# Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimal Control Applications to Scheduling in Manufacturing Systems

- ${M}_{g}$—dynamic model of MS elements and subsystems motion control;
- ${M}_{k}$—dynamic model of MS channel control;
- ${M}_{o}$—dynamic model of MS operations control;
- ${M}_{f}$—dynamic model of MS flow control;
- ${M}_{p}$—dynamic model of MS resource control;
- ${M}_{e}$—dynamic model of MS operation parameters control;
- ${M}_{c}$—dynamic model of MS structure dynamic control;
- ${M}_{\nu}$—dynamic model of MS auxiliary operation control.

_{α}(t), ρ

_{β}(t) can be determined through the following expressions:

_{l}are elements of a general state vector, ψ

_{l}are elements of an adjoint vector. Additional transversality conditions for the two ends of the state trajectory should be added for a general case:

_{0}under the conditions

**h**

_{0}(

**x**(T

_{0})) ≤

**0**and for the given value of $\mathsf{\psi}({T}_{0})$ should return the maximum to the Hamilton’s function:

_{1}= T

_{0}+ $\tilde{\mathsf{\delta}}$ ($\tilde{\mathsf{\delta}}$ is a step of integration). The process of integration is continued until the end conditions ${\mathbf{h}}_{1}\left(\mathbf{x}({T}_{f})\right)\le \overrightarrow{O}$ are satisfied and the convergence accuracy for the functional and for the alternatives is adequate. This terminates the construction of the optimal program control ${\mathbf{u}}^{*}(t)$ and of the corresponding state trajectory ${\mathbf{x}}^{*}(t)$.

_{0}. The end point of the trajectory and the time interval are fixed:

## 3. Classification and Analyses of Optimal Control Computational Algorithms for Short-Term Scheduling in MSs

_{r}(0 ≤ γ

_{r}≤1) is selected to meet the constraint:

_{(r)}= 1 is tried. Then values 1/2, 1/4, and so on until (14) is true. The selection of γ

_{(r)}is being performed during all iterations.

_{u}is a given accuracy.

_{i}are positive coefficients. If the coefficients are sufficiently large, the minimal value of the functional is received in the case of ρ

_{i}(T

_{f}) = 0. Therefore, the following algorithm can be used for the solving of the boundary problem. The control program u

_{(r)}(t) is searched during all iterations with fixed C

_{i}(Newton’s method can be used here, for example). If the accuracy of end conditions is insufficient, then the larger values of C

_{i}are tried. Otherwise, the algorithm is terminated and a solution is received. Although the method seems deceptively simple, it does not provide an exact solution. Therefore, it is advisable to combine it with other methods.

_{u}can be used:

_{(r)}determines the value of the shift in direction of ∆

_{(r−1)}. In the subgradient, methods vectors ∆

_{(r−1)}in (16) are some subgradients of the function ∆

_{u}.

_{<i,r>}can be selected according to some rule, for example:

_{i}= a

_{i}− x

_{<i,r>}is a residual for end conditions at the iteration r. Similarly, dx1

_{i}= a

_{i}− x

_{<i,(r−1)>}and dx2

_{i}= a

_{i}− x

_{<i,(r−1)>}are residual at iterations (r − 1) and (r − 2). The main advantage of these algorithms over classical gradient algorithms is a simpler calculation of the direction vector during all iterations. However, this results in slower convergence (sometimes in divergence) of the general gradient (subgradient) methods as compared with classical ones. The convergence of all gradient methods depends on the initial approximation ${\mathsf{\psi}}_{(0)}({T}_{0})$.

_{f}. The solution of the problem with a free right end is some approximation ${\tilde{\psi}}_{(r)}({T}_{0})$ (r = 1, 2, …).

_{0}, T

_{f}], ${\mathbf{u}}_{d}(t)$ ∈ M is selected.

_{0}, T

_{f}].

_{f}to t = T

_{0}under the end conditions:

_{<i,d>}of time and particularly in ψ

_{<i,d>}(T

_{0}).

_{0}to t = T

_{f}.

_{1}and ε

_{2}define the degree of accuracy, then the optimal control ${\mathbf{u}}_{(r)}^{*}(t)$ = ${\mathbf{u}}_{(r)}(t)$ and the vector ${\tilde{\psi}}_{(r)}({T}_{0})$ are received at the first iteration. If not, we repeat

**Step 3**and so on.

_{<g,Θ>}), then all the components of the model М (M

_{<o,Θ>}, M

_{<k,Θ>}, M

_{<p,Θ>}, M

_{<n,Θ>}, M

_{<e,Θ>}, M

_{<c,Θ>}, M

_{<ν,Θ>}) will be finite-dimensional, non-stationary, linear dynamical systems or bi-linear M

_{<k,Θ>}dynamic systems. In this case, the simplest of Euler’s formulas can be used for integration.

_{0}, T

_{f}], and not over the whole interval, i.e.,

_{0}, T

_{f}], in order to meet the condition:

_{r}for the iteration (r + 1) is formed during iteration r during the maximization of Hamiltonian function (19). The set includes the time points at which the operations of model М are interrupted. This idea for interval (t′, t′′] determination was used in a combined method and algorithm of MS OPC construction. The method and the algorithm are based on joint use of the SSAM and the “branch and bounds” methods.

## 4. Combined Method and Algorithm for Short-Term Scheduling in MSs

- models M
_{<o>}of operation program control; - models M
_{<ν>}of auxiliary operations program control.

**Theorem**

**1.**

- (a)
- If the problem P does not have allowable solutions, then this is true for the problem Г as well.
- (b)
- The minimal value of the goal function in the problem P is not greater than the one in the problem Г.
- (c)
- If the optimal program control of the problem P is allowable for the problem Г, then it is the optimal solution for the problem Г as well.

**Proof.**

- (a)
- If the problem P does not have allowable solutions, then a control $\mathbf{u}(t)$ transferring dynamic system (1) from a given initial state $\mathbf{x}({T}_{0})$ to a given final state $\mathbf{x}({T}_{f})$ does not exist. The same end conditions are violated in the problem Г.
- (b)
- It can be seen that the difference between the functional J
_{p}in (23) and the functional J_{ob}in the problem P is equal to losses caused by interruption of operation execution. - (c)
- Let ${\mathbf{u}}_{p}^{*}(t)$, ∀ t ∈ (T
_{0}, T_{f}] be an MS optimal program control in P and an allowable program control in Г; let ${\mathbf{x}}_{p}^{*}(t)$ be a solution of differential equations of the models M_{<o>}, M_{<ν>}subject to $\mathbf{u}(t)$=${\mathbf{u}}_{p}^{*}(t)$. If so, then ${\mathbf{u}}_{p}^{*}(t)$ meets the requirements of the local section method (maximizes Hamilton’s function) for the problem Г. In this case, the vectors ${\mathbf{u}}_{p}^{*}(t)$, ${\mathbf{x}}_{p}^{*}(t)$ return the minimum to the functional (1).

_{0}, T

_{f}] (an arbitrary allowable control, in other words, allowable schedule) is selected. The variant ${\mathbf{u}}_{g}(t)\equiv 0$ is also possible.

_{f}, then the record value J

_{p}= ${J}_{p}^{(o)}$ can be calculated, and the transversality conditions (5) are evaluated.

_{f}to t = T

_{0}. For time t = T

_{0}, the first approximation ${\mathsf{\psi}}_{i}^{(o)}({T}_{0})$ is received as a result. Here the iteration number r = 0 is completed.

_{0}onwards, the control ${u}^{(r+1)}(t)$ is determined (r = 0, 1, 2, … is the number of iteration) through the conditions (19). In parallel with the maximization of the Hamiltonian, the main system of equations and the adjoint system are integrated. The maximization involves the solving of several mathematical programming problems at each time point.

_{0}, T

_{f}] if the operation ${D}_{\mathsf{\ae}}^{(i)}$ is being interrupted by the priority operation ${D}_{\xi}^{(\omega )}$. In this case, the problem Г is split into two sub-problems (${\mathsf{{\rm P}}}_{\mathsf{\ae}}^{(i)}$,${\mathsf{{\rm P}}}_{\xi}^{(\omega )}$).

_{p}is updated if ${J}_{p0}^{(1)}$ < J

_{p}or/and ${J}_{p1}^{(1)}$ < ${J}_{p}^{(1)}$ [we assume that the functional (23) is minimized]. If only ${J}_{p0}^{(1)}$ < ${J}_{p}^{(1)}$, then J

_{p}= ${J}_{p0}^{(1)}$. Similarly, if only ${J}_{p1}^{(1)}$ < ${J}_{p}^{(1)}$, then J

_{p}= ${J}_{p1}^{(1)}$. If both inequalities are true, then J

_{p}= min{${J}_{p0}^{(1)}$, ${J}_{p1}^{(1)}$}. In the latter case, the conflict resolution is performed as follows: if ${J}_{p0}^{(1)}$ < ${J}_{p1}^{(1)}$, then during the maximization of (19) ${D}_{\mathsf{\ae}}^{(i)}$ is executed in an interrupt-disable mode. Otherwise, the operation ${D}_{\mathsf{\ae}}^{(i)}$ is entered into the Hamiltonian at priority ${D}_{\xi}^{(\omega )}$ at arrival time $\tilde{t}$.

_{1}is a given value, r = 0, 1, … If the condition (24) is not satisfied, then the third step is repeated, etc.

_{0}, T

_{f}] = (0, 14], ${a}_{i\mathsf{\ae}}^{(o)}$ = 2 (I = 1,2; æ = 1, 2, 3); Θ

_{i}

_{æj}(t) = 1 ∀ t; ε

_{11}(t) = 1 at t ∈ (0, 14], ε

_{21}(t) = 0 for 0 ≤ t < 1, ε

_{21}(t) = 1 for t ≥ 1. In this case, the model can be described as follows:

_{i}

_{æ}(t) are given time functions denoting the most preferable operation intervals: γ

_{11}= 15γ

_{+}(6 − t), γ

_{12}= 10γ

_{+}(9 − t), γ

_{13}= 10γ

_{+}(11 − t), γ

_{21}= 20γ

_{+}(8 − t), γ

_{22}= 15γ

_{+}(8 − t), γ

_{23}= 30γ

_{+}(11 − t). Here, γ

_{+}(α) = 1 if α ≥ 0 and γ

_{+}(α) = 0 if α < 0. The integration element in (27) can be interpreted similarly to previous formulas through penalties for operations beyond the bounds of the specified intervals. The scheduling problem can be formulated as follows: facilities-functioning schedule [control program $\overrightarrow{u}(t)$] returning a minimal value to the functional (27) under the conditions (26) and the condition of interruption prohibition should be determined.

## 5. Qualitative and Quantitative Analysis of the MS Scheduling Problem

- the total number of operations on a planning horizon;
- a dispersion of volumes of operations;
- a ratio of the total volume of operations to the number of processes;
- a ratio of the amount of data of operation to the volume of the operation (relative operation density).

## 6. Conclusions

_{f}.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AS | Attainability Sets |

OPC | Optimal Program Control |

MS | Manufacturing System |

SSAM | Successive Approximations Method |

## Appendix A. List of Notations

_{f}is a dynamic model of MS flow control,

_{p}is a dynamic model of MS resource control,

## References

- Blazewicz, J.; Ecker, K.; Pesch, E.; Schmidt, G.; Weglarz, J. Scheduling Computer and Manufacturing Processes, 2nd ed.; Springer: Berlin, Germany, 2001. [Google Scholar]
- Pinedo, M. Scheduling: Theory, Algorithms, and Systems; Springer: New York, NY, USA, 2008. [Google Scholar]
- Dolgui, A.; Proth, J.-M. Supply Chains Engineering: Useful Methods and Techniques; Springer: Berlin, Germany, 2010. [Google Scholar]
- Werner, F.; Sotskov, Y. (Eds.) Sequencing and Scheduling with Inaccurate Data; Nova Publishers: New York, NY, USA, 2014. [Google Scholar]
- Lauff, V.; Werner, F. On the Complexity and Some Properties of Multi-Stage Scheduling Problems with Earliness and Tardiness Penalties. Comput. Oper. Res.
**2004**, 31, 317–345. [Google Scholar] [CrossRef] - Jungwattanakit, J.; Reodecha, M.; Chaovalitwongse, P.; Werner, F. A comparison of scheduling algorithms for flexible flow shop problems with unrelated parallel machines, setup times, and dual criteria. Comput. Oper. Res.
**2009**, 36, 358–378. [Google Scholar] [CrossRef] - Dolgui, A.; Kovalev, S. Min-Max and Min-Max Regret Approaches to Minimum Cost Tools Selection 4OR-Q. J. Oper. Res.
**2012**, 10, 181–192. [Google Scholar] [CrossRef] - Dolgui, A.; Kovalev, S. Scenario Based Robust Line Balancing: Computational Complexity. Discret. Appl. Math.
**2012**, 160, 1955–1963. [Google Scholar] [CrossRef] - Sotskov, Y.N.; Lai, T.-C.; Werner, F. Measures of Problem Uncertainty for Scheduling with Interval Processing Times. OR Spectr.
**2013**, 35, 659–689. [Google Scholar] [CrossRef] - Choi, T.-M.; Yeung, W.-K.; Cheng, T.C.E. Scheduling and co-ordination of multi-suppliers single-warehouse-operator single-manufacturer supply chains with variable production rates and storage costs. Int. J. Prod. Res.
**2013**, 51, 2593–2601. [Google Scholar] [CrossRef] - Harjunkoski, I.; Maravelias, C.T.; Bongers, P.; Castro, P.M.; Engell, S.; Grossmann, I.E.; Hooker, J.; Méndez, C.; Sand, G.; Wassick, J. Scope for industrial applications of production scheduling models and solution methods. Comput. Chem. Eng.
**2014**, 62, 161–193. [Google Scholar] [CrossRef] - Bożek, A.; Wysocki, M. Flexible Job Shop with Continuous Material Flow. Int. J. Prod. Res.
**2015**, 53, 1273–1290. [Google Scholar] [CrossRef] - Ivanov, D.; Dolgui, A.; Sokolov, B. Robust dynamic schedule coordination control in the supply chain. Comput. Ind. Eng.
**2016**, 94, 18–31. [Google Scholar] [CrossRef] [Green Version] - Giglio, D. Optimal control strategies for single-machine family scheduling with sequence-dependent batch setup and controllable processing times. J. Sched.
**2015**, 18, 525–543. [Google Scholar] [CrossRef] - Lou, S.X.C.; Van Ryzin, G. Optimal control rules for scheduling job shops. Ann. Oper. Res.
**1967**, 17, 233–248. [Google Scholar] [CrossRef] - Maimon, O.; Khmelnitsky, E.; Kogan, K. Optimal Flow Control in Manufacturing Systems; Springer: Berlin, Germany, 1998. [Google Scholar]
- Ivanov, D.; Sokolov, B. Dynamic coordinated scheduling in the manufacturing system under a process modernization. Int. J. Prod. Res.
**2013**, 51, 2680–2697. [Google Scholar] [CrossRef] - Pinha, D.; Ahluwalia, R.; Carvalho, A. Parallel Mode Schedule Generation Scheme. In Proceedings of the 15th IFAC Symposium on Information Control Problems in Manufacturing INCOM, Ottawa, ON, Canada, 11–13 May 2015. [Google Scholar]
- Ivanov, D.; Sokolov, B. Structure dynamics control approach to supply chain planning and adaptation. Int. J. Prod. Res.
**2012**, 50, 6133–6149. [Google Scholar] [CrossRef] - Ivanov, D.; Sokolov, B.; Dolgui, A. Multi-stage supply chains scheduling in petrochemistry with non-preemptive operations and execution control. Int. J. Prod. Res.
**2014**, 52, 4059–4077. [Google Scholar] [CrossRef] - Pontryagin, L.S.; Boltyanskiy, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F. The Mathematical Theory of Optimal Processes; Pergamon Press: Oxford, UK, 1964. [Google Scholar]
- Athaus, M.; Falb, P.L. Optimal Control: An Introduction to the Theory and Its Applications; McGraw-Hill: New York, NY, USA; San Francisco, CA, USA, 1966. [Google Scholar]
- Lee, E.B.; Markus, L. Foundations of Optimal Control Theory; Wiley & Sons: New York, NY, USA, 1967. [Google Scholar]
- Moiseev, N.N. Element of the Optimal Systems Theory; Nauka: Moscow, Russia, 1974. (In Russian) [Google Scholar]
- Bryson, A.E.; Ho, Y.-C. Applied Optimal Control; Hemisphere: Washington, DC, USA, 1975. [Google Scholar]
- Gershwin, S.B. Manufacturing Systems Engineering; PTR Prentice Hall: Englewood Cliffs, NJ, USA, 1994. [Google Scholar]
- Sethi, S.P.; Thompson, G.L. Optimal Control Theory: Applications to Management Science and Economics, 2nd ed.; Springer: Berlin, Germany, 2000. [Google Scholar]
- Dolgui, A.; Ivanov, D.; Sethi, S.; Sokolov, B. Scheduling in production, supply chain Industry 4.0 systems by optimal control: fundamentals, state-of-the-art, and applications. Int. J. Prod. Res.
**2018**. forthcoming. [Google Scholar] - Bellmann, R. Adaptive Control Processes: A Guided Tour; Princeton University Press: Princeton, NJ, USA, 1972. [Google Scholar]
- Maccarthy, B.L.; Liu, J. Addressing the gap in scheduling research: A review of optimization and heuristic methods in production scheduling. Int. J. Prod. Res.
**1993**, 31, 59–79. [Google Scholar] [CrossRef] - Sarimveis, H.; Patrinos, P.; Tarantilis, C.D.; Kiranoudis, C.T. Dynamic modeling and control of supply chains systems: A review. Comput. Oper. Res.
**2008**, 35, 3530–3561. [Google Scholar] [CrossRef] - Ivanov, D.; Sokolov, B. Dynamic supply chains scheduling. J. Sched.
**2012**, 15, 201–216. [Google Scholar] [CrossRef] - Ivanov, D.; Sokolov, B. Adaptive Supply Chain Management; Springer: London, UK, 2010. [Google Scholar]
- Ivanov, D.; Sokolov, B.; Käschel, J. Integrated supply chain planning based on a combined application of operations research and optimal control. Central. Eur. J. Oper. Res.
**2011**, 19, 219–317. [Google Scholar] [CrossRef] - Ivanov, D.; Dolgui, A.; Sokolov, B.; Werner, F. Schedule robustness analysis with the help of attainable sets in continuous flow problem under capacity disruptions. Int. J. Prod. Res.
**2016**, 54, 3397–3413. [Google Scholar] [CrossRef] - Kalinin, V.N.; Sokolov, B.V. Optimal planning of the process of interaction of moving operating objects. Int. J. Differ. Equ.
**1985**, 21, 502–506. [Google Scholar] - Kalinin, V.N.; Sokolov, B.V. A dynamic model and an optimal scheduling algorithm for activities with bans of interrupts. Autom. Remote Control
**1987**, 48, 88–94. [Google Scholar] - Ohtilev, M.Y.; Sokolov, B.V.; Yusupov, R.M. Intellectual Technologies for Monitoring and Control of Structure-Dynamics of Complex Technical Objects; Nauka: Moscow, Russia, 2006. [Google Scholar]
- Krylov, I.A.; Chernousko, F.L. An algorithm for the method of successive approximations in optimal control problems. USSR Comput. Math. Math. Phys.
**1972**, 12, 14–34. [Google Scholar] [CrossRef] - Chernousko, F.L.; Lyubushin, A.A. Method of successive approximations for solution of optimal control problems. Optim. Control Appl. Methods
**1982**, 3, 101–114. [Google Scholar] [CrossRef] - Ivanov, D.; Sokolov, B.; Dolgui, A.; Werner, F.; Ivanova, M. A dynamic model and an algorithm for short-term supply chain scheduling in the smart factory Industry 4.0. Int. J. Prod. Res.
**2016**, 54, 386–402. [Google Scholar] [CrossRef] - Hartl, R.F.; Sethi, S.P.; Vickson, R.G. A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev.
**1995**, 37, 181–218. [Google Scholar] [CrossRef] - Chernousko, F.L. State Estimation of Dynamic Systems; Nauka: Moscow, Russia, 1994. [Google Scholar]
- Gubarev, V.A.; Zakharov, V.V.; Kovalenko, A.N. Introduction to Systems Analysis; LGU: Leningrad, Russia, 1988. [Google Scholar]

Results and Their Implementation | The Main Results of Qualitative Analysis of MS Control Processes | The Directions of Practical Implementation of the Results | |
---|---|---|---|

No | |||

1 | Analysis of solution existence in the problems of MS control | Adequacy analysis of the control processes description in control models | |

2 | Conditions of controllability and attainability in the problems of MS control | Analysis MS control technology realizability on the planning interval. Detection of main factors of MS goal and information-technology abilities. | |

3 | Uniqueness condition for optimal program controls in scheduling problems | Analysis of possibility of optimal schedules obtaining for MS functioning | |

4 | Necessary and sufficient conditions of optimality in MS control problems | Preliminary analysis of optimal control structure, obtaining of main expressions for MS scheduling algorithms | |

5 | Conditions of reliability and sensitivity in MS control problems | Evaluation of reliability and sensitivity of MS control processes with respect to perturbation impacts and to the alteration of input data contents and structure |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sokolov, B.; Dolgui, A.; Ivanov, D.
Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems. *Algorithms* **2018**, *11*, 57.
https://doi.org/10.3390/a11050057

**AMA Style**

Sokolov B, Dolgui A, Ivanov D.
Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems. *Algorithms*. 2018; 11(5):57.
https://doi.org/10.3390/a11050057

**Chicago/Turabian Style**

Sokolov, Boris, Alexandre Dolgui, and Dmitry Ivanov.
2018. "Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems" *Algorithms* 11, no. 5: 57.
https://doi.org/10.3390/a11050057