Dynamic programming approach for solving the open shop problem

Abstract

This paper deals with the open shop scheduling problem (OSP) with makespan minimization. An exact dynamic programming algorithm is proposed for solving the OSP to optimality. This approach is applied to the OSP for the first time. Computational results show that the proposed algorithm is able to solve moderate benchmark instances.

Appendix

Table 7 Processing times of benchmark instance tai4-1

We study an example, which explains how to create forbidden intervals. Similarly as by Guéret and Prins (1999), we investigate benchmark instance j3-per0-1, see Table 7. Also, we set $bound=1174$.

The algorithm selects operation $O_{2,2}$, which has the longest processing time $p^*=511$. Due to a symmetry, $O_{2,2}$ cannot start later than

$$\begin{aligned} \dfrac{bound-1-p^*}{2}=331. \end{aligned}$$

Thus, the forbidden interval of operation $O_{2,2}$ is [332, 1174).

Due to the symmetry, operations $O_{1,2},O_{2,1},O_{2,3},O_{3,2}$ cannot be completed in the interval $[332,p^*)$. Thus, we have For example, if $O_{1,2}$ started at 23, then this operation would be completed at 332, which belongs to the forbidden interval.

Operations	Forbidden intervals due to a symmetry
$O_{1,2}$	[23, 511)
$O_{2,1}$	[232, 511)
$O_{2,2}$	[332, 1174)
$O_{2,3}$	[0, 511)
$O_{3,2}$	[152, 511)

The subset-sum values are:

Name	Subset-sums
$S_{M_1}$	0	100	436	464	536	564	900	1000
$S_{M_2}$	0	180	309	489	511	691	820	1000
$S_{M_3}$	0	227	384	389	611	616	773	1000
$S_{J_1}$	0	227	309	464	536	691	773	1000
$S_{J_2}$	0	100	389	489	511	611	900	1000
$S_{J_3}$	0	180	384	436	564	616	820	1000

Subset-sum values in bold can be removed due to a symmetry.

Using algorithm proposed by Guéret and Prins (1999), we obtain forbidden intervals for both the starting and completion times. These intervals have directly been derived from the subset-sum values.

Table 8 Forbidden intervals derived from the subset-sum values

By joining all previously obtained forbidden intervals, we obtain the resulting forbidden intervals for the starting times of operations.

Table 9 Forbidden intervals of benchmark instance j3-per0-1

For example, the forbidden intervals are obtained for the starting time of operation $O_{3,2}$ in the following way. The interval [152, 511) is forbidden due to a symmetry. From Table 8 it follows that

$$\begin{aligned}{}[174,180)\cup [483,511)\cup [685,691)\cup [994,1000) \end{aligned}$$

and

$$\begin{aligned}{}[174,180)\cup [354,384)\cup [790,820)\cup [994,1000) \end{aligned}$$

are also the forbidden intervals.

Since $p_{3,2}=180$ and Table 8 also represents the forbidden intervals for the completion times, we have that intervals

$$\begin{aligned}{}[174,204)\cup [610,640)\cup [814,820) \end{aligned}$$

and

$$\begin{aligned}{}[303,331)\cup [505,511)\cup [814,820) \end{aligned}$$

are forbidden.

Finally, since operation $O_{3,2}$ has to be completed before $bound=1174$, we have that [994, 1174) is also forbidden. Thus, we have obtained the full forbidden intervals for operation $O_{3,2}$, see Table 9.

See Table 10.

Table 10 Detailed results of the effectiveness of the dominance test