Scheduling to maximize the weighted number of on-time jobs on parallel machines with bounded job-rejection

Abstract

We study a scheduling problem on parallel identical machines, where the objective function is maximizing the weighted number of jobs completed exactly at their due-dates. The scheduler may reject a subset of the jobs, and the total permitted rejection cost is assumed to be bounded. Thus, when a decision has to be made regarding a certain job, the following options need to be considered: either ($\mathrm{i}$) identify the set of machines on which the job can be scheduled on time, and assign the job to one of these machines, or ($\mathrm{ii}$) reject the job (if possible), or ($\mathrm{iii}$) delay the job. A pseudo-polynomial dynamic programming algorithm is introduced for this NP-hard problem. Medium size problems are solved in reasonable running times. We then study a different version of the problem, in which job-dependent due-windows are considered, and the job processing times are assumed to be identical. This version is proved to be NP-hard, and a pseudo-polynomial dynamic programming algorithm is introduced as well. Our numerical study indicates that medium size instances can be handled for this version. In addition, for both problems, an alternative solution procedure based on integer-linear-programming formulation is introduced.

Appendices

Appendix A1

Mixed Integer Linear Programming (MILP1) formulation for Problem P1.

Jobs are assumed to be sorted in a non-decreasing order of ${d}_{j}$ and are numbered accordingly.

We define the following variables:

• $Y_{r} = \left\{ {\begin{array}{*{20}c} {1, {\text{if}}\,{\text{job r is not completed on its due date}}} \\ {0, {\text{ otherwise}}} \\ \end{array} } \right.,r = 1, \ldots ,n$

• $Z_{r} = \left\{ {\begin{array}{*{20}c} {1, {\text{if job r is processed on machine 1 or rejected}}} \\ {0, {\text{ if job r is processed on machine 2 or rejected}}} \\ \end{array} } \right., r = 1, \ldots ,n$

• $S_{1,r} = {\text{the starting time of job r on machine}} 1,r = 1, \ldots ,n$

• $S_{2,r} = {\text{the starting time of job r on machine}} \,2,r = 1, \ldots ,n$

• $E_{r} = \left\{ {\begin{array}{*{20}c} {1, {\text{if job r is rejected}}} \\ {0, {\text{otherwise}}} \\ \end{array} } \right.,r = 1, \ldots ,n$

MILP1 is the following:

$$ {\text{Min }}\left\{ {\mathop \sum \limits_{r = 1}^{n} w_{r} \cdot Y_{r} } \right\} $$

S.T.

• 1. $M \cdot \left( {Y_{r} + E_{r} + 1 - Z_{r} } \right) + S_{1,r} \ge S_{1,r - i} + p_{r} - M \cdot \left( {1 - Z_{r - i} + E_{r - i} + Y_{r - i} } \right),\, = 2, \ldots ,n,\,i = 1, \ldots ,r - 1$

• 2. $M \cdot \left( {Y_{r} + E_{r} + Z_{r} } \right) + S_{2,r} \ge S_{2,r - i} + p_{r} - M \cdot \left( {Z_{r - i} + E_{r - i} + Y_{r - i} } \right),\,r = 2, \ldots ,n,\,i = 1, \ldots ,r - 1$

• 3. $M \cdot \left( {E_{r} + Y_{r} + 1 - Z_{r} } \right) \ge d_{r} - \left( {S_{1,r} + p_{r} } \right),\,r = 1, \ldots ,n$

• 4. $M \cdot \left( {E_{r} + Y_{r} + Z_{r} } \right) \ge d_{r} - \left( {S_{2,r} + p_{r} } \right),\,r = 1, \ldots ,n$

• 5. $M \cdot \left( {E_{r} + Y_{r} + 1 - Z_{r} } \right) \ge \left( {S_{1,r} + p_{r} } \right) - d_{r} ,\,r = 1, \ldots ,n$

• 6. $M \cdot \left( {E_{r} + Y_{r} + Z_{r} } \right) \ge \left( {S_{2,r} + p_{r} } \right) - d_{r} ,\,r = 1, \ldots ,n$

• 7. $\mathop \sum \limits_{j = 1}^{n} E_{j} r_{j} \le U$

  $$\begin{aligned} & S_{(1,r)} \ge 0,S_{(2,r)} \ge 0;r = 1, \ldots ,n;\\ & Y_{r}^{E} ,Y_{r}^{T} ,Z_{r} ,E_{r} \,{\text{binary}}\,r = 1, \ldots ,n \end{aligned}$$

($M$ is a large number.)

The objective function is the minimum weighted number of jobs completed either early or tardy.

Constraint 1 guarantees a feasible schedule on machine 1. The starting time of each job which is not tardy and is processed on machine 1 is not smaller than the completion time of all jobs which were completed before it on the same machine.

Constraint 2 is similar to constraint 1, but guarantees a feasible schedule on machine 2.

Constraint 3 guarantees that if job $r$ is processed on machine 1 prior to its due-date, ${Y}_{r}$ must be 1. For each job processed on machine 1, if the job is completed prior to its due-date, then the right-hand-side of the constraint is strictly positive. This implies that the left-hand-side must be positive. If job $r$ is processed on machine 2 then ${Z}_{r}=0$ and the left-hand-side is positive and the inequality holds. Otherwise (${Z}_{r}=1$), the left-hand-side is positive if and only if either ${Y}_{r}=1$ (i.e., the job is completed prior to its due-date) or ${E}_{r}=1$ (i.e., the job is rejected).

Constraint 5 guarantees that if job $r$ is processed on machine 1 after its due-date, ${Y}_{r}$ must be 1. As above, for each job processed on machine 1, if the job is completed after its due-date, then the right-hand-side of the constraint is strictly positive. Hence, the left-hand-side has to be positive. Again, this implies that either ${Y}_{r}=1$ (i.e., the job is completed after its due-date) or ${E}_{r}=1$ (i.e., the job is rejected). Otherwise (both ${Y}_{r}=0$ and ${E}_{r}=$ 0), it follows that ${Z}_{r}=0$, i.e., the job is processed on machine 1.

Constraints 4 and 6 guarantee the same as Constraints 3 and 5, respectively, for machine 2.

Constraint 7 guarantees that the total rejection cost is not larger than its upper bound.

Appendix A2

Mixed Integer Linear Programming (MILP2) formulation for Problem P2.

We assume that the jobs are sorted in a non-decreasing order of ${d}_{i}^{1}$ and are numbered accordingly.

We define the following variables:

• $Y_{r}^{E} = \left\{ \begin{gathered} 1, {\text{if}}\,{\text{job r completed before the window starting time}} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}} \hfill \\ \end{gathered} \right.,r = 1, \ldots ,n$

• $Y_{r}^{T} = \left\{ \begin{gathered} 1, {\text{if job r is completed after the window completion time}} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}} \hfill \\ \end{gathered} \right., r = 1, \ldots ,n$

The definitions of the variables ${Z}_{r}, {S}_{1,r}, {S}_{2,r}$ and ${E}_{r}$ are identical to those in MILP1.

MILP2 is the following:

$$ {\text{Min}} \left\{ {\mathop \sum \limits_{r = 1}^{n} w_{r} \left( {Y_{r}^{E} + Y_{r}^{T} } \right)} \right\} $$

S.T.

• 1. $M \cdot \left( {Y_{r}^{T} + E_{r} + 1 - Z_{r} } \right) + S_{1,r} \ge S_{1,r - i} + p - M \cdot \left( {1 - Z_{r - i} + E_{r - i} + Y_{r - i}^{T} } \right),\,r = 2, \ldots ,n,\,i = 1, \ldots ,r - 1$

• 2. $M \cdot \left( {Y_{r}^{T} + E_{r} + Z_{r} } \right) + S_{2,r} \ge S_{2,r - i} + p - M \cdot \left( {Z_{r - i} + E_{r - i} + Y_{r - i}^{T} } \right),\,r = 2, \ldots ,n,\,i = 1, \ldots ,r - 1$

• 3. $M \cdot \left( {E_{r} + Y_{r}^{E} + 1 - Z_{r} } \right) \ge d_{r}^{1} - \left( {S_{1,r} + p_{r} } \right),\,r = 1, \ldots ,n$

• 4. $M \cdot \left( {E_{r} + Y_{r}^{E} + Z_{r} } \right) \ge d_{r}^{1} - \left( {S_{2,r} + p_{r} } \right),\,r = 1, \ldots ,n$

• 5. $M \cdot \left( {E_{r} + Y_{r}^{T} + 1 - Z_{r} } \right) \ge \left( {S_{1,r} + p} \right) - d_{r}^{2} ,\,r = 1, \ldots ,n$

• 6. $M \cdot \left( {E_{r} + Y_{r}^{T} + Z_{r} } \right) \ge \left( {S_{2,r} + p} \right) - d_{r}^{2} ,\,r = 1, \ldots ,n$

• 7. $\mathop \sum \limits_{j = 1}^{n} E_{j} r_{j} \le U$

  $$\begin{aligned}& S_{(1,r)} \ge 0,S_{(2,r)} \ge 0;r = 1, \ldots ,n;\\ & Y_{r}^{E} ,Y_{r}^{T} ,Z_{r} ,E_{r} \,{\text{binary}}\,r = 1, \ldots ,n\end{aligned} $$

($M$ is a large number)

The objective function remains minimum weighted number of jobs completed either early or tardy. The constraints are only slightly modified when comparing to MILP1: in Constraints 1, 2, 5 and 6 ${Y}_{r}^{T}$ replaces ${Y}_{r}$; in Constraints 3 and 4, ${Y}_{r}^{E}$ replaces ${Y}_{r}$.