1. Project selection with sets of mutually exclusive alternatives Harald Minken, TOI ITEA Conference, Oslo June 2015

2. Side Independent projects, budget constraint  The linear programming problem  The solution, if b 1 /c 1  b 2 /c 2  …  b n /c n : 2

3. Side A reformulation  Set  Reformulated problem  «Same» solution 3

4. Side Mutually exclusive alternatives  At an early stage, projects usually have mutually exclusive alternative concepts, route options etc.  Different route options (different places to cross a fjord, through or around a small town, etc.)  Different concepts for solving a transport problem, different designs of a transport facility  Mutually exclusive alternatives in the form of all combinations of small groups of interdependent projects  Mixed cases 4

5. Side The simplest case 5

6. Side The choice between projects ABE and BCEF  The best alternative is the one with the largest volume above the k line:  ABE > BCEF  AB > C   Our choice indicator in cases like this should be: 6

7. Side The general case  The problem 7

8. Side Lagrangian relaxation The new objective function consists of a constant plus a sum of w ij indicators of the same form as in the former example: 8

9. Side Solution  Assume k is given, and maximise V(k) by - For each project j, select the i that maximises w ij  Then find the minimal k that keeps the budget constraint: - If the sum of costs c ij at the initial k exceeds a, increase k and repeat (or if it is much smaller than a, decrease k and repeat)  Stop when sum of costs c ij is just below a.  We use the fact that by construction, k is both a Lagrangian multiplier and the BCR of the last project that fits within the budget constraint 9

10. Side E39 Aksdal-Bergen. Cost c, benefit cost ratio h and the indicator w(k) at different values of k 10 K2K3K4AK4CK4DK5AK5B c5,332,513,829,113,628,023,4 h (=NNB)01,0 1,71,11,7 w(0) 0331429233133 w(0,25) 241022202427 w(0,5) -316715161721 w(0,75) -4837131015 w(1) -50001039 w(1,5) -8-16-7-153-11-2 w(2) -11-33-14-29-4-25-14

11. Side Conclusion  We have shown that the BCR rule breaks down if there are many alternative designs of one or more of the project proposals.  We have also shown how to select projects and project alternatives for the optimal plan in this case.  In outline, this was shown already in 1955 by Lorie and Savage, but has nor to my knowledge been used in pratice, at least not in transport. 11