U.S Aircrafts Carriers Arrangement Scheduling

1. U.S Aircrafts Carriers Arrangement Scheduling Dongxu Wu SiningGu Hong Ouyang

2. The Map

3. Distance between Dif. Spots

4. Basic Stand-by Strategy • Each base can accommodate at most 2 carriers • Each carrier can only stay on one hot-spot\ • 12 hotspots. Gen. 12 different strategies that minimize the total response time of aircraft carriers for each hot post. • Compare and analyze, then get the best solution. S1. Goal: Minimize Response Time

5. For each strategy: || • Let Tij be the time needed for aircraft carrier i to travel from spot j to destination. • Let Cij be the placement of aircraft carrier i at spot j (binary). • Min S.T. , All variables St1. Formulation

6. # File Project.mod # model for final project param T{iin 1..12, j in 1..18}; varc{iin 1..12, j in 1..18} integer >=0; minimize Response: sum{i in 1..12, j in 1..18} T[i,j]*c[i,j]; subject to Sixbase: sum{i in 1..12, j in 1..12} c[i,j] = 6; subject to Sixhotspot: sum{i in 1..12, j in 13..18} c[i,j] = 6; subject to maxbase {j in 13..18}: sum{i in 1..12} c[i,j] <= 2; subject to maxhotspot {j in 1..12}: sum{i in 1..12} c[i,j] <= 1; subject to oneship {i in 1..12}: sum{j in 1..18} c[i,j]=1; S1.amplmod. Code

7. St1. Data file

8. St1. Results

9. We get the U.S navy 12 carriers response ability is as right side • (based on 6 existed bases and 12 hot spots) St1.Conclusion

10. No 1-12 carriers are located at the 12 hotspots respectively. Each hotspot has one carrier. • All the carriers are now performing on some tasks. Once they receive the order to return to bases, they should first accomplish the task and then return. Goal: • Find the shortest time needed for all the 12 carriers to return to a base. Strategy 2. Assumptions

11. Qm|rj|Cmax. • Qm = 12 carriers with different speed • rj= the time each carrier spend on continue finishing the task • Cmax = minimize the length of time needed for all the 12 carriers to return to the bases St2. Formulation

12. St2. Data

13. paramnumcarriers = 12; paramnumbases = 6; param T {1..numcarriers, 1..numbases} >=0; var X {1..numcarriers, 1..numbases} binary; minimize Makespan {i in 1..numcarriers}: max { sum {j in 1..numbases} T[i,j]*X[i,j]}; subject to Carrier_Constraints {i in 1..numcarriers} : sum {j in 1..numbases} X[i,j] = 1; subject to Base_Constraints {j in 1..numbases} : sum {i in 1..numcarriers} X[i,j] = 2; St2. cplexamp Code

14. Cmax = 19.1 St2. Result

15. Job Shop Scheduling Problems •  3 jobs (carriers), 4machines(hotspots) • Each carrier follows a predetermined route • (a sequence of hotspots) Strategy 3: • Operation (i,j) : Processing of carrier j on hotspot i • Processing time pij • Assume that carriers do not recirculate • Minimize Cmax Strategy 3, Job Shop

16. St3. data

17. paramnumhotspots = 3; • param carrier = 3; • param M = 10000; • param C >= 0; • param T [i in 1..numhotspots, j in 1.. carrier]; • param P [i in 1..numhotspots, j in 1.. carrier]; • var X[ i in 1..numhotspots, j in 1..carrier, k in 1.. carrier] binary; • minimize Makespan: C; • subject to makespan1: C>=T[3,1] + P[3,1]; • subject to makespan2: C>=T[3,2] + P[3,2]; • subject to makespan3: C>=T[4,3] + P[4,3]; • subject to job1:t21 ≥ t11 + p11;  • subject to job2:t31 ≥ t21 + p21; • subject to job3:t12 ≥ t22 + p22; • subject to job4:t42 ≥ t12 + p12; • subject to job5:t32 ≥ t42 + p42; • subject to job6:t23 ≥ t13 + p13; • subject to job7:t43 ≥ t23 + p23 • subject to constraint {i in 1..numhotspots,}: • {j in 1.. carrier, j<k}  • { •    T[i,1]+P[i,1] <= T[i,2]+M*(1-X[i,1,2]); •    T[i,2]+P[i,2] <= T[i,1]+M*X[i,1,2]; •    T[i,1]+P[i,1] <= T[i,3]+M*(1-X[i,1,3]); •    T[i,3]+P[i,3] <= T[i,1]+M*X[i,1,3]; •    T[i,2]+P[i,2] <= T[i,3]+M*(1-X[i,2,3]); •    T[i,3]+P[i,3] <= T[i,2]+M*X[i,2,3]; • } St3. Code

18. Optimal solution with makespan 60 St 3. Result

19. Carriers Arrange Schedule for 10 Hotspots • Assume there are 10 carriers (C0–C9) available and there are also 10 hotspots(H1–H10) need to be watched. Each time only one carrier watching one hotspot. • Every hotspot must be processed on each of the 10 carriers in a predefined sequence. • The objective is to minimize the completion time of the last Hotspot(job) to be processed (Makespan). • The hotspots are described in the data set “arr” by using the following statements. Strategy 4. A Bigger Job Shop

20. /* Hotspots specification */ data arr(drop=i mid); do i=1 to 10; input mid _DURATION_ @; _RESOURCE_=compress(‘C'||put(mid,best.)); output; end; datalines; SAS/OR Code

21. /* create the Activity data set */ data act (drop= i j); format _ACTIVITY_ $8. _SUCCESSOR_ $8.; set arr; _QTY_ = 1; i=mod(_n_-1,10)+1; j=int((_n_-1)/10)+1; _ACTIVITY_ = compress('J'||put(j,best.)||'P'||put(i, best.)); JOB=j; TASK=i; if iLT 10 then _SUCCESSOR_ = compress('J'||put(j,best.)||'P'||put((i+1),best.)); else _SUCCESSOR_ = ' '; output; run; The minimum makespan would be 617—the time it takes to go to Hotspot 1.(finish hotspot 1 job) St4. Infeasible Schedule(Uncontrained)

22. procconst domain=[0,842] actdata=actdata schedout=sched_jobshop dpr=50 restarts=150 showprogress; schedule finish=842 edgefindernf=1 nl=1; run; Result: 842 days St 4. Add an Edge-finder algorithm(Get a Contrained Schedule)

23. http://www.fas.org/sgp/crs/weapons/RS20643.pdf • http://www.fas.org/programs/ssp/man/uswpns/navy/aircraftcarriers/cvn68nimitz.html • http://www.wired.com/dangerroom/2013/03/replacing-aircraft-carriers/all/ • http://www.jmp.com/community/ • http://www.columbia.edu/~cs2035/courses/ieor4405.S14/index.html Sources

24. Thank you!