1. Evaluation of Different solution methods Joonkyu Kang

2.  Based on typical Fedex delivery situation ◦ Machines: delivery trucks ◦ Jobs: packages to be delivered  Processing time: traveling time to the destination  Deadline: different for each type of package  Constrained to Manhattan, New York area only  Objective: two possible objectives ◦ Minimizing tardiness of delivery ◦ Maximizing the number of delivery  Quite similar, but can be different  Will concentrate on minimizing tardiness

3.  Simplifying Manhattan area into a graph ◦ Within the same zip code area, the traveling time is less than five minutes ◦ Define vertices as different zip codes, and edges as distance between center of each area ◦ Create edges only between adjacent areas  Since complete graph would take much more computational time  Packages of the same type and delivery trucks have identical characteristics  No further resource constraints  Ignore distance required to come back to the center

4.  Map ◦ Start at Fedex Ship Center at 25 W, 45 th st, New York, NY, 10036 ◦ Use Google Map to calculate cost of edges  Packages ◦ Based on the revenues obtained, calculate the approximate percentage of each package ◦ Cost of each package delivery is 1 ◦ Set deadlines based on package types ◦ In this simulation, we will use 200 packages.  Trucks ◦ Normally, the number of trucks are very small compared to the number of packages ◦ Therefore here we use 10 trucks here.

5.  If we have sufficient number of delivery trucks, we can assign one for each package ◦ Total delivery time: 415.88 (Since every zipcode is in delivery locations) ◦ Total lateness: 16.20  Also, the largest lateness of all packages is 4.310, but it is not as good as above(with delivery time 14.31)

6.  First notice this is a NP Problem  To minimize lateness, the simplest approach would be using EDD rule  The algorithm is: ◦ First align jobs according to the due date ◦ Assign each job to each empty truck  A greedy algorithm in terms of lateness

7.  …for i=1:Count  [TotalTime Assignment] = min(TruckTime);  CurrentLoc = TruckLocation(Assignment);  MatLoc = find(Distance==CurrentLoc);  Destination = Packages(i,4);  DestLoc = find (Distance==Destination);  CurrentTime = Distance(MatLoc(1), DestLoc(1));  TruckTime(Assignment)=TruckTime(Assignment)+CurrentTime;  TruckLocation(Assignment)=Destination;  JobAssign(Assignment,i)=i;  Tmp = TruckTime(Assignment)-Packages(i,3);  disp(Tmp);  Lateness = Lateness + max(TruckTime(Assignment)- Packages(i,3),0);  End…

8.  Processing time of 1,638.2, Lateness of 12,322  Problems ◦ Extremely high lateness compared to the lower bound we obtained  Need to find a better lower bound  Need to compare to other methods ◦ No consideration of processing time ◦ Many factors affect the performance of the algorithm in this problem; not simply deadline.

9.  Here we process each package in order  However, at each turn the truck closest to the next package is assigned for delivery  Repeat until all packages are assigned

10.  Destination = Packages(i,4);  DestLoc = find (Distance==Destination);  MinTruck = 0;  MinTruckNo = 0;  MinDistance = 99999;  for j=1:MNo  TempTruck = TruckLocation(j);  TempTruckLoc = find (Distance==TempTruck);  TempDist = Distance(DestLoc(1), TempTruckLoc(1));  if TempDist < MinDistance  MinTruckNo = j;  MinTruck = TempTruckLoc(1);  MinDistance = TempDist;  end  CurrentTime = Distance(MinTruck, DestLoc(1));  TruckTime(MinTruckNo)=TruckTime(MinTruckNo)+CurrentTime;  TruckLocation(MinTruckNo)=Destination;  JobAssign(MinTruckNo,i)=i;  Lateness = Lateness + max(TruckTime(MinTruckNo)-Packages(i,3),0);

11.  Processing time of 480.68 total, with lateness of 2770.6  Problems ◦ Much better result, but with some lateness ◦ The closest truck to the next delivery location is not always the best choice for tardiness ◦ Need to incorporate choice of packages ◦ Also, the completion time of trucks vary; therefore extra lateness occurs

12.  By using a greedy algorithm, we could obtain somewhat satisfying schedule in terms of processing time  However, we are not completely sure if the large lateness is because of the number of trucks or the problem of algorithms  We can expect to obtain the tardiness closer to the lower bound by making the completion time of each truck even

13.  Critical Path Method: Applying Graph Theory ◦ Define an appropriate sink node and apply the following algorithm:  For each truck, find a critical path to the sink  Delete the path previously used ◦ Problems: is independent of the characteristics of packages. Data size gets enormous.  Check sensitivity of algorithms by changing factors  In order to have completion time even, mix different algorithms

14.  Fedex Website(www.fedex.com/us )  Google Map(maps.google.com)  Class Notes on NP-Completeness