Question $1$: Hand$-$Simulation $(60$ points$)$

Consider a warehouse where trucks arrive to either pick$-$up or deliver items. There are two workers $($W$1,$

W$2)$ for transferring the items between the trucks and the warehouse, and one manager $($M$)$ keeps track of

the incoming and outgoing item amounts and types.

Trucks arrive for one of two purposes: to pick up some items $($pick$-$up trucks$-$ PT$)$ or to deliver some items

$($delivery trucks$-$ DT$).$ Pick$-$up trucks do not need the manager and are processed by either one of the two

identical workers since the amount and types of items to be sent are already decided and approved by the

manager on the previous day. Delivery trucks, however, need to get the manager's approval before leaving

the warehouse. For delivery trucks, firstly, one of the workers takes whatever items are to be delivered to

the warehouse, and then the manager inspects the items and gives approval. Upon this inspection, the

truck can leave the warehouse.

In this system, since the trucks are loaded or unloaded from the rear $($behind$),$ a last$-$come$-$first$-$serve $($LIFO$)$

queue discipline is used. This is because only the rear of the last arriving truck is available. If$,$ however, a

truck arrives while another is being processed, the arriving truck leaves some space between the truck that

is in process and joins the queue. After a truck is done, the first idle worker moves on to the truck that has

joined the queue most recently $($the last truck in the queue$).$

It takes $30$ minutes to load or unload the truck, so whenever a worker starts working on a truck, it will be

done in $30$ minutes. The manager's approval $($which is only necessary for delivery trucks that are done with

the unloading process$)$ takes $20$ minutes. The manager's queue follows a FIFO discipline. Each worker and

the manager can work on a single truck at a time, and trucks arrive one by one in the system.

The system is empty, and all resources are idle at time $0.$ The manager is given the list of trucks that are

planned to arrive tomorrow. She or he wants to calculate several performance measures on this system.

Help the warehouse by creating a hand$-$simulation table and running the simulation until all entities leave

the system.

A$.$ List all the primary events in the system.

B$.$ Note that there is one manager and two workers. If we want to find the utilization of each person

separately, what variables should we have? Clearly state what values they can have.

C$.$ Run the simulation until all entities leave the system. Construct the simulation table like we did in

the class and recitations, including the event calendar. Consider the parts D$-$J and only keep the

necessary variables and statistical accumulators. For each part, explain which variables and

statistical accumulators you kept in the table so it will be easier for us to follow your solution.

D$.$ What is the average total time spent in the system over all $($completed$)$ trucks?

E$.$ What is the average waiting time for trucks to start loading$/$unloading $($in the worker queue$)?$

F$.$ Draw $Q(t)$ graph for the number of boxes that are waiting in the worker queue.

G$.$ What is the average queue length at the worker queue $($queue that trucks wait for the loading and

unloading$)?$

H$.$ Draw the $B(t)$ graph for the manager.

I. In what percentage of the time does the manager actively work?

J$.$ Specify the type of statistical accumulators in parts D$,$ E$,$ G and I.

Note: The described system is inspired by the article "The vehicle routing problem with simultaneous pickup

and delivery and handling costs" by Richard P$.$ Hornstra, Allyson Silva, Kees Jan Roodbergen,

and Leandro C$.$ Coelho. If you are interested in reading the article, you can find it with the following doi:

https:$//$doi$.$org$/10\mathrm{.}1016/$j$.$cor$\mathrm{.}2019\mathrm{.}104858$ I WANT TO SEE THE EXCEL TABLE $!!!!!!!$