Recursive solution fig] denote the fastest possible time to get a chassis from the starting point through station ei denote an entry time for the chassis to enter assembly line denote an exit time for the completed auto to exit assembly line i iy denote the assembly time required at station S denote the time to transfer a chassis away from assembly line i after through station S Our ultimate goal is f=mingi[n]+x,f2[n]+x2)
Recursive solution denote the fastest possible time to get a chassis from the starting point through station Sij. Our ultimate goal is: f * = min(f1 [ n] + x 1, f2 [ n] + x 2 ). denote an entry time for the chassis to enter assembly line i. denote an exit time for the completed auto to exit assembly line i. denote the assembly time required at station Sij. denote the time to transfer a chassis away from assembly line i after through station Sij. fi [j ] ei xi ai,j ti,j
Recursive solution(cont We obtain the recursive equations e,t a f min(1-11+a1/27-1+t21+a1,)ifj22 if j=I falL min(-1]+a2yV-1]+11+a2)ifj22 Lili denote the line number i, whose station j-1 is used in a fastest way through station s
Recursive solution (cont.) We obtain the recursive equations f1 [j] = e 1 + a1,1 if j = 1, min(f1 [j − 1] + a1,j, f2 [j − 1] + t2,j − 1+ a1,j) if j ≥ 2. f2 [j] = e 2 + a2,1 if j = 1, min(f2 [j − 1] + a2,j, f1 [j − 1] + t1,j − 1+ a2,j) if j ≥ 2. denote the line number i, whose station j − 1 is used in a fastest way through station Sij. li [j ]
Computing the fastest times Station Station Station StationStation Station S Assembly line 1 4 p Chassis Completed enters auto exits Assembly line 2(8(5(64 Station Station Station Station Station Station j123 23456 f lUI
Computing the fastest times 7 9 3 4 3 2 2 4 8 4 8 5 6 4 5 7 2 2 3 1 1 2 3 2 4 1 Chassis enters Assembly line 1 Completed auto exits Assembly line 2 Station S1,1 Station S1,2 Station S1,3 Station S1,4 Station S1,5 Station S1,6 Station S2,1 Station S2,2 Station S2,3 Station S2,4 Station S2,5 Station S2,6 12345 6 2345 6 f1[j] j f2[j] l1[j] j l2[j]
Computing the fastest times Station Station Station StationStation Station S Assembly line 1 4 Chassis Completed enters auto exits Assembly line 2(8(5(64 Station Station Station Station Station Station j123 23456 fun lUI
Computing the fastest times 7 9 3 4 3 2 2 4 8 4 8 5 6 4 5 7 2 2 3 1 1 2 3 2 4 1 Chassis enters Assembly line 1 Completed auto exits Assembly line 2 Station S1,1 Station S1,2 Station S1,3 Station S1,4 Station S1,5 Station S1,6 Station S2,1 Station S2,2 Station S2,3 Station S2,4 Station S2,5 Station S2,6 9 12345 6 2345 6 f1[j] j f2[j] l1[j] j l2[j]
Computing the fastest times Station Station Station StationStation Station S Assembly line 1 4 p Chassis Completed enters auto exits Assembly line 2(8(5(64 Station Station Station Station Station Station j123 23456 lUI
Computing the fastest times 7 9 3 4 3 2 2 4 8 4 8 5 6 4 5 7 2 2 3 1 1 2 3 2 4 1 Chassis enters Assembly line 1 Completed auto exits Assembly line 2 Station S1,1 Station S1,2 Station S1,3 Station S1,4 Station S1,5 Station S1,6 Station S2,1 Station S2,2 Station S2,3 Station S2,4 Station S2,5 Station S2,6 9 12345 6 12 2345 6 f1[j] j f2[j] l1[j] j l2[j]