Optimization Methods for the Single-Machine Problem

Optimization Methods for the Single-Machine Problem

Chapter 3

Elements of Sequencing and Schedulingby Kenneth R. Baker

Byung-Hyun Ha

2

Outline

Introduction

Adjacent pairwise interchange methods

A dynamic programming approach

Dominance property

A branch and bound approach

A mixed integer programming approach 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. 1 T-problem by ??

General purpose optimization methods Dynamic programming approach Branch and bound approach Mathematical programming approach ...

4

Introduction

A general setting gj(t) -- a penalty function, which is incurred when job j completes at time t

• assuming nondecreasing with regard to t

Typical scheduling problems Maximum penalty problem

• to minimize the maximum gj(t)

Total penalty problem• to minimize the sum of gj(t)

Theorem 1 When the objective is to minimize the maximum penalty, job i may be as

signed 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}

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Adjacent Pairwise Interchange Methods

Thrust of adjacent pairwise interchange methods Finding a sequence of which every adjacent pairwise interchange leads t

o poorer performance F-problem optimal by the method T-problem ??

Three-job example Job j 1 2 3

pj

dj

14

22

33

1-3-2 3-1-2

2-1-3 2-3-1

3-2-11-2-3

T = 5 T = 4

T = 4T = 3

T = 4 T = 5

local optimum

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Additive form of regular measures Z = j=1

n gj(Cj)

gj(Cj) = max{0, Cj – dj} -- total tardiness

We can find an optimal sequence with a dynamic programming

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) = minjJ {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).

Dynamic Programming Approach

j... ...

p(J)

J X – J

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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

Job j 1 2 3

pj

dj

14

22

33

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Computational efforts In proportion to n2n

U-problem -- O(nlogn) All enumerations -- in proportion to n!

Exercise X = {1, 2, 3, 4}, gj(Cj) = max{0, Cj – dj}

Job j 1 2 3 4

pj

dj

59

67

911

813

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Computer implementation Labeling scheme

• lk = 2k-1, lJ = kJ lk 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(lJ) = minjJ {G(lJ – {j}) + gj(p(J))}

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Dominance Properties

Dominance properties involving schedules Schedules without preemption and inserted idle times constituted domin

ant set They reduces the number of alternatives

Dominance properties involving 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)• In the T-problem, there is an optimal schedule in which job j follows job i if on

e of the following conditions is satisfied:

(a) pi pj and di max{dj, p(Bj) + pj}

(b) di dj and dj p(Ai') – pj

(c) dj p(Ai')

where

• Ai -- set of jobs to follow job i in an optimal solution; 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 solution; i.e., “before” set

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Dominance Properties

Computer implementation Modified labeling scheme

• Nj = {i | i < j}

• lj' = lNj + 1 – lBjNj , lJ' = kJ lk'

Overall procedure1. Set b(i) = 0 for i = 1..|X|, G(0) = 0

2. Loop begin

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, if j Ai , b(j) = 0

2-4. Let J = {j | b(j) = 1}

2-5. Set G(lJ') = minjJ {G(lJ – {j}') + gj(p(J))}

Example of p. 3.12 -- 5 jobs• A1 = {2}, A2 = , A3 = {4, 5}, A4 = , A5 =

l1' = 1, l2' = 1, l3' = 3, l4' = 3, l5' = 6

Number of candidates are not 25 = 32 but 14!

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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(2)P(1) P(n)...

...... ...

... ......

P(32)P(12) P(n2)

P(S)

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Bounding Curtailing enumeration process, by calculating a lower bound on the opti

mal solution of a given subproblem• Suppose Z is known performance measure and lower bound of a certain sub

problem 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 subprobl

em 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 subp

roblems ranked by lower bound

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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

• p(s) = js' pj ; p(js) = p(s) – pj

• P(s) -- subproblem with a sequence which ends with partial sequence s• P(0) -- original problem

• vs = js Tj ; vjs = max{0, p(s) – dj} + vs

• bs -- lower bound of subproblem P(s)

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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() = j pj .

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.

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Example -- T-problem Job j 1 2 3 4 5

pj

dj

45

36

78

28

217P(0)

P(1) P(2) P(3) P(4) P(5)

P(35) P(45)P(25)P(15)

P(245) P(345)P(145)P(435)P(135) P(235)P(354)P(254)P(154)P(253)P(153)

P(1435) P(2435)

P(12435)

P(54)P(53)

1

1

23

4

5

6

7

8

10

11

13 12 10

10 10 12 11 9 9

14 13 11 19 18 16 13 12 10 18 17 15

12 11

11

10

9

P(1453) P(2453)

P(453)

15 16

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A branch and bound procedure for T-problem (cont’d) Options

• Lower bounds

• bs = vs , or

• bs = vs + minjs' {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• ...

• ...

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Mixed Integer Programming Approach

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; 0, otherwise

• tj -- tardiness of job j

Formulation• Minimize

j=1n tj

• Subject to j=1

n xjk = 1, positions k

k=1n xjk = 1, jobs j

j=1n pj

u=1k xju – j=1

n dj xjk tk , positions k

• xjk = 0 or 1, jobs j, positions k

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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 optimal)

• x11 = 0, x21 = 0, x31 = 1, x12 = 1, x22 = 0, x32 = 0, x13 = 0, x23 = 1, x33 = 0

Job j 1 2 3

pj

dj

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22

33

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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; 0, otherwise, for jobs i j

• sj , tj – start time and tardiness of job j

• M -- big number

Formulation• Minimize

j=1n tj

• Subject to

• si + pi sj + M(1 – yij), jobs i j

• sj + pj si + Myij, jobs i j

• sj + pj – dj tj , jobs j

• xij = 0 or 1, jobs i j

Exercise: instantiate T-problem with 3 jobs.Job j 1 2 3

pj

dj

14

22

33

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Team Assignment

TEAM HOMEWORK #1 Implement a dynamic programming approach for solving T-problem.

TEAM HOMEWORK #2 Implement a branch and bound approach for solving T-problem.

TEAM HOMEWORK #3 Solve T-problem using commercial MIP (mixed integer problem) solver.

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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

• Efficient computer implementation• Labeling schemes and set generation algorithms

• Dominance properties Tw-problem up to 30 jobs (Schrage and Baker, 1978)

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

Optimization Methods for the Single-Machine Problem

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