Minmax scheduling and due-window assignment with position-dependent processing times and job rejection

Abstract

We study scheduling and due-window assignment problems with the objective function of a minmax type, i.e., the goal is to minimize the largest cost among all scheduled jobs. We assume that the processing times of jobs are position-dependent in the most general way. For a single machine and for a proportionate flow shop environment we present polynomial time solution procedures that are based on solving a linear assignment problem as a subroutine. We further extend the single machine model by allowing job-rejection, provided that the processing times deteriorate, i.e., the position-dependent processing times are non-decreasing functions of the job position. For this setting, the scheduler may decide not to process certain jobs, and each rejected job is penalized accordingly. The problem with the objective that additionally involves a maximum rejection cost component is also shown to be solvable in polynomial time.

Appendix A: Numerical examples

Example 1

Consider a 7-job problem. The position-dependent processing times are given in Table 2. Assume: $\alpha = 3, \beta = 1.2, \gamma = 2, \delta = 1.5$. Following Algorithm  $Due\_Window\_p_{jr}$, Case (i) is optimal. It follows that $d_{1} = d_{2} = 0,$ the optimal job sequence is $\left( {7,2,1,5,6,4,3} \right)$, $C_{max} = 32$, and $Z = \beta C_{max} = 38.4$.

Table 2 Job-position processing times for Examples 1–4

Example 2

The job-position processing times are those of Example 1. Assume: $\alpha = 2, \beta = 3.2, \gamma = 0.6, \delta = 1.8$. Now Case (ii) is optimal. After solving seven assignment problems ($ASSIGN\left( k \right), k = 1, \ldots ,7$), we obtain the following optimal solution (achieved for $k = 2$): $d_{1} = d_{2} = 33.69,$ the optimal job sequence is $\left( {2,7,1,5,6,4,3} \right)$, $C_{ \text{min} } = 22, C_{ \text{max} } = 41$, and $Z = \frac{{\beta \left( {\alpha + \gamma } \right)}}{\alpha + \beta }C_{ \text{max} } - \frac{{\alpha \left( {\beta - \gamma } \right)}}{a + \beta }C_{ \text{min} } = 43.6$.

Example 3

The job-position processing times are those of Example 1, and $\alpha = 2, \beta = 3, \gamma = 1.5, \delta = 2$. Case (iii) is optimal. As in Example 2, we solve seven assignment problems. The optimal solution is obtained for $k = 7$: $d_{1} = C_{ \text{min} } = 6; d_{2} = C_{ \text{max} } = 32,$ the optimal job sequence is $\left( {7,2,1,5,6,4,3} \right)$, and $Z = \delta C_{ \text{max} } - \left( {\delta - \gamma } \right)C_{ \text{min} } = 61$.

Example 4

The job-position processing times are those of Example 1. Assume: $\alpha = 3, \beta = 2, \gamma = 2.5, \delta = 1$. Case (iv) is optimal. As in Example 1, we solve a single assignment problem in order to minimize the makespan. The optimal solution is: $d_{1} = 0; d_{2} = C_{ \text{max} } = 32,$ the optimal job sequence is $\left( {7,2,1,5,6,4,3} \right)$, and $Z = \delta C_{ \text{max} } = 32$.

Example 5

Consider a 5-machine proportionate flow shop 4-job problem. The position-dependent processing times are given in Table 3. The cost parameters are those given in Example 2: $\alpha = 2, \beta = 3.2, \gamma = 0.6, \delta = 1.8$. Again, Case (ii) is optimal, and thus $d_{1} = d_{2} = \frac{{\alpha C_{ \text{min} } + \beta C_{ \text{max} } }}{\alpha + \beta }$, and $Z = \frac{{\beta \left( {\alpha + \gamma } \right)}}{\alpha + \beta }C_{ \text{max} } - \frac{{\alpha \left( {\beta - \gamma } \right)}}{a + \beta }C_{ \text{min} }$. We first solve the bottleneck problem, and obtain an optimal value of 7. Hence, only the job-positions with processing time of 7 and above should be considered as potential pseudo job-positions. Following Algorithm $Due\_Window\_p_{jr} \_Flowshop$, we obtain: Job 3 is scheduled first; the pseudo job-position is (4, 4) (its value is $5 \times 7 = 35)$; the optimal job sequence is (3, 2, 1, 4); $C_{max} = 46$, $C_{min} = 30$ (obtained by delaying the operations on machine 5 to eliminates idle times), and $Z = 43.6$.

Table 3 Job-position processing times for Example 5

Example 6

Consider a 7-job problem. The (deteriorating) job-position processing times are given in Table 4. The rejection costs (after sorting) are: $e_{1} = 42, e_{2} = 59, e_{3} = 67, e_{4} = 78, e_{5} = 82, e_{6} = 90, e_{7} = 95$. Assume that $\alpha = 3, \beta = 1.2, \gamma = 2, \delta = 1.5$. Given these cost parameters, Case (i) is optimal. The initial values for the binary search are: $k_{ \text{min} } = 0$ (no jobs are rejected), $k_{ \text{max} } = 7$ (all jobs are rejected). Thus, $k = \frac{{k_{ \text{min} } + k_{ \text{max} } }}{2} = 4$, and we solve the due-window assignment problem for jobs $\left\{ {5,6,7} \right\}$. The resulting makespan is $C_{ \text{max} } = 34$. Since $Z^{\left( 4 \right)} = \beta C_{ \text{max} } = 40.8 < 78 = e_{4}$, we check the option of rejecting less jobs. In the next iteration $k_{ \text{min} } = 0$ and $k_{ \text{max} } = 4$, implying $k = 2$. We solve the due-window assignment problem for jobs $\left\{ {3,4,5,6,7} \right\}$. The resulting makespan is $C_{ \text{max} } = 84$. In this case: $Z^{\left( 2 \right)} = \beta C_{ \text{max} } = 100.8 > 59 = e_{2}$, and we check the option of rejecting more jobs. Now $k_{ \text{min} } = 2$ and $k_{ \text{max} } = 4$, implying $k = 3$. We solve the due-window assignment problem for jobs $\left\{ {4,5,6,7} \right\}$. The resulting makespan is $C_{ \text{max} } = 55$, and $Z^{\left( 3 \right)} = \beta C_{ \text{max} } = 66 < 67 = e_{3}$. We check again the option of rejecting less jobs: $k_{ \text{min} } = 2$ and $k_{ \text{max} } = 3$. Thus, the next $k$-value to be checked is 3, which is equal to $k_{ \text{max} }$ and the procedure is finished. The optimal solution consists of rejecting jobs $\left\{ {1,2,3} \right\}$, and processing the remaining jobs: $\left\{ {4,5,6,7} \right\}$. The due-window is reduced to a due-date at time zero ($d_{1} = d_{2} = 0$), the optimal job sequence is $\left\{ {4,7,5,6} \right\}$ with maximum (tardiness) cost of 66. Thus, the optimal minmax cost is 67, obtained by the rejection cost of job 3.

Table 4 Job-position processing times for Example 6