I need a detailed solution about this. AMS 341 Operational research 1 6. (15 poi

Question

I need a detailed solution about this. AMS 341 Operational research 1

6. (15 points) A bus company has a total of 4 busses it can use on its daily routes. There are 3 possible round trip routes (NY-Atlantic City, NY-Boston, NY-Stony Brook). The table below shows the profit given how many busses are assigned to each route: (Note, if no busses are sent on a particular route, the profit is zero for that route.) 1 bus 2 busses3 busses 4 busses NY-Atlantic City100 NY-Boston NY-Stony Brook 120 150 160 140 180 200 160 200 210 170 80 Assume that each bus is assigned to a single route. The company wishes to maximize its profit To solve the problem using Dynamic Programming define fi(s)-the maximum profit stages i and above and state s Solve the problem. Make sure to state at the end how many busses are assigned to each route. (A solution by guessin get no credit, I want to see your computations using f(s) with the stages and states you defined.)

Explanation / Answer

Basically, in a Dynamic programming, we divide the problem into many subproblems(Stages) and then use the solution of the sub-problems to calculate the overall solution to the problem.

Hence, in this case, we divide our problem to allocate buses to different routes into 3 states:

Stage 1: Next we try to figure out solution for each state F1(s)= Maximize(State profits)

State 1: All 4 buses assigned to one route: F1(s)= Maximize(State 1 profits)

The route can be Atlantic, Boston or Stony Book and we would select Boston as it gives the max profit =210

State 2: 3 buses assigned to one route and 1 bus to second route: F1(s)= Maximize(State 2 profits)

Hence the route for 3 buses would be Boston giving us 200 and the route for 1 bus would be stony brook giving us 120. Hence the total profit would be 320

State 3: 2 buses assigned to one route and 1 each assigned to one route: F1(s)= Maximize(State 3 profits)

The maximum profit for assigning 2 buses to one route is Boston and the profit is 160 and if we have to assign 1 bus each to 2 routes, we would pick the top 2 profitable routes which would be Stony Brook (120) and Atlantic City (100) which gives us total profit of 160+120+100 = 380

Stage 2: F2(s)= Maximize(State 1 profit, State 2 profit, State 3 profit)

Now that we have maximized the 3 options, we compare the values in all and choose the most profitable one which would be state 3 that is Assigning 2 buses to Boston and 1 each to Stony Brook and Atlantic city.

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