October 1997 Presentation Topic (continued)

Observe that routes have been numbered according to the statement of the problem.

Also observe that in order to make shipping a continuous steady state process, to have one ship per day arriving (or leaving) a given port, there will have to be a queue of ships spaced one day apart along the adjoining route. Furthermore, once a ship has finished a cargo route, it may have to travel empty to another port in order to begin another route.

For example, in order to have 1 ship per day shifting from route 2 to route 1, there will need to be a queue of 10 ships spaced out along the itinerary MA -> IS ->DH. This is because it takes a ship 10 days from the time it begins route 2 until the time it is ready to begin route 1: 1 day loading cargo at Marseilles, 3 days travelling along route 2 to Istanbul, 1 day unloading at Istanbul, and 5 days travelling empty from Istanbul to Dhahran in preparation for beginning route 1.

In general, in order to have 1 ship per day shifting from route *i* to
route *j*, there must always be a queue of ships involved in
servicing route *i* then travelling
to begin route *j*, where is 2 plus the time taken to travel route
*i* plus the time taken to travel from the endpoint of route *i* to the
starting point of route *j*.

The values of are tabulated below:

If we let be the number of ships shifting from route *i*
to route *j* daily, then the total number of ships servicing
route *i* at any one time (where "servicing route *i*" includes
getting ready for the next route after finishing route *i*) is

Our goal is to choose the so as to minimize the total number of ships, which is

The only constraints are that the daily number of ships completing a route must equal the daily number of ships beginning it, and this must equal the number specified in the problem (3 for route 1, 2 for route 2, and 1 for routes 3 and 4).

Under our definition for ,
yields the daily number of ships completing route *i*. This gives
constraints
,
,
,
.

Similarly
yields the daily number of ships beginning route *i*. This gives
constraints
,
,
,
.

Eureka! a transportation problem of modest size. Its solution is

Which we interpret as follows:

Of the three ships servicing route 1 daily, one goes back to begin route 1 again, one goes on to route 1, and one goes on to route 4. Referring back to the table for , this means that at any given time

- 36 ships are occupied travelling route 1 (DH -> NY) then returning empty in preperation for another route 1 trip.
- 32 ships are occupied travelling route 1 then going empty to MA in preparation for a route 2 (MA -> IS) cargo journey;
- 19 ships are occupied travelling route 1 then reloading in preparation for a loaded journey to MA (route 4).

Similarly, we find that, at any given time,

- 10 ships are occupied travelling route 2 (MA -> IS) then going empty to DH in preparation for a route 1 (DH -> NY) cargo journey;
- 7 ships are occupied travelling route 2 then going empty to NA in preparation for a route 3 (NA -> BO) cargo journey;
- 12 ships are occupied travelling route 3 (NA -> BO) then going empty to DH in preparation for a route 1 (DH -> NY) cargo journey;
- 15 ships are occupied travelling route 4 (NY -> MA) then reloading in in preparation for a route 2 (MA -> IS) cargo journey.

Thus, 131 ships are needed in all.

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