Optimization Methods for the Single-Machine Problem

1. Optimization Methods for the Single-Machine Problem Chapter 3 Elements of Sequencing and Schedulingby Kenneth R. Baker Byung-Hyun Ha R4

2. Outline • Introduction • Adjacent pairwise interchange methods • A dynamic programming approach • Dominance property • A branch and bound approach • Mixed integer programming formulation • from Baker and Trietsch, 2009 • Summary

3. Introduction • Different scheduling procedures for different measures • F-problem and L-problem by SPT sequencing • U-problem by Algorithm 1 of Ch. 2 • T-problem by ?? • NP-hard (currently no efficient algorithms known) • Optimization methods • Dynamic programming approach • Branch and bound approach • Approximation schemes, ... • Using optimization problem solvers • Problems are mostly described by mathematical programming model. • The solvers (e.g., CPLEX, LINDO) employ branch-and-cut, branch-and-price algorithms, etc.

4. Introduction • Typical scheduling problems • Penalty function, usual • gj(t) -- a penalty incurred when job j completes at time t • assuming nondecreasing with regard to t • Maximum penalty problem • to minimize the maximum gj(t), e.g., Tmax-problem • Total penalty problem • to minimize the sum of gj(t), e.g., T-problem • Maximum penalty problems • Theorem 1 • When the objective is to minimize the maximum penalty, job i may be assigned the last position in sequence if gi(P)  gk(P) for all jobs k  i, where P is the total processing time of jobs. • Application of Theorem 1 • Tmax-problem -- gj(t) = max{0, t – dj} • Weighted quadratic tardiness -- gj(t) = wj(max{0, t – dj})2

5. Adjacent Pairwise Interchange Methods • Thrust of adjacent pairwise interchange methods • Finding a sequence of which every adjacent pairwise interchange leads to poorer performance • F-problem  optimal by the method • T-problem  not sufficient to identify optimality! • Three-job example  There does not exist transitive (optimal) sequencing rule for T-problem. local optimum 1-3-2 3-1-2 T = 5 T = 4 1-2-3 3-2-1 T = 4 T = 5 2-1-3 2-3-1 T = 3 T = 4

6. Dynamic Programming Approach • Additive form of regular measures • Z = j=1n gj(Cj) • For total tardiness: gj(Cj) = max{0, Cj – dj} • We can easily formulate a dynamic programming for optimal sequence. • Notation • X -- set of all jobs, J -- subset of X • p(J) -- total time required to process jobs in J • Dynamic programming formulation • G(J) = minjJ{G(J – {j}) + gj(p(J))}, where G() = 0 • minimum penalty for the subproblem consisting of the jobs in J • gj(p(J)) -- penalty incurred by job j • G(J – {j}) -- minimum penalty incurred by remaining jobs • Then, minimum total penalty will be G(X). J X – J ... ... j p(J)

7. Dynamic Programming Approach • Example -- X = {1, 2, 3}, gj(Cj) = max{0, Cj – dj} • G(X) = G({1,2,3}) = minj{1,2,3}{G({1,2,3} – {j}) + gj(p({1,2,3}))} = min{G({2,3}) + g1(p({1,2,3})), G({1,3}) + g2(p({1,2,3})), G({1,2}) + g3(p({1,2,3}))} = min{2 + 2, 0 + 4, 0 + 3} = 3  Sequence: 2-1-3 • G({2,3}) = minj{2,3}{G({2,3} – {j}) + gj(p({2,3}))} = min{G({3}) + g2(p({2,3})), G({2}) + g3(p({2,3}))} = min{0 + 3, 0 + 2} = 2 • G({1,3}) = minj{1,3}{G({1,3} – {j}) + gj(p({1,3}))} = min{G({3}) + g1(p({1,3})), G({1}) + g3(p({1,3}))} = min{0 + 0, 0 + 1} = 0 • G({1,2}) = minj{1,2}{G({1,2} – {j}) + gj(p({1,2}))} = min{G({2}) + g1(p({1,2})), G({1}) + g2(p({1,2}))} = min{0 + 0, 0 + 1} = 0 • G({1}) = minj{1}{G({1} – {j}) + gj(p({1}))} = G() + g1(p({1})) = g1(p1) = 0 • G({2}) = minj{2}{G({2} – {j}) + gj(p({2}))} = G() + g2(p({2})) = g2(p2) = 0 • G({3}) = minj{3}{G({3} – {j}) + gj(p({3}))} = G() + g3(p({3})) = g3(p3) = 0

8. Dynamic Programming Approach • Exercise (p. 3.6) • X = {1, 2, 3, 4}, gj(Cj) = max{0, Cj – dj} • Computational efforts • In proportion to n2n • Number of subsets to be considered – 2n • Finding G(J) for each subset J -- O(n) times evaluation • U-problem -- O(nlogn) • All enumerations -- in proportion to n!

9. Dynamic Programming Approach • Computer implementation • Labeling scheme • Label of a set of jobs J -- L(J) = kJLk , where Lk = 2k-1 is label of job k • Overall procedure 1. Set b(i) = 0 for i = 1..|X|, G(0) = 0 2. Loop 2-1. Find smallest integer j for which b(j) = 0. • If all b(j) = 1, then stop. 2-2. Set b(j) = 1. 2-3. For all i < j, set b(i) = 0. 2-4. Let J = {j | b(j) = 1}. 2-5. Set G(L(J)) = minjJ{G(L(J – {j})) + gj(p(J))}.

10. Dominance Properties • Dominance properties reduces the number of alternatives to be considered. • Those regarding schedules • schedules without inserted idle times constituted dominant set • Those involving relationship between jobs • Theorem 2 • In the Tw-problem, suppose that there exists a job k for which dk p(X). Then job k may be assigned the last position in sequence. • Theorem 3 (Emmons, 1969) (HW 3) • In the T-problem, there is an optimal schedule in which job j follows job i if one of the following conditions is satisfied: (a)pi pjand di max{dj, p(Bj) + pj} (b)di djand dj p(Ai') – pj (c)dj  p(Ai') where • Ai -- set of jobs to follow job i in an optimal schedule; i.e., “after” set • Ai' -- complement set of Ai , i.e., Ai' = X – Ai • Bi-- set of jobs to precede job i in an optimal schedule; i.e., “before” set

11. Dominance Properties • Computer implementation • Modified labeling scheme • Renumbering jobs so that i < j whenever job i dominates job j • Nj = {i | i < j} • L'(J) = kJL'k , whereL'k = L'(Nk) – L'(Bk Nk) + 1 • Overall procedure 1. Set b(i) = 0 for i = 1..|X|, G(0) = 0 2. Loop 2-1. Find smallest integer j for which b(j) = 0. • If all b(j) = 1, then stop. 2-2. Set b(j) = 1. 2-3. For i = j–1, ..., 1, if b(i) = 1 and b(k) = 0 for all kAi , set b(i) = 0. 2-4. Let J = {j | b(j) = 1}. 2-5. Set G(L'(J)) = minjJ:AjJ= {G(L'(J – {j})) + gj(p(J))}. • Example of p. 3.12 -- 5 jobs • A1 = {2}, A2 = , A3 = {4, 5}, A4 = , A5 =  • L'1 = 1, L'2 = 1, L'3 = 3, L'4 = 3, L'5 = 6 Number of subsets to be evaluated are not 25 = 32 but 15.

12. Branch and Bound Approach • Iterations of two fundamental procedures • Branching and bounding • Branching • Partitioning a large problem into subproblems, i.e., replacing an original problem by a set of new problems that are (a) mutually exclusive and exhaustive subproblems of the original, (b) partially solved version of the original, and (c) smaller problems than the original P(0) ... P(1) P(2) P(n) ... ... ... P(12) P(32) P(n2) ... ... ... P(S)

13. Branch and Bound Approach • Bounding • Curtailing enumeration process, by calculating a lower bound on the optimal solution of a given subproblem • Suppose Z is known performance measure and lower bound of a certain subproblem is not less than Z, then the subproblem need not be considered any further. • Fathomed branches • Branches of a subproblem that, no matter how the remainder of the subproblem is solved, the resulting solution can never have a value better than known solution • Active subproblems • Subproblems that have been encountered in branching process but that have not been eliminated by dominance properties and whose own subproblems have not yet been generated • Termination condition • A solution appears at the head of active list, which is a list of active subproblems ranked by lower bound

14. Branch and Bound Approach • Example: a branch and bound procedure for T-problem • Notation • s -- partial sequence of jobs from among n jobs originally in problem • js -- partial sequence in which s is immediately preceded by job j • s' -- the complement of s • js' -- the complement of js • p(s') = js'pj ; p(js') = p(s') – pj • P(s) -- subproblem with a sequence which ends with partial sequence s • P(0) -- original problem • vs = jsTj ; vjs = max{0, p(s) – dj} + vs • bs -- lower bound of subproblem P(s)

15. Branch and Bound Approach • Example: a branch and bound procedure for T-problem (cont’d) • Algorithm 1 -- Branch and Bound 1. (Initialization) Place P(0) on the active list. Set v0 = 0 and p() = jpj . 2. Remove the first subproblem, P(s), from the active list.Let k denote the number of jobs in the partial sequence s.If k = n, stop: the complete sequence s is optimal.Otherwise test Theorem 2 for P(s). If the property holds, go to Step 3; otherwise go to Step 4. 3. Let job j be the job with the latest due date in s'. Create the subproblem P(js) with p(js') = p(s') – pj ; vjs = vs ; and bjs = vs Place P(js) on the active list, ranked by its lower bound.Return to Step 2. 4. Create (n – k) subproblem P(js), one for each j in set s'. For P(js), let p(js') = p(s') – pj ; vjs = vs + p(s') – dj ; and bjs = vjs Now place each P(js) on the active list, ranked by its lower bound bjs .Return to Step 2.

16. Branch and Bound Approach • Example -- T-problem P(0) 7 5 1 P(1) P(2) P(3) P(4) P(5) 13 12 10 10 1 6 3 2 8 P(53) P(54) P(15) P(25) P(35) P(45) 10 10 12 11 9 9 9 4 P(153) P(253) P(453) P(154) P(254) P(354) P(135) P(235) P(435) P(145) P(245) P(345) 14 13 11 19 18 16 13 12 10 18 17 15 10 P(1453) P(2453) P(1435) P(2435) 15 16 12 11 11 P(12435) 11

17. Branch and Bound Approach • A branch and bound procedure for T-problem (cont’d) • Options • Lower bounds • bs = vs , or • bs = vs + minjs'{max{0, p(s) – dj}}, or • ... • Trial solutions for bounding • a schedule obtained during branching, or • one obtained by pursuing tree to bottom as rapidly as possible, or • one by heuristics, such as MDD rule, or • ... • Branching • jumptracking, or • backtracking, or • ... • ...  Enumeration of all the feasible solutions, implicitly!

18. Mixed Integer Programming Formulation • Example: T-problem by sequence-position decisions • Notation • n -- number of jobs • pj , dj -- processing time and due date of job j • xjk = 1, if job j is assigned to kth position in sequence;xjk = 0, otherwise • tk -- tardiness of kth job in sequence • Formulation • Minimize • k=1ntk • Subject to • j=1nxjk = 1  positions k • k=1nxjk = 1  jobs j • j=1npju=1kxju– j=1ndj xjk tk positions k • xjk= 0 or 1  jobs j positions k

19. Mixed Integer Programming Formulation • Example: T-problem by sequence-position decisions (cont’d) • Instantiation of T-problem with 3 jobs • Minimize • t1 + t2 + t3 • Subject to • x11 + x21 + x31 = 1, x12 + x22 + x32 = 1, x13 + x23 + x33 = 1 • x11 + x12 + x13 = 1, x21 + x22 + x23 = 1, x31 + x32 + x33 = 1 • 1x11 + 2x21 + 3x31 – (4x11 + 2x21 + 3x31)  t1 • 1(x11 + x12)+ 2(x21 + x22) + 3(x31 + x32) – (4x12 + 2x22 + 3x32)  t2 • 1(x11 + x12 + x13)+ 2(x21 + x22 + x23) + 3(x31 + x32 + x33) – (4x13 + 2x23 + 3x33)  t3 • x11, x21, x31, x12, x22, x32, x13, x23, x33 = 0 or 1 • A solution (may not be optimal) • x11 = 0, x21 = 0, x31 = 1, x12 = 1, x22 = 0, x32 = 0, x13 = 0, x23 = 1, x33 = 0

20. Mixed Integer Programming Formulation • Example: T-problem by precedence decisions • Notation • n -- number of jobs • pj , dj -- processing time and due date of job j • yij = 1, if job i is scheduled before job j in sequence; for jobs ijyij = 0, otherwise • sj, tj -- start time and tardiness of job j • M -- big number • Formulation • Minimize • j=1ntj • Subject to • si + pi  sj + M(1 – yij)  jobs i  j • sj + pj  si + Myij  jobs i  j • sj + pj – dj tj jobs j • yij= 0 or 1 jobs i  j • Exercise: instantiate T-problem with 3 jobs.

21. Summary • Sequencing and scheduling, notoriously difficult problems • Relatively few situations that can be analyzed by special structure • General purpose techniques for optimal solutions, in this chapter • Heuristic method for relatively good solutions, in the next chapter • Some options for efficiency of general methods • Dynamic programming approach • Importance of efficient computer implementation • Labeling schemes and set generation algorithms • Dominance properties • Solving Tw-problem up to 30 jobs (Schrage and Baker, 1978) • Solving T-problem up to 100 jobs (Potts and Van Wassenhove, 1982) • Branch and bound approach • Lower bound calculation • Initial trial solutions • Dominance check • Branching mechanism • Mixed integer programming approach