Objective Functions for Minimizing Rescheduling Changes in Production Control

Abstract

This paper presents an advanced rescheduling approach that jointly applies two sets of objective functions within a novel multi-objective search algorithm and a production simulation of the manufacturing system. The role of the first set of objective functions is to optimize the performance of production systems, while the second newly proposed set of objective functions aims to minimize the intervention changes from the original schedule, thereby supporting schedule stability and smooth manufacturing processes. The combined use of these two objective sets is ensured by a flexible candidate-qualification method, which allows for priorities to be assigned to each objective function, offering precise control over the rescheduling process. The applicability of this approach is presented through an example of an extended flexible flow shop manufacturing system. A new test problem containing 16 objective functions has been developed. The effectiveness of the proposed new objective functions and rescheduling method is validated by a simulation model. The obtained numerical results are also presented in this paper. The aim of this study is not to compare different search algorithms but rather to demonstrate the beneficial impact of change-minimizing objective functions within a given search framework.

1. Introduction

Rescheduling plays a very important role in the automation, management, execution, and shop floor control systems of discrete manufacturing processes. Solving automation, production planning, and control tasks is a challenging endeavor, as it requires considering various constraints, many decision variables, and multiple optimization objectives simultaneously. Serving specific customer needs, expectations, and anticipated demands based on preliminary calculations generates increasingly complex scheduling tasks.

Research on the development of scheduling models and methods is very important from both theoretical and practical perspectives. The fundamental classification of scheduling tasks can be found in [1]. Optimization tasks related to this topic are closely linked with problems in various research areas. These domains include optimization problems related to manufacturing [2], logistics [3], resource planning [4], and business processes [5]. To efficiently utilize the available resources in manufacturing systems, precisely developed technological process plans and finely tuned production schedules are necessary. The basic information required for scheduling and rescheduling is provided by digitalization, information and communication technologies, and specific systems falling under the scope of Industry 4.0. By processing the information extracted from production systems, sufficiently sophisticated scheduling models can be created and utilized.

For processing more detailed and accurate information than previously available, advanced decision support models are also needed. To analyze and evaluate the consequences of decisions, simulation models are often used. Based on their results, the efficiency of production system management can be further enhanced and partially automated. The impact of new methods may necessitate corrections, refinements, or major transformations in both production structures and management philosophies.

Two important requirements for rescheduling models are that the results of intervention decisions must comply with the new set of conditions and fully support the new objective system. The most important characteristics of rescheduling concepts can be found, for example, in publications [6,7,8].

Rescheduling focuses on updating and modifying the actual production schedule to respond to unexpected events and disturbances that occur in the executing environment. These unfavorable events can lead to delays and decreased performance. Correcting or creating new versions of schedules can be necessary. The rescheduling procedure is crucial in dynamic manufacturing systems where unexpected events and disturbances frequently occur [8].

Numerous papers have focused on the challenges of solving production rescheduling problems. For instance, Dong and Jang [9] introduced algorithms to minimize mean tardiness in job shop rescheduling due to machine breakdowns. They focused on adjusting the processing sequence to utilize idle machine time effectively. Similarly, Kundakci and Kulak [10] developed a hybrid genetic algorithm to solve dynamic job shop scheduling problems where breakdowns and new orders occur, demonstrating its ability to deliver high-quality solutions efficiently.

Artificial intelligence has emerged as a promising approach for optimizing rescheduling problems. Zhang et al. [11] proposed a fuzzy neural network to adapt rescheduling decisions based on current system states and disturbances, although its practical application in manufacturing systems remains an area for further exploration. Li et al. [12] integrated machine learning classification and optimization algorithms to identify rescheduling patterns in flexible shop environments, emphasizing the potential of artificial intelligence to improve rescheduling efficiency. Wu et al. [13] developed a neural network with reinforcement learning for emergency production scheduling, highlighting its ability to generate schedules quickly, although reinforcement learning algorithms need improvement. Bo et al. [14] examined the rescheduling problem of a two-tier supply chain after major supply disruptions. Their innovation lies in proposing a two-stage genetic algorithm for rescheduling the manufacturing. Lv et al. [15] emphasized the importance of rescheduling in energy-efficient manufacturing processes, especially for manufacturing systems operating in dynamic and non-deterministic environments. Their innovation lies in proposing rescheduling decision mechanisms that respond to two typical dynamic events and utilize alternative process plans for energy-efficient flexible job shops. Yang et al. [16] developed a new measure (RMc) using an extreme learning machine (ELM), which evaluates schedule robustness by considering the probability of machine breakdowns and the location of float time. They combine this with an improved genetic algorithm to solve scheduling problems under uncertainties in flexible job shop environments. They use only two objective functions.

Our research can contribute to advancing well-known production rescheduling approaches by proposing new components for metaheuristic search algorithms. Our approach includes a relative qualification model and new objective functions to minimize changes to the original schedule. Search algorithms integrated with our new qualification method achieve better rescheduling in extended flexible shop environments affected by machine failures and changes in machine abilities or order attributes. Our proposals address the limitations of traditional search methods, aiming to minimize the distance between the rescheduled solution and the original one in complex manufacturing environments.

2. Description of the Investigated Problem

2.1. Background and Characteristics of the Problem

The motivation for our research originates from industrial enterprises where many product types with variable series are produced in flexible workshop environments. Investigating and modeling the problem, developing a software to support rescheduling activities, and integrating the new system with other components of enterprise information systems lie in the background of the research.

In [17], the present authors presented a manufacturing system characterized by four manufacturing phases for producing final products. Considering the resource groups formed within the shop floor system, the manufacturing primary processes can be described using the extended flexible flow shop (EFFS) model.

In the manufacturing system, internal manufacturing orders (IMOs) are generated by production planning. Each IMO may consist of multiple jobs. A job encompasses the manufacturing operations of a given number of identical products. Each operation can be carried out using any items of the given group of suitable machines (workplaces). There are technological route alternatives for each type of product to be produced. Each IMO has its own dedicated deadline, which is inherited by the associated jobs. Machines and other manufacturing equipment must be setup before executing the operations of different product types. The setup times depend on the sequence of jobs according to their product types. Machines and workstations are available only during predefined time intervals and cannot perform operations outside these time frames.

In scheduling the processes to be executed in this examined discrete manufacturing system, it is assumed that the current state of the manufacturing system and the set of valid constraints are given. The set of given restrictions include, for example, primary technological and other resources, their limited capabilities and capacities, availability intervals, definitions of operation sequences, and additional special conditions. In the basic predictive scheduling task, to fulfill the issued internal orders, manufacturing jobs and operations need to be defined, and suitable resources need to be assigned to them. Start and finish times need to be assigned to each manufacturing event in a way that the key performance indicators (KPIs) representing the performance of the manufacturing process are quasi-optimal while adhering to the constraints. The KPIs express the management’s formulated objectives in this optimization. The applied enterprise-specific performance indicators appear in our model in the form of optimization objective functions.

In this article, the following selected objective functions are used:

• Minimize the number of orders (IMOs) completed after the deadline ($f_1$).
• Minimize the number of jobs finished after the deadline ($f_2$).
• Minimize the sum of job tardiness ($f_3$).
• Minimize the maximum job lateness ($f_4$).
• Minimize the number of machine setups ($f_5$).
• Minimize the finish time of the last operation ($f_6$), aiming to make it as early as possible.

A predictive schedule alone does not guarantee optimal execution of the production process within the scheduled period, as real-time continuous control of production is necessary due to the inevitable uncertain behavior of both the internal manufacturing environment and the external business environment. Unexpected events can include machine breakdowns, changes in operation capabilities, variations in raw material arrival times, deadline modifications, and so on. To minimize or reduce the economically detrimental effects of such unexpected events, numerous corrective intervention decisions need to be made. Situations requiring such interventions necessitate the review and modification of the previously well-planned and released schedule that is in progress, thus resulting in the rescheduling of activities.

Rescheduling involves updating, correcting, and redesigning the initiated and ongoing production schedule, serving as one of the intervention functions of the control system.

2.2. Measuring the Impact of Unexpected Events

To demonstrate the rescheduling problem, a simulation model of a manufacturing system has been developed. The system consists of 124 machines organized into 10 machine groups. To produce the final product, four consecutive technological operation sequences need to be performed. In the current period, there are 502 orders (IMOs) to be fulfilled, comprising a total of 5231 jobs. The IMOs are categorized into eight types based on the required operation sequences. The machine setup types were classified into four categories. The setup durations of the machines are asymmetric and unique to each machine. The machines have unique operational characteristics and shift arrangements. Not all machines can work on every IMO, and the operation capabilities of the machines are also unique, resulting in the execution time of operations depending on the specific machine and IMO.

Our rescheduling system includes an interface that can display the impact of unexpected events in the form of enhanced Gantt charts.

Figure 1 illustrates a machine-oriented Gantt chart, where machine M3 and M4 are displayed. Each machine has two scheduling rows. The upper row shows the current state and the expected actual states in the future. The bottom row displays the originally planned schedule.

In this example, two unexpected events were simulated related to the machines. The operational intensity of the machine identified as M4 decreased from its initial value of 9 units to 6 units. As a result, each processing time increased from its original value to one and a half times its original value. The gray rectangles represent the availability intervals (shifts) of the machines. The colored smaller rectangles symbolize the duration of the activity to be performed. It is evident that the load originally filling two shifts would fill three shifts due to the unexpected intensity reduction event. Due to the slowdown, the operations originally finishing on time would exceed the allowed deadlines. These changes are highlighted with red rectangles on the diagram. The light green rectangle represents timely completion, while the red color denotes exceeding the deadline. The orange-colored rectangles indicate machine setup activities.

For machine M3, we simulated an error occurring at the beginning of the second shift, and its existence renders the machine unusable during the second shift. Consequently, the planned operations are shifted to the right in time.

Figure 2 depicts an enhanced order-oriented Gantt chart. We simulated two additional new unexpected events. The release time constraint of order O480 increased significantly from the original 91 units to 500 units. This indicates that the arrival or preparation of some raw material will be delayed compared to the originally planned schedule. The gray rectangle on the diagram symbolizes the constrained time window for IMO fulfillment. The light green or red rectangles represent the duration between the start and finish of the jobs. Light green indicates on-time completion, while red indicates a deadline overrun. The simulation clearly shows that, due to the unexpected event, the execution of order O480 would exceed the expected completion deadline.

Regarding order O481, we simulated an unexpected event that significantly reduced its original deadline from 878 time units to 600 time units. This event represents an external impact occurring in the business process. As a result of this deadline change, the original schedule would cause a delay in execution. This is also indicated by the red color of the rectangle.

The simulation results also indicate the unfavorable impact of unexpected events on the objective function values, which primarily depend on the extent of changes in the recalculated start and finish times.

Table 1. The impact of unexpected events on the objective function values.

Objective Function	Previous Planned Value	Changed Value
$f_1$	0	8
$f_2$	0	28
$f_3$	0	3513
$f_4$	0	356
$f_5$	230	230
$f_6$	2012	2012

summarizes the original values of the objective functions according to the original schedule and the modified values due to the occurrence of unexpected events. As a result of the unexpected events, the work according to the original schedule would complete eight orders (IMOs) and 28 of their jobs after their deadlines, while all orders (IMOs) and their jobs would have been completed on time according to the original schedule without unexpected events. The number of setups ($f_5$) did not change because, after repairing machine M3, it can continue to operate with the previous settings at the originally planned start of the third shift (which, due to the machine breakdown, became the second active shift.

The simulation results clearly show that the four unexpected events appearing in the example have a very unfavorable impact on the manufacturing system’s performance within the given time horizon.

3. Advanced Rescheduling Method

3.1. Fundamental Components of the Rescheduling System

Our rescheduling approach is based on a new combined usage of extended multi-objective searching strategies, modifying operators, locking techniques, and a relative qualification model. The rescheduling extension is achieved by formulating new constraints and introducing new objective functions.

In the rescheduling task, the current production schedule serves as input data. It represents the planned execution of manufacturing processes. We also consider the planned performance indicator values associated with the current schedule as input. Together, they form the initial state description. This information set becomes the starting point and serves as the basis for comparing modifications. Additionally, as further input baseline information, we assume the availability of the current state precisely described schedule. This represents the events that have occurred and the resulting changes in state. All this information, along with real-time data, is available.

As a fundamental constraint, we introduced that the new schedule candidates resulting from rescheduling must necessarily preserve the parts of the original schedule completed up to the time of rescheduling. This modification prohibition also applies to the already started but not yet completed elementary parts of the original schedule. Every operation that cannot be interrupted is considered an elementary part.

Furthermore, we defined an additional time interval, which extends from the moment of rescheduling to a given later point in time, and this interval is also subject to the prohibition of changes. We named this interval the reaction period, expressing in the model that the system will still operate according to the original schedule between the start of rescheduling and the release of the new schedule. Operations, jobs, and IMOs that start before the end of the reaction period will directly affect the new schedule as well. For example, if a specific product type has already started on a production route originally chosen for it, then after the reaction period, any yet-to-start production operations can only be redirected to a subset of the initial resource alternatives due to dependencies. In such cases, intervention opportunities decrease as time progresses. This implies that the longer the reaction period, the higher the likelihood of increased dependencies.

During the rescheduling intervention, we incorporated new objective functions into the evaluation criteria of the modified schedule. These new KPI elements express the extent of modifications compared to the original schedule. We defined these new KPI metrics to be minimized. With the introduction of these metrics, we fulfill the previously formulated requirement that the rescheduling method achieves maximum improvement with minimal changes. Thus, stable, transparent, and traceable planning can be achieved, ensuring production without significant fluctuations or jerks.

With these extensions added to the original scheduling tasks, we obtained the general model of rescheduling problems. For the software implementation of the solution, we relied on our previously developed multi-objective search methods [17,18,19]. These search algorithms supporting simulation and multi-objective optimization were flexible enough to meet the specific requirements of rescheduling with appropriate enhancements, as demonstrated by the presented approach.

The core component of our scheduling software is an advanced multi-objective, multi-operator search algorithm (AMOMOSA). The pseudocode of this algorithm is presented in Algorithm 1. AMOMOSA operates iteratively within the virtual space of feasible solutions, progressing from a current base schedule $s_0$ to a candidate neighboring schedule $s_c$, located in the neighborhood of $s_0$, until a stopping criterion is met. In order to explore previously unvisited regions of the search space, AMOMOSA dynamically adjusts the neighborhood structure of each base schedule as the search advances. To avoid being trapped in local optima, AMOMOSA maintains a tabu list that records schedules visited in the recent past. The maximum number of entries in this list is determined by a predefined parameter. Schedules contained in the tabu list are excluded from the current neighborhood of the base schedule. New neighboring candidates are generated randomly through neighborhood operators, which modify resource allocations and job sequences in accordance with the characteristics of the problem space. These variables represent the primary decision factors in the scheduling problem. During the search, the best solution in the current neighborhood is denoted by $s_n$, while the best solution found throughout the entire search is represented by $s^{*}$.

The proposed neighboring operators are as follows:

• Operator $N_1$ removes a randomly selected production order from the schedule and then inserts all its jobs as a matching series. These jobs follow the same route and visit the same machines. On each selected machine, the jobs are executed in a predefined sequence.
• Operator $N_2$ removes a randomly selected late production order from the schedule and inserts all its jobs in the same way as operator $N_1$.
• Operator $N_3$ modifies the job sequence on a randomly selected machine using a permutation cycle of random length.
• Operator $N_4$ removes a randomly selected late job from the schedule, redefines its manufacturing tasks by assigning resources, and finally inserts the job into a random position on each relevant machine.
• Operator $N_5$ works similarly to operator N₄, but the target job is selected from the entire set of jobs.

These neighboring operators generate new candidate schedules by modifying the values of the primary decision variables of the base schedule. Our proposed algorithm combines the advantages of variable neighborhood structures and tabu search.

Algorithm 1: AMOMOSA

{

s0 ← Create an initial schedule.

s* ← s0;

Tabu_List ← NULL;

while ( Stop criterion is not satisfied )

{ while ( Neighbouring criterion is satisfied )

{

Nc ← Choose the actual neighboring operator(priority_list);

sc ← Create a neighbour schedule by modifying the base schedule( s0 , Nc);

if ( Tabu_List does not strore the schedule ( sc ) )

{

Insert the new tabu schedule into the first position of Tabu_List ( sc );

if ( The number of the stored tabu schedules > Maximum number )

Delete the tabu schedule from the last position of Tabu_List;

if ( Is this the first neighbour schedule ( sc ) ) sn ← sc;

else if ( sc < sn ) sn ← sc;

}

}

s0 ← sn;

if ( sn < s* ) s* ← sn;

}

return s*;

}

The new enhancements are summarized as follows:

• Modeling the current state and implementing data loading.
• Storing and loading the current scheduling issued for a given time period.
• Advancing the simulation model.
• Expanding the set of objective functions to minimize the extent of changes.
• Introducing locking techniques to handle prohibitions arising from dependencies and ensure limited modifications.
• Developing new search operators to consider locks and improve the extended KPI metrics.

The multi-objective search method iteratively performs a rapid simulation-based problem space transformation. During the search for the best solution, a simplified schedule is generated based on the values of decision variables. This simplified solution representation captures the technological route of jobs, the selected and allocated resources, and the execution sequence of operations (elementary activity units) assigned to each resource. Starting from these decision variables, the software simulation of the entire schedule calculates the start and finish times for each element. This simulation-based problem transformation method operates during rescheduling by using locking technique data to consider the already recorded time data for past production events through a simulation algorithm based on a rule base. Then, it only computes the time data for future planned events with unknown timings.

We utilized our multi-objective search method applied to predictive scheduling, as presented in [18], and incorporated the new elements of rescheduling into it. Based on the weighted sum of the relative changes in objective functions, we could clearly determine which of two candidate solutions is more favorable overall. Using this qualification principle, among many candidate solutions, the best solution can be selected through pairwise comparison.

To minimize the extent of rescheduling changes, we defined new objective functions. The following list explains the newly introduced stability-oriented objective functions used in the rescheduling framework. These objectives quantify the extent to which the revised schedule deviates from the original one. Each function captures a different aspect of stability, such as route changes, machine reassignments, lateness variations, and setup modifications. The following explanations provide a detailed account of the specific aspects quantified by each objective function.

• Number of jobs completed on the modified route according to the schedule ($g_1$): This objective function counts the number of jobs that are executed on a different route compared to the original schedule. A change in the main assigned machine group indicates that the job has been rerouted, potentially requiring additional attention in the production process.
• Number of jobs scheduled on a different machine from the original schedule ($g_2$): This objective function counts the number of jobs that have at least one operation assigned to a different machine than originally specified in the baseline schedule. Such deviations may entail additional control and logistical efforts during execution.
• Number of jobs with a completion status different from the originally planned status ($g_3$): This objective function quantifies the number of jobs whose completion status—classified as either on time or delayed—has changed due to rescheduling. It reflects whether the revised schedule has led to previously timely jobs becoming delayed or delayed jobs being completed on time.
• Number of jobs becoming delayed due to rescheduling ($g_4$): This objective function focuses specifically on jobs that were originally on time but have become delayed as a result of rescheduling. It identifies new delays introduced by the changes.
• Number of orders (IMOs) with jobs satisfying the condition described in the first point ($g_5$): This objective function counts the number of individual orders (IMOs) that include at least one job whose main route has changed compared to the original schedule. Each order is counted only once, regardless of how many jobs it contains.
• Number of orders (IMOs) with jobs satisfying the condition described in the second point ($g_6$): This objective function quantifies the number of orders that include at least one job whose machine assignment differs from that specified in the original schedule. It captures the propagation of job-level machine changes to the order level.
• Number of orders (IMOs) with a changed completion status compared to the original plan ($g_7$): This objective function counts the number of orders for which the completion status of at least one job has changed. It aggregates job-level status changes (as captured in g₃) to the order level.
• Number of orders (IMOs) becoming delayed due to rescheduling ($g_8$): This objective function captures the number of orders that have become delayed due to at least one newly delayed job. It focuses on the propagation of job delays to the order level.
• Number of machines affected by changed setups due to rescheduling ($g_9$): This objective function counts the number of machines for which the number of required setup operations differs between the original and rescheduled plans. It reflects changes in machine workload and preparation requirements.
• Number of machines burdened with new setups due to rescheduling ($g_{10}$): This objective function counts the number of machines that require at least one new setup operation in the rescheduled plan that was not present in the original schedule. A setup is considered necessary when the next job on a machine requires a different configuration than the previous one, based on the execution sequence.

The newly defined objective functions play a crucial role in ensuring schedule stability. In manufacturing environments, rescheduling is often unavoidable; however, it is essential that the modifications affect the original plan only to the extent absolutely necessary. Without explicitly addressing stability, a revised schedule may deviate substantially from the original, leading to increased complexity in production control, higher logistical overhead, and reduced operational transparency. The purpose of these objectives is to minimize the extent of changes in job sequences, machine assignments, completion statuses, and setup requirements. These stability-oriented goals help ensure that rescheduling does not result in a completely new plan but rather a controlled adaptation that preserves as much of the original structure as possible while responding effectively to the new situation.

The objective functions expressing the changes ($g_1$–$g_{10}$) and the production efficiency quantifying objective functions ($f_1$–$f_6$) collectively form the objective function system, which evaluates the modified production schedule. To express the importance of different objective functions, we use priority values that can be adjusted by the user. The search engine applies to a comparative and result-evaluating method based on relative changes according to these priorities [17].

The formal description of the relative quality model is as follows:

Given K objective functions to be minimized, as shown in (1),

$$q_k:S\to {\mathbb{R}}^{+}\cup \left\{0\right\},\hspace{0.33em}\forall k\in \left\{\mathrm{1,2},\dots ,K\right\},q_k\to min$$

(1)

the following two functions are used for multi-objective relative comparison:

$$D:{\mathbb{R}}^2\to \mathbb{R},D\left(a,b\right)=\left\{\begin{array}{c}0,\hspace{1em}if\hspace{0.33em}\mathit{max}\left(a,b\right)=0\\ {\displaystyle \frac{b-a}{\mathit{max}\left(a,b\right)}},\hspace{1em}otherwise\end{array}\right.,$$

(2)

$$F:S^2\to \mathbb{R},F(s_x,s_y)={\sum }_{k=1}^K\left(w_k\cdot D\left(q_k\left(s_x\right),q_k\left(s_y\right)\right)\right),$$

(3)

where

$S$—the set of feasible solutions;

$q_k$—the kth objective function to be minimized;

$K$—the number of objective functions;

$s_x$, $s_y$—two given solutions;

$w_k$—the priority of the kth objective function;

$F(s_x, s_y)$—the relative quality of the solution $s_y$ compared to the solution $s_x$.

Using the signed return value of the function $F(s_x, s_y)$, we extend the interpretation of relational operators to the solutions $s_x$ and $s_y$ in the set $S$. The definition of this operator overloading is as follows:

$$\left(s_y\hspace{0.33em}?\hspace{0.33em}{s}_x\right):=\left(F(s_x,s_y)\hspace{0.33em}?\hspace{0.33em}0\right)$$

(4)

where $?$ indicates any of the relational operators $(<, \le ,>, \ge ,=, \ne )$.

The relationship between $F(s_x, s_y)$ and zero expresses the relative ranking between solution $s_y$ and solution $s_x$. For example, $s_y$ is considered better than $s_x (s_y<s_x) if F(s_x, s_y)$ is less than zero. This relative evaluation model effectively supports the comparison of candidate solutions within the search algorithm. Therefore, the proposed AMOMOSA method is capable of performing multi-objective optimization while taking into account the actual requirements of production management.

Rescheduling starts from the production schedule describing the current situation and creates the new schedule through a series of elementary modifications. Thus, the multi-objective search techniques used for proactive scheduling (such as tabu search, simulated annealing, genetic algorithm, etc.) can also be used for rescheduling. However, during rescheduling, it must be considered that the past part of the initial schedule and the part falling into the time interval required to create and start the new schedule cannot be modified. To increase the speed of rescheduling, prohibited modifications are analyzed before the search. The permissible alternatives are incorporated into the model’s network in an organized form. Thus, during decision-making situations in the search process, only options that fully satisfy the new conditions are available. We implemented the extension of rescheduling constraints in this direction using locking techniques. Locking rules can apply to jobs, orders, and machines:

• For any job, it can be specified from which operation onwards changes are allowed.
• For any machine, it can be specified from which operation onwards changes can be made in its execution sequence.
• For any order, the degree and form of locking can be clearly deduced from the attributes of the locked jobs associated with it. These include the applicable routes, the affected machine groups, and the set of specific machines within the machine group for the given order (IMO).

The neighborhood or modifying operators used for proactive scheduling have been enhanced to accommodate locking considerations. As a result, the multi-objective heuristic search algorithm has become suitable for solving rescheduling tasks.

3.2. Minimizing the Impact of Unexpected Events Appearing in the Illustrative Example Through Rescheduling

We applied the proposed AMOMOSA rescheduling method to the situation described above. In this example, we limited the size of the tabu list to 150 solutions. The maximum number of search steps was set to 2000. We allowed the examination of 100 neighboring solutions in each step. The end of the locked interval was set to 100 time units. We did not change the priorities of the objective functions $(f_1,f_2\dots f_6)$ expressing manufacturing efficiency compared to the original schedule. The priority of each objective function $(g_1, g_2\dots g_{10})$ minimizing rescheduling changes was set to 5.

The result of the rescheduling is presented in

Table 2. The effect of rescheduling on the objective function values.

Objective Function	Value Before Rescheduling	Value After Rescheduling
$f_1$	8	0
$f_2$	28	0
$f_3$	3513	0
$f_4$	356	0
$f_5$	230	230
$f_6$	2012	2012

. With rescheduling, we were able to eliminate the threatening deadline violations. The values of the objective functions quantifying delays and tardiness $(f_1–f_4)$ were reduced to zero. By rescheduling the jobs, the number of machine setups did not increase. The values of objective function $f_5$ remained 230, and the completion time of the last operation in the busy period ($f_6$) remained at 2012 time units.

Table 3. The values of rescheduling objective functions quantifying the extent of modifications to the original schedule.

Objective Function	Value
$g_1$	0
$g_2$	21
$g_3$	28
$g_4$	0
$g_5$	0
$g_6$	5
$g_7$	8
$g_8$	0
$g_9$	0
$g_{10}$	0

contains the numerical values of deviations from the original schedule. During rescheduling, none of the jobs had to be redirected to a modified route; all of them followed the original sequence of machine groups. Only five orders (IMOs), totaling 21 jobs, were transferred to a different machine of the same group than originally planned to eliminate the delay of 28 jobs and eight orders (IMOs). The rescheduling method did not cause any new delays for the remaining jobs and orders. None of the machines were affected by changes in setups. Machines did not require the insertion of new setup activities.

The presented rescheduling method, using the enhanced searching operators, locking techniques, and new objective functions, effectively solved rescheduling problems.

Selected results of the rescheduling are illustrated in Figure 3, Figure 4, and Figure 5. Considering Figure 3, the impacts of unexpected events affecting machines M3 and M4 on the original schedule (bottom rows) can be compared with the results of the rescheduling (upper rows).

Figure 3 also shows that the rescheduling algorithm only retained the jobs on the affected machines that could still meet the deadlines under the changed conditions. The remaining jobs were transferred to alternative machines of the given group. Machine M4 still anticipates increased operational time due to the persistent fault, while machine M3 is not burdened by the missing shift but schedules full-intensity work for the other shifts.

Figure 4 depicts that orders O480 and O481 can be completed on time because of the rescheduling. The modified schedule ensures that, despite the supply delay of raw materials for order O480, the order can still be fulfilled. Despite a drastic reduction in the deadline for order O481, the rescheduling process ensures its on-time completion.

Figure 5 illustrates the details of fulfilling the two selected orders. Compared to the diagram shown in Figure 4, this more sophisticated second diagram not only displays the order fulfillment time window but also shows all the jobs associated with the given orders. The light blue rectangle represents the duration between the start and finish. The colored small rectangles represent the execution of jobs associated with the IMO. These jobs represent smaller batches of the ordered final product. For example, a larger order can be fulfilled with several smaller batches that are scheduled independently of each other.

To emphasize the importance of the proposed new objective functions ($g_1$–$g_{10}$) that minimize changes, we also present the results of a rescheduling process (R2) where only the production performance oriented objective functions ($f_1$–$f_6$) were used by setting the priority of each change-minimizing objective function to zero.

The comparison of the R1 and R2 rescheduling results can be found in

Table 4. Comparison of the values of rescheduling objective functions using two different priority settings.

Objective Function	$\mathbf{Value} \mathbf{of} \mathbf{R}1 \mathbf{Using} \mathit{f}$$\mathbf{and} \mathit{g}$	$\mathbf{Value} \mathbf{of} \mathbf{R}2 \mathbf{Using} \mathbf{Only} \mathit{f}$
$f_1$	0	0
$f_2$	0	0
$f_3$	0	0
$f_4$	0	0
$f_5$	230	226
$f_6$	2012	1977
$g_1$	0	60
$g_2$	21	985
$g_3$	28	28
$g_4$	0	0
$g_5$	0	28
$g_6$	5	269
$g_7$	8	8
$g_8$	0	0
$g_9$	0	14
$g_{10}$	0	5

. The numerical results show that both rescheduling variants successfully eliminated all delays, but there were significant differences in terms of minimizing changes.

During the R1 rescheduling, both the $f$ and $g$ sets of objective functions were used. The results indicate that the R1 rescheduling eliminated all delays. Additionally, the number of machine setups remained unchanged, meaning that no setup modifications were necessary compared to the original schedule. The routes of orders and jobs also remained unchanged, with only 21 cases of jobs being assigned to different machines. This means that the $g$ objective functions in the R1 rescheduling had a favorable impact by minimizing changes and maintaining stability in the schedule.

In contrast, the R2 rescheduling used only the $f$ objective functions. While the results show that all delays were eliminated, significant modifications were required compared to the original schedule. The number of machine setups decreased from 230 to 226, which is a positive outcome, and the finish time also decreased from 2012 to 1977. However, the R2 rescheduling involved assigning new routes to 28 orders and 60 jobs and assigning different machines to 269 orders and 985 jobs. These would require a very drastic reorganization of material handling as well as the supply of tools and equipment within the manufacturing system. Additionally, the original setups were modified on 14 machines, with 5 of these machines requiring new setups. These changes would give a significant amount of extra work and burden to logistics and shop floor control.

In summary, the R1 rescheduling, which used both the $f$ and $g$ objective functions, resulted in more balanced and stable outcomes with fewer changes and modifications to the schedule. The R2 rescheduling, which used only the $f$ objective functions, while effectively eliminating delays and reducing the finish time, required substantial changes compared to the original schedule. This underscores the importance of the $g$ objective functions in minimizing changes and maintaining stability in the schedule.

To evaluate the runtime performance of the proposed rescheduling approach, the same rescheduling problem (R1) instance was solved five times using identical algorithmic settings. The total search process included 2000 iterations, and in each iteration, 100 neighboring solutions were evaluated. The measured runtime durations were as follows:

1 min 28 s, 1 min 28 s, 1 min 27 s, 1 min 29 s, and 1 min 34 s, respectively.

Notably, the values of all tardiness-related objective functions were reduced to zero within the first 120–160 iterations, corresponding to only 5–6 s of execution time. The remaining computation time was primarily used for searching for further improvements in solution quality, and this phase can be easily customized by applying improvement-oriented termination criteria. The investigation focused on execution time, so the search process was allowed to continue until the step limit was reached. However, it is also possible to configure the algorithm to stop automatically if none of the objective functions improve within a given number of iterations.

The problem instance contained 502 orders, 5231 jobs, and 124 machines, which needed to be simulated for every candidate solution. Given this complexity, the observed runtimes can be considered highly efficient and practically acceptable.

All experiments were executed on a standard consumer-grade laptop with the following specifications:

• Operating System: Microsoft Windows 11 Home (Build 26100)
• Processor: Intel(R) Core(TM) Ultra 5 135U, 1.6 GHz, 12 cores, 14 logical threads
• RAM: 16 GB
• System Model: Lenovo ThinkPad L13 2-in-1 Gen 5

No specialized hardware acceleration (e.g., GPU computation) or cloud infrastructure was used during the experiments. These results demonstrate the method’s suitability for deployment in real-world environments with typical hardware configurations.

In addition to the presented example, we also tested the proposed rescheduling method with various problem instances and different settings. The results of the experiments confirmed the flexibility of the method. By adjusting the priorities of the objective functions, the rescheduling can be effectively regulated. This allows for users to balance manufacturing efficiency and the extent of modifications even in situations where meeting deadlines requires significant rescheduling. Due to the flexibility of our software implementation, the proposed rescheduling method can serve as an effective tool in production management. It provides quick and easy-to-use support for both planners and operational managers.

In this paper, our aim was not to compare various search algorithms but rather to demonstrate the advantages of applying new objective functions that support the minimization of schedule changes during rescheduling tasks. The focus of this study is on the proposed g-type objective functions, which are specifically designed to ensure that modifications to the ongoing schedule are limited to the minimum necessary to resolve disruptions or eliminate delays.

Experimental results clearly show that the use of g objective functions offers significant benefits: rescheduling interventions introduce only minimal changes to the existing schedule—just enough to address the underlying problems. In contrast, traditional approaches relying solely on, for example, f-type objective functions, while capable of eliminating delays and improving performance indicators, often lead to extensive reorganizations in production. These excessive changes place a considerable burden on logistics, material handling, and shop-floor operations, thereby complicating production support.

In light of these findings, the application of the proposed g-type objective functions is both justified and advantageous. They not only enable the resolution of scheduling issues but also preserve the stability and continuity of the original schedule with minimal intervention. This is particularly important in dynamic, real-time manufacturing environments where rapid yet predictable rescheduling is essential.

4. Summary and Conclusions

This paper presents an advanced model to solve the rescheduling problems of a flexible manufacturing system where alternative manufacturing routes, machine groups, and flexible technological equipment are key components. To examine the efficiency of the manufacturing process, we developed a simulation model programmed in C++ and equipped with a graphical user interface for ease of use. To improve the performance of discrete manufacturing processes, we developed a novel multi-objective rescheduling method that generates near-optimal solutions through iterative refinement of detailed schedules, aiming to minimize the adverse effects of unexpected events in the manufacturing and business environment.

Utilizing the presented rescheduling concept, both proactive production scheduling and rescheduling tasks can be efficiently managed. The proposed method relies on problem space transformation based on execution-simulation, locking techniques, and multi-objective search algorithms.

In this paper, we primarily focused on demonstrating the role of change-minimizing objective functions in the rescheduling system. Through a specific case study, we illustrated the benefits of their usage.

The proposed rescheduling method, using both sets of performance-oriented and change-minimizing objective functions, effectively eliminated delays while maintaining schedule stability. This highlights the advantage of incorporating the proposed new objective functions to ensure achieving a balanced, stable schedule. The flexibility of this method also allows for efficient rescheduling tailored to specific needs, making it a valuable tool for production management. The developed software has demonstrated that the outlined model, using the proposed new objective functions, effectively supports the management of technological alternatives, resource allocation, and the scheduling and rescheduling of manufacturing tasks.

In addition to these results, the proposed set of g-type objective functions represents a novel scientific contribution to the field of rescheduling. These functions are specifically designed to minimize changes to the running schedule, ensuring that rescheduling interventions modify the original plans only to the extent strictly required by the current disruption. This capability supports schedule stability and operational continuity, addressing a key challenge in dynamic manufacturing environments.

Furthermore, another important contribution of this research is the introduction of a relative change-based evaluation method for assessing multi-objective solutions. This approach is capable of handling any number, dimension, and value domain of objective functions, enabling unified and flexible evaluation in multi-objective optimization. Unlike traditional Pareto-based approaches, the proposed method provides a more adaptable framework that can be effectively combined with any search algorithm. It thus offers a new, general-purpose tool for solving complex rescheduling problems where trade-offs between performance and stability must be carefully managed.