스케줄 이론 Ch 5 Earliness Tardiness Costs

1. 스케줄 이론 Ch 5 Earliness Tardiness Costs 박진우(서울대) 배준수(전북대)

2. 1. Introduction • (Notation)

3. 2. Minimizing Deviations from a Common Due Date • Minimizing the sum of absolute deviations of the job completion times from a Common Due Date. • The case with    is called the basic E/T problem. • Common sense suggests that in order to minimize the sum of Earliness and Tardiness, it will be desirable to have the due date in the middle of the jobs. • But, if the due date is too tight, then it will not be possible to fit enough jobs in front of the due date. • So the basic E/T problems can be divided into the Restricted version and Unrestricted version depending on the tightness of the due dates. • If the due date is too tight, then we have a restricted version of E/T problem. • A good dividing point for restrictedness is (for the moment). • We consider the unrestricted version first.

4. 2. Minimizing Deviations from a Common Due Date • (Theorem 5.1) • In the basic E/T model, schedules without inserted idle time constitute a dominant set. • (Theorem 5.2) • In the basic E/T model, jobs that complete on or before the due date can be sequenced in LPT order, while jobs that start late can be sequenced in SPT order. • (Theorem 5.3) • In the basic E/T model, there is an optimal schedule in which some job completes exactly at the due date. Cj Sj d d Sj Cj Sj Cj d d

5. 2. Minimizing Deviations from a Common Due Date • B: The set of jobs completing on or before the due date, b = |B|, the cardinality • A: The set of jobs completing after the due date, a = |A| • Bi: The index of the ith job in B, Aj: The index of the jth job in A, j Now consider the costs: • (Algorithm 5.1) Solving the Basic E/T Problem • Step 1. Assign the longest job to set B. • Step 2. Find the next two longest jobs. Assign one to B(the Before set) and on to A(the After set). • Step 3. Repeat Step 2 until there are no jobs left, or until there is one job left, in which case assign this job to either A or B. Finally order the jobs in B by LPT and the jobs in A by SPT. • The complexity of the algorithm is O(n log n). The V-shaped schedule constitutes a dominant set.

6. 2. Minimizing Deviations from a Common Due Date • Example Problem • (Algorithm 5.1*) Secondary objective: Minimize total processing time in set B • Assign the shorter job to B at Step 2 • if n is even, assign the shortest job to A in Step 3

7. 2. Minimizing Deviations from a Common Due Date • (Theorem 5.4) • In the basic E/T model, there is an optimal schedule in which the bth job in sequence completes at time d, where b is the smallest integer greater than or equal to n/2. • Application • If we set  as the sum of the processing times of the Before set, indexed in SPT order, then total processing time •  is the smallest value of d at which the optimal penalty attains its minimum in unrestricted version of the problem. • d   : Unrestricted problem • d <  : Restricted problem • For cases where the due date is treated as a decision variable, the unrestricted versions of the problem gives rise to the constant objective function value even if the due dates are varied.

8. 3. The Restricted Version • The straddling job • A job that starts before the due date and completes after the due date. • There is no simple way to find an optimal solution to the restricted version of the basic E/T problem. In fact, it is NP-hard. Inserted idle time may help. • A specialized DP technique devised by Hall, Kubiak and Sethi(1991) can solve several hundred job problem in modest amounts of computer time. • A heuristic developed by Sundararaghavan and Ahmed(1984) solves the problem very efficiently. • Example Problem 5.3(V shaped schedule with L=available time before due date, R=..after due date If L>R insert next job into the first position, otherwise insert into the last position) • (Formulation) • Let e denote the number of jobs that finish before the due date. Equivalently, (n-e) is the number of jobs that finish on or after the due date. Suppose we delay the start of the schedule by a small amount, . • The delay leads to a reduction in the total penalty if . Thus, if more than half the jobs are early, then the start of the schedule should be delayed, at least long enough to make the last early job complete exactly at the due date.

9. 4. Different Earliness and Tardiness Penalties • Different Earliness and Tardiness Penalties • may represent a holding cost while represents a tardiness penalty. These are likely to be different. • : tends to be determined by endogenous factors • : tends to be determined by exogenous values • Let    , then in the unrestricted version, an optimal solution has the properties similar to Theorems 5.1~5.3 in an optimal schedule. • (Theorem 5.1’~5.3’) • There is no inserted idle time. • Jobs that complete on or before the due date are sequenced in LPT order, while jobs that start late should be sequenced in SPT order. • One job completes at the due date. • Objective function: CB + CA

10. 4. Different Earliness and Tardiness Penalties • (Algorithm 5.2) Solving the E/T Problem with Different E/T Penalties • Step 1. Initially, sets B and A are empty, and the jobs are in LPT order. • Step 2. If , then assign the next job to B; otherwise, assign the next job to A. • Step 3. Repeat Step 2 until all jobs have been scheduled. • Example Problem 5.1 with different Earliness and Penalty Costs. • (Theorem 5.4a) • In the basic E/T model with different Earliness and Tardiness penalties, there is an optimal schedule in which the bth job in sequence completes at time d, where b is the smallest integer greater than or equal to .

11. 5. Quadratic Penalties • Quadratic Penalties • In spite of the problem's equivalence to ‘the completion time variance problem’ ( where ), the quadratic E/T problem is not easily solved. • For the unrestricted problem, Theorems 5.1 and 5.2 hold, but the Theorems 5.3 and 5.4 do not. • Only enumerative methods have been developed, with minor success in heuristic solutions.

12. 6. Job-dependent Penalties • Job-dependent Penalties • For this type of problem, even the unrestricted version of this case is NP hard. However, Hall and Posner(1991) describe a dynamic programming algorithm and show that it is capable of solving problems containing hundreds of jobs in modest run times. Following Theorems hold. • (Theorem 5.1’’-5.3’’) • There is no inserted idle time. • Jobs that complete on or before the due date can be sequenced in non-increasing order of the ratio , and the jobs that start late can be sequenced in non-decreasing order of the ratio . • One job completes at the due date. • (Theorem 5.4b) : no proof • In an optimal schedule the bth job in sequence completes at time d, where b is the smallest integer satisfying the inequality

13. 7. Distinct Due Dates •     problem is harder than the problems discussed so far. However, if the due dates are treated as decision variables, the problem turns out to be relatively simple. Seidmann, Panwalker and Smith(1981) developed a very efficient method for solving such kind of problem.  • However, the general problem is NP-hard, and even ‘Theorems 5.1 and 5.2’ do not hold in general. In particular, inserted idle time may be desirable.  • Some researchers developed a B&B method, and Fry, Armstrong and Blackstone(1987) developed a pretty good heuristic solution.

14. 8. Summary • We need solution procedures for more general setting than just for a single machine sequencing problem. • Just on Time Schedules? • Earliness and Tardiness for Job Shop Problem