Track Inspection Planning Cooperate has recently completed a project for the Dep

Question

Track Inspection Planning Cooperate has recently completed a project for the Department of Transportation, which was for optimizing the track inspection planning on the railroad network. In this question, you are asked to formulate a simpler version of the track inspection planning problem. In particular, suppose that there 5 rail tracks that you can inspect. Each track has different inspection importance and each track has different inspection times. The table below gives the importance level and inspection time for each track. Track 1 10 5 hours Track 2 Track 3 13 8 hours Track 4 12 7 hours Track 5 Importance Level Inspection time 4 hours 6 hours As the inspection planner, you want to determine which tracks to inspect such that you maximize the total importance level of the inspections. However, you have one day, i.e., 24 hours available for inspections. That is, total inspection time cannot exceed 24 hours. a) (10 points) Formulate a binary linear programming model for the above inspection planning problem by defining you decision variables and writing the objective and objective function and the constraints in terms of your decision variables. Combine everything to get the final model. b) (15 points, 3 points each) Mathematically formulate the following restrictions as constraints independent of each other and the constraints in part a. You should formulate a single constraint for each part. i. If you inspect track 1, then you have to inspect track 2. ii. You can either inspect track 3 or track 4, but not both. ii. If you inspect both track 1 and track 2, then you have to inspect track 4. iv. You cannot inspect track 3 unless you inspect track 4. v. You cannot inspect track 1 unless you inspect track 3; and, you cannot inspect track 3 unless you inspect track 1.

Explanation / Answer

(a)

Let Yi be the set of binary integers such that Yi=1 when Track-i is inspected and Yi=0 otherwise. i=1,2,..5

Objective Function: Maximize. Z = 10Y1 + 11Y2 + 13Y3 + 12Y4 + 9Y5
Subject to,
5Y1 + 4Y2 + 8Y3 + 7Y4 + 6Y5 <= 24 (hours restriction)
Yi = {0,1}

(b)

(i) Y2 - Y1 >= 0

(ii) Y3 + Y4 = 1

*(iii) Y1 + Y2 - Y4 <= 1

(iv) Y4 - Y3 >= 0

(v) Y1 - Y3 = 0

Explanation of (iii)

We avoid the barred (i.e. Y4=0 even when Y1=Y2=1) by limiting the constraint's RHS to 1 and not 2.

Y1 Y2 Y4 Y1+Y2 - Y4 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 1 -1 1 1 0 2 (barred)

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