigned to a workstation, then the AGV that is currently transporting the rack will carry it from the current location to a workstation. When an AGV carrying a rack arrives at the workstation, it will park in a berth near the workstation. If the workstation has no empty berth at the moment, no rack is assigned to this workstation. A typical objective is to minimise the total distance travelled by the AGVs over a typical day. However, doing this would require to include a time dimension $($i$.$e$.$ orders arrive over time, hence one will need to re$-$optimise many times over a time period$)$ as well as a forecasting component $($i$.$e$.$ dispatching decisions need to be taken at regular time intervals, without knowing the future orders$).$ To keep things simple, we consider a time$-$snapshot of the warehouse operation, where we know the positions of the racks, AGVs, current orders to be fulfilled and berths available. Given that typically we will not have sufficient available berths and AGVs to fulfill all current orders, our objective function will be some weighted combination of the total distance and the number of orders left unfulfilled.

$2$ Data description:The warehouse is a $50$m$\backslash$times $50$m square with four workstations located at each corner. There are $40$ SKUs, $20$ AGVs, and $80$ racks. The data is provided in the file Project$\_$data.xlsx$.$ Below is a summary of the data contained

in each tab of the file.

$$Coordinates and status$$: this contains the $$x$,$y$$ coordinates of the racks, workstations, and AGVs. For the racks, numbered $1$ through $80,$ the $$status$$ columns indicates whether the rack is stationary or moving. If the rack is moving, then the $$x$,$y$$ coordinates are not provided but we indicate, in the column $$AGV assigned$,$ the AGV currently transporting the rack.

$$Inventory matrix$$: here we show the stock level of each SKU in each rack $($note that, since each rack has $10$ storage grids, each rack carries at most $10$ distinct types of SKU$).$

$$Demands$$: this tab provides the number of orders of each SKU to be dispatched to each workstation.

$$Berths$$: number of open berths for each workstation.

$3$ Project aim: Since the AGVs move only along the x$,$y$-$axes, the distance between two locations in the warehouses is given by the $$Manhattan distance$,$ i$.$e$.$ the $`1$ norm, where the distance between two points $($x$(1),$ y$(1))$ and $($x $(2),$ y$(2))$ is defined as $||($x$(1),$ y$(1))($x$(2),$ y$(2))||=|$x$(1)$ x$(2)|+|$y$(1)$ y$(2)|.$

Our objective, which we intend to minimise, is a weighted combination of the total distance of the rack from their assigned AGV $($ weight $\backslash$alpha $1=1),$ the total distance of the rack from their assigned workstation $($ weight $\backslash$alpha $1=1\mathrm{.}3),$ and the total number of orders that will be left unfulfilled $($weight $\backslash$alpha $1=3).($Note that it makes sense to assign larger weight in the objective to the distance the

AGV travel when carrying a rack, since they will be slower and consume more energy.$)$

$4$ Building an integrated orderdispatching system: Currently House.AI has a legacy system to assign orders to workstations. They would like to explore the benefit of integrating the order$-$dispatching decision into the AGV$-$dispatching model. In this

second part of the project, the task is to include the assignment of orders to workstations in the optimisation model. How does the integrated model compare to the previous one? What is the difference in cost$?$ How many more orders are fulfilled?