Problem 44718. Optimize the Tollbooths

Created by Aamod Garg

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<div style = "text-align: start; line-height: 20px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; "><div style="display: block; min-width: 0px; padding-top: 0px; transform-origin: 332px 171px; vertical-align: baseline; perspective-origin: 332px 171px; "><div style="font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; perspective-origin: 309px 42px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; "><span style="">Your company has recently built its own highway from which they hope to generate some revenue. The highway has no branches or intersections, it is a simple line segment. Your company also has access to simple data from some potential customers that describe their start and endpoint locations and their budget.</span></span></div><div style="font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; perspective-origin: 309px 42px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; "><span style="">Tollbooths are also located on various locations on the highway. The total toll a customer pays is the sum of all tolls on the tollbooths that lie between their start and end locations. If a customer cannot afford the total toll they must pay, then they simply don't make the trip and end up paying nothing.</span></span></div><div style="font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 52.5px; white-space: pre-wrap; perspective-origin: 309px 52.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; "><span style="">The customer wants to start or end their destination at the precise location of a tollbooth, so we can represent the problem as follows. We have a graph that is a simple path with N nodes. Each node represents a potential start or end location of a customer and there is a single tollbooth located on each node. So the total toll a customer pays is the sum of the tolls on the nodes they cross to reach their endpoint. Both the start point and end point are also toll booths.</span></span></div><div style="font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 21px; white-space: pre-wrap; perspective-origin: 309px 21px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; "><span style="">Your task is to set the tolls on each of the tollbooths to generate the</span></span><span style="display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; "><span style=""> </span></span><span style="display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; "><span style="font-weight: bold; ">maximum revenue</span></span><span style="display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; "><span style=""> for your company. The answer is the pricing of each toll booth.</span></span></div></div></div>

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11 Solutions
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Last Solution submitted on Aug 23, 2020

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J. S. Kowontan on 4 Aug 2018

Hi, Aamod, this is quite an interesting problem. I'll appreciate your clarification on test case 2. The toll pricing strategy that maximized the profit was stated as [10, 5, 0]. . Hence, for the given option and budget, the total profit will be 30. But a quick check shows that a toll pricing strategy of [5, 10, 0], which is valid for the given option and budget, also gives a total profit of 30. Unfortunately, the test suite assumes uniqueness of the profit maximizing toll strategy. In such situation, how are we to proceed? Thanks.

Alfonso Nieto-Castanon on 23 Aug 2020

I agree, the problem as stated has multiple valid solutions. Could you please change the testsuite to accommodate that?: e.g. by changing "assert(isequal(toll_pricing_strategy(option,budget),y_correct))" to "y=toll_pricing_strategy(option,budget); assert(sum((option*y').*(option*y'<=budget))==sum(option*y_correct'));" which checks whether the solution results in the same revenue as your "optimal" solution...

goc3 on 29 Sep 2020

@Alfonso, I tested your suggestion against the reference solution provided by Aamod Garg (problem author). His solution worked for test cases 2 and 3, but failed 1, 4, and 5 due to that modification (therefore, I reverted to its original state). Do you have a further suggestion to fix this?

Alfonso Nieto-Castanon on 29 Sep 2020

@goc3, I would suggest "y=toll_pricing_strategy(option,budget); revenue=@(x)sum((option*x(:)).*((option*x(:))<=budget)); assert(revenue(y)>=revenue(y_correct))". That should allow equivalent or better alternatives to the author's reference solutions (if I am not mistaken this is necessary because in problems 4 and 5 the reference solution does not appear to be optimal; e.g. in problem 5, tolls [18 33 9 14 13 11 12 10] result in 330 revenue, while the reference solution results in 280 revenue)

goc3 on 3 Oct 2020

@Alfonso, the test suite has been updated per your suggestion. It does work against the author's reference solution.

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Problem 44718. Optimize the Tollbooths

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