The Multiresource Flexible Job-Shop Scheduling Problem with Early Resource Release

Abstract

In this work, we study the multiresource flexible job-shop scheduling problem (MRFJSSP), which relaxes the standard “simultaneous occupation” policy described in the literature. This policy implies that a job operation starts only when all its assigned necessary resources are available and releases them simultaneously. In contrast, our approach assumes that a job operation begins simultaneously across all assigned resources, although these resources may not be occupied for the same duration. This variant (which we will call “early resource release”) was first formally proposed in the scheduling literature more than twenty years ago, but to the best of our knowledge, it has not been empirically tested. Thus, to tackle this problem, we formulate a constraint programming (CP) model adopting a multi-mode resource-constrained project scheduling problem (MMRCPSP) representation. We tested our approach on 65 instances of the MRFJSSP where the precedence relationships between operations of a given job follow a linear order. We prove optimality in 44 instances with an average optimality gap of 9.37%. Additionally, we contributed eight new lower bounds for the same set of instances in the literature when considering the simultaneous occupation policy.

1. Introduction

The job-shop scheduling problem (JSSP) has been one of the most widely studied problems in workshop scheduling research over the past decades. It is characterized by the need to sequence job tasks (operations) for completion using limited, renewable resources, such as machines or workers. In a JSSP environment, tasks must be performed in a given order and processed on a specific machine. The most common regular goal is to minimize the makespan ($C_{max}$), that is, the total time required to complete a set of jobs or tasks from start to finish. The JSSP under makespan minimization can be represented as $J_m\left|\right|C_{max}$ according to the well-known three-field notation $\alpha \left|\beta \right|\gamma$ introduced by [1].

A well-known extension of the JSSP is the flexible job-shop scheduling problem (FJSSP). Unlike a JSSP, in an FJSSP, each operation has a subset of machines on which it can be processed, adding an assignment layer to the scheduling problem. Indeed, as mass customization and automation advance, the FJSSP is no longer considered merely an extension of the JSSP but has emerged as a standalone shop-scheduling problem. In this context, various variants of the FJSSP have been introduced successively. We refer the reader to [2] for a comprehensive review of the literature related to the FJSSP in the last decades.

This paper focuses on a specific under-studied extension of the FJSSP that accounts for the use of multiple necessary resources, such as machines and human labor, for specific tasks. This problem, known as the multiresource flexible job-shop scheduling problem (MRFJSSP), frequently appears in practical applications. The potential applications of the MRFJSSP are broad and evident, in both manufacturing and the service industry, where alternative resources are available to process certain work operations and are additionally required simultaneously at the start of the operation. In this context, a common assumption for the MRFJSSP is that an operation starts simultaneously on all the resources assigned to it and that these resources are occupied for the same duration. This policy is referred to as “simultaneous occupation” by [3]. However, in practice, an operation may not finish simultaneously on all its assigned resources, as explicitly recognized more than 20 years ago by [4]. An illustrative example of this scenario can be found in an aircraft engine production line, where the assembly of certain subcomponents requires the concurrent use of multiple resources (both human and machine, such as skilled workers, tools, and cranes) that may differ in their operating rates and participation times. Additional potential applications of the MRFJSSP are presented in Section 2.

Thus, in this work, we propose an approach to handle the non-simultaneous use of resources for the first time. Here, even though we assume that a job operation starts simultaneously on all assigned resources, such resources are not necessarily occupied for the same time. That is, a resource that completes its task (often human) can be released early without waiting for the other resources assigned to the same operation to finish. A priori, this policy could improve the utilization of bottleneck resources, enhancing the overall efficiency of the production process. Hereafter, we refer to this variant as the MRFJSSP with early resource release.

To address this problem, we extend the constraint programming (CP) formulation of the multi-mode resource-constrained project scheduling problem (MMRCPSP) proposed in [5] for the FJSSP, incorporating the necessary constraints to execute the MRFJSSP with early resource release. This novel CP formulation for the MRFJSSP with early resource release incorporates dummy operations to model the problem’s multi-resource characteristics. Then, we test our approach using 65 available instances used in [3], in which the precedence relationships between operations of a given job follow a linear order. Finally, we compare our results with those reported in [3] and with our own implementation of the MRFJSSP using the simultaneous occupation scheme. In this way, our objective is to gain insight into the impact of relaxing the simultaneous-occupation policy in favor of one that permits early resource release.

All in all, the scientific contributions of this article can be summarized as follows:

• We propose a CP formulation, based on the MMRCPSP, to address the MRFJSSP with early resource release, a variant that more accurately reflects most industrial applications.
• Using this approach, we derive eight new lower bounds for the MRFJSSP under the standard simultaneous resource occupancy assumption.
• We show that the proposed formulation can be adapted to the classical MRFJSSP with simultaneous occupation.
• Computational experiments for both variants demonstrate the competitiveness of our method with respect to the best bounds reported in the literature.

The remainder of the paper is structured as follows. Section 2 presents an in-depth study of the state-of-the-art extensions of multiple resource FJSSP, as well as the motivation underlying this research. Section 3 presents the problem definition, our CP formulation for the MRFJSSP with early resource release, an illustrative small-sized instance, and an extension of our proposed formulation to tackle the case of simultaneous occupation. The experimental setup, computational results, a sensitivity analysis of the results for different time limits, and some concluding remarks are provided in Section 4. Finally, conclusions and future work are presented in Section 5.

2. Literature Review

In the JSSP, we are given n jobs and m machines. Each job j consists of a set of operations $O_j$, each performed on one of the m machines. The operations of a job j must be performed sequentially in a predefined order (a fixed route), and pre-emption is not allowed (an operation cannot be interrupted once started). The route that each job follows through the machines may differ.

In terms of complexity, ref. [6] demonstrated that the minimum-makespan scheduling problem for an m-machine JSSP is NP-complete for $m\ge 2$. In this context, the JSSP is recognized as one of the most challenging combinatorial problems and has been the subject of research for decades. Despite this, many classical $J_m\left|\right|C_{max}$ instances remain open, meaning that their optimality has not yet been proven (see [7]).

A highly relevant extension to the classical JSSP is the FJSSP, which allows an operation to be assigned to different resources. This problem has been studied since the last century, with pioneering works including [8,9]. This extension introduces flexibility into the predefined route, allowing each operation to be processed on any machine within a subset of the available resources, thereby expanding the allocation possibilities. Extensions of the FJSP are critical in several real-world applications. In [10,11], the Dynamic Flexible Job-Shop Scheduling Problem (DFJSP) is presented in relevant scenarios nowadays. Ref. [10] studies its application to a dynamic scheduling problem in a job shop robotic cell. In their study, Fatemi-Anaraki et al. consider that multiple robotic arms are responsible for material handling. Intermediate buffers are positioned between pairs of consecutive robots. The dynamic nature arises from the arrival of new jobs at unpredictable times, necessitating the rescheduling of the current state. Ref. [11] presents a genetic algorithm coupled with simulated annealing and variable neighborhood search to address dynamic scenarios involving single-machine breakdowns, single-job arrivals, multiple-machine breakdowns, and multiple-job arrivals. We encourage the reader to consult [2] for a comprehensive survey of the FJSSP and their variants.

Over the past decades, various models and extensions of the JSSP and FJSSP have been applied across industrial sectors where efficient resource management is critical. We have focused our analysis on variants that implement simultaneous occupation but could also incorporate an early resource release policy. In this regard, we have mainly considered applications related to the Multi-Resource Flexible and the Dual-Resource Constrained Flexible extensions. In these variants, each operation may require a set of resources. In most industrial settings, some of these resources can be released early, potentially enabling more efficient solutions. We have also identified industrial applications of these variants.

In this context, the MRFJSSP was first introduced in [12]. The authors considered three extensions: (1) a resource may be selected from a given set, (2) a nonlinear (a job operation may have more than one predecessor or more than one successor) routing of operations, and (3) operations may need several resources at the same time to be performed (multiresource). The authors introduce a new version of the problem, extend the disjunctive graph representation, and propose a tabu search approach that uses a well-defined exploration neighborhood. Simultaneous occupancy policy is implemented here.

In [13], the authors propose the Multiresource Shop Scheduling with Resource Flexibility and Blocking (MRSSRFB). In their extension, operations can be performed in different modes and require different sets of resources depending on the selected mode. Moreover, blocking constraints require holding resources used for an operation until resources needed for the next operation of the same job are available. An explicit simultaneous occupation extension is defined in this version of the problem. A shortest-path approach and a greedy heuristic are employed to schedule sets of jobs, which are then optimized using a tabu search algorithm to solve instances with up to 20 jobs and 10 machines. Applications of this problem extension can be found in shipbuilding manufacturing [14], systems that require highly coordinated manufacturing, food canning operations, and several metal processing production lines [15].

Dual-resource constraints have also been extensively studied in the literature [16]. In these extensions, two resources are required for operations: machines and workers. The Dual-Resource Constrained Flexible Job Shop Problem (DRCFJSP) was first proposed in [17]. Since its introduction, most of their approaches have considered machine and worker flexibility. In these studies, each machine must be operated by a specific worker, selected from the pool. The processing time for each operation performed by a worker on a machine is fixed according to the operation time [16]. In these studies, decisions are related to workers operating tasks in machines, but not the time workers are effectively required [18].

There are interesting applications of these DRCFJSP that involve specific considerations regarding resource-occupancy time requirements. For example, in [19], the authors propose planning medical exams that account for the assignment of material and human resources across different locations, considering a given horizon. In this case, transportation times for human resources are considered, but they should still be available throughout the entire exam.

In [20], the Multi-resource Partial-Ordering Flexible Job-shop Scheduling problem (MRPOFJSP) was introduced. In this extension, the partially ordered features imply that some operations of a job can be executed in any order. This scenario could be considered for the natural incorporation of early release occupancy protocols.

More recently, in [21], the FJSSP with arbitrary precedence graphs was studied. A mixed-integer linear programming (MILP) model, a CP model, and an evolutionary algorithm (EA) are proposed to solve large-scale problems. Moreover, in [3], the MRFJSSP with arbitrary precedence graphs was studied. A MILP model and a CP model are formulated and used for computational experiments. Again, the decisions in their formulation concern the allocation of resources to operations. Different resource release times are not accounted for. In [22], authors propose the MRFJSSP with Partially Necessary Resources (MRFJSSPwPNR) that explicitly divides operations into sub-operations according to the resources they require. The approach is though for industrial environments where sub-operations such as setup, processing, and cleanup can be clearly separated. These sub-operations are considered different explicit sequential operations.

In [5], a CP model is developed for the Multi-Mode Resource-Constrained Project Scheduling Problem (MMRCPSP), allowing the resolution of the FJSSP under the makespan minimization criterion for both linear and non-linear routes.

Table 1 summarizes the main extensions considered in the literature. We list some relevant references for each variant, the problem acronym, the solution approaches, the objective optimized, the multi-resource treatment, and the policy implemented. First, with respect to the Multi-resource treatment, in the MRFJSSP proposal, it is assumed that all resources are utilized throughout the entire operation time. In [12,23], the general formulation is solved, while ref. [24] considers tri-resource constraints of fixture-pallet-machine, and ref. [25] studies scenarios with reconfigurable machine tools assisted by auxiliary modules. In all studies, the simultaneous occupancy is assumed. In the DRCFJSP versions, it is explicitly indicated that the time workers are required to be in machines corresponds to the operation times, i.e., considering simultaneity. In [26] different workers using machines can be considered, while ref. [27] explicitly considers workers boredom to schedule their labors and ref. [28] consider workers experience. In the MRSSRFB, there is explicit resource blocking that implies longer occupation times. In this case, extended simultaneous occupancy is applied. In the MRPOFJSP, again, it is assumed that all resources are utilized for the full operating time; however, early resource release could have a greater impact, given the flexibility of task orders. Only simultaneous occupation has been considered. Lastly, in the MRFJSSPwPNR, operations are divided into subtasks to more precisely estimate occupation times. However, these subtasks are still treated under a simultaneous-occupation policy.

Table 1. Multi-resource and time occupation policy summary. MRFJSSP: Multiresource flexible job-shop scheduling problem, MRSSRFB: Multiresource shop scheduling with resource flexibility and blocking, DRCFJSP: Dual-resource constrained flexible job shop problem, MRPOFJSP: Multi-resource partial-ordering flexible job-shop scheduling problem, MRFJSSPwPNR: MRFJSSP with partially necessary resources. SO: Simultaneous occupation, E-SO: Explicit simultaneous occupation.

Ref.	Problem	Approaches	Objective	M-R Usage	Policy
[12]	MRFJSSP	Tabu search	makespan	No explicit	SO
[23]	MRFJSSP	A shuffled multi-swarm micro-migrating birds optimization	makespan	No explicit	SO
[24]	MRFJSSP	MILP and an Improved genetic algorithm hybrid with feasibility correction strategy and self-learning variable neighbourhood search	makespan	No explicit	SO
[25]	MRFJSSP	MILP and an adaptive collaborative evolution-based estimation of distribution algorithm	makespan and total cost	No explicit	SO
[26]	DRCFJSP	Quantum genetic algorithm	makespan	No explicit	SO
[27]	DRCFJSP	Bi-level lexicographic model and a multi-objective particle swarm optimization	completion of tasks and worker satisfaction	No explicit	SO
[28]	DRCFJSP	MILP and a migratory bird optimization algorithm	setup and processing times	No explicit	SO
[13]	MRSSRFB	Low-level shortest path, greedy heuristic, and high-level tabu search.	makespan	Explicit-block of resources	E-SO
[20]	MRPOFJSP	Answer set programming and two multi-shot solving strategies	average total tardiness	Partial order of operations	SO
[22]	MRFJSSPwPNR	MILP and a simulated annealing	mean tardiness of all jobs	Sub-tasks definition	SO

Research Motivation

As previously established, under the simultaneous occupation policy, the MRFJSSP assumes that an operation begins when all assigned necessary resources are available and then releases them simultaneously. This means that the processing time of an operation equals the longest processing time across all allocated resources. However, this is not always the practice case, where a resource (often human) might be released and start another operation before the operation is complete [4]. This is especially evident when a resource that can be released early is a bottleneck.

Thus, the motivation for this research study stems from relaxing the common assumption of simultaneous occupation proposed in the literature for more than two decades, but, to the best of our knowledge, has not yet been empirically studied. In the following, we assume that a job operation starts simultaneously on all assigned resources, even though these resources may not be occupied for the same timeframe. Indeed, we firmly believe that this scenario is more common in productive environments than its counterpart, which assumes simultaneous resource occupation.

Finally, we decided to approach this problem using the CP paradigm, given the method’s competitiveness in tackling a wide range of scheduling problems (see [29]) and its ability to provide lower bounds that can serve as a basis for future research.

3. A CP Formulation for the MRFJSSP with Early Resource Release

We split this section into four parts. First, in Section 3.1, we formally define the MRFJSSP adopting the nomenclature of the MMRCPSP. Second, in Section 3.2, we adapt a CP formulation for the FJSSP from [5] to tackle the MRFJSSP with early resource release, incorporating new constraints. Third, in Section 3.3, we describe the procedure used to represent an MRFJSSP instance from [12] under the assumption of early resource release. For clarity, we illustrate our description with a small example. Finally, for benchmark purposes, in Section 3.4, we adapt the formulation of Section 3.2 to tackle the MRFJSSP under the simultaneous occupation policy.

3.1. Problem Definition

We now proceed to define the MRFJSSP with linear precedence constraints using the nomenclature of the well-known MMRCPSP. Here, we can omit the definition of the job set, as representing the problem’s characteristics in terms of operations or tasks is sufficient.

Thus, the MRFJSSP can be described as follows: Let $T^{\prime }=\{1,\dots ,N\}$ be a set of the original tasks and $R=\{1,\dots ,NR\}$ be a set of renewable resources or machines. A renewable resource $r\in R$ (a resource with limited capacity per unit of time that becomes available again after use) can only perform one operation at a time, and pre-emption is not allowed. Each task $t\in T^{\prime }$ requires $R_t$ necessary resources to be performed. Then, for every $t\in T^{\prime }$, if $R_t>1$, we add $R_t-1$ dummy tasks that are grouped into the set $TaskId{s}_t$, so that $NT=N+{\sum }_{t\in T^{\prime }}(R_t-1)$ corresponds to the total number of tasks (originals and dummies) that must be scheduled, that is $T=\{1,\dots ,NT\}$. Dummy tasks are created to differentiate the use of resources for each task. The idea is that dummy tasks associated with each task begin processing simultaneously but can finish as soon as the need for the resource ends. For example, given $T_1$ an original task that involves the use of $R_1=3$ resources or machines. For this original task, $R_1-1=2$ dummy tasks will form the set $TaskId{s}_1$ and will be added to the set T to be scheduled by the model. Different times are associated with each of these resource utilization, even when they work together to perform a unique original task. The set of tasks T reflects the full set of tasks-resources and allows us to start all the use of resources at the same time, and also end tasks, and liberate resources as soon as they end.

Every task $t\in T$ must be performed in exactly one necessary resource $r\in R$ that belongs to one of the $m_t$ optional modes or available resources. The processing time of a scheduled task $t\in T$ in mode m is $p_{tm}$. In an MRFJSSP with linear precedence constraints, each task $t\in T^{\prime }$ has at most one (original task) predecessor and one (original task) successor. Finally, task $t\in T^{\prime }$ and its associated $TaskId{s}_t$ must start processing simultaneously on all assigned resources, sharing the same set of predecessors and successors. However, we relax the assumption that these assigned resources are occupied for the same duration. This is what we refer to as early resource release.

It is worth noting that in the classical MRFJSSP definition, which assumes n jobs with o operations per job, the total number of tasks (operations), N, is $N=n\times o$. However, in the more general case where the number of operations per job varies, N is given by $N={\sum }_{j\in J}{O}_j$, where $O_j$ represents the number of operations in job j.

3.2. CP Formulation

Now we proceed to describe the elements of the proposed CP model. This model extends an MMRCPSP formulation previously proposed by [5] for the FJSSP. This extension is necessary because the new problem requires modeling its multi-resource characteristics and enforcing the simultaneous start of the resources selected to perform a given task. Next, the nomenclature of the IBM ILOG CP Optimizer is used.

• Indices, Sets and Parameters:

• t: Index for tasks (operations);
• r: Index for renewable resources (machines);
• N: Number of original tasks;
• $NT$: Number of tasks (originals + dummies);
• $NR$: Number of renewable resources;
• $T^{\prime }$: Set of originals tasks $\left\{1,\dots ,N\right\}$;
• T: Set of tasks $\left\{1,\dots ,NT\right\}$;
• $TaskId{s}_t$: Set of dummy tasks associated with task t;
• R: Set of renewable resources $\left\{1,\dots ,NR\right\}$;
• $A_r$: Availability of the resource $r\in R$.

• Tuples:

• $Task$: A tuple with the following fields: {id: identifier of the task, succs: set of immediately successors of the task};
• $Mode$: A tuple with the following fields: {taskId: identifier of the task, id: identifier to determine the mode type, pt: processing time for the mode, $dm{d}_R$: number of resources that belong to the set R used per time period during the execution of the mode}.

• Tuplesets:

• $Tasks$: A tupleset that stores instances of the Task tuple;
• $Modes$: A tupleset that stores instances of the Mode tuple.

• Decision Variables:

• $tas{k}_{t\in Tasks}$: Interval variable between the start and the end of the Tasks t;
• $mod{e}_{m\in Modes}$: An optional interval variable of size $pt$ if $tas{k}_t$ is performed under $mod{e}_m$.

• Cumul function expression:

A cumul expression is a built-in function available in the Optimization Programming Language (OPL), part of the IBM ILOG CPLEX software package, that can be used to model and track cumulative resource consumption. This feature is handy in the context of the MRFJSSP, as it prevents overlapping tasks on a resource, since each machine can process only one task at a time. Thus, we define the following function:

$$Usag{e}_r=\sum _{minModes}pulse(mod{e}_m,m.dm{d}_r),m.dm{d}_r>0$$

where $Usag{e}_r$ tracks the cumulative consumption over time of the resource $r\in R$ through the sum of individual contributions of the optional interval variables $mod{e}_m$.

• Constraint Programming formulation:

$$\min \left\{\underset{t\in Tasks}{\max }\mathit{endOf}\left(tas{k}_t\right)\right\}$$

(1)

subject to:

$$\begin{array}{c}\hfill Usag{e}_r\le A_r\forall r\in R\end{array}$$

(2)

$$\begin{array}{c}\hfill \mathit{alternative}(tas{k}_t,mod{e}_m)\forall t\in Tasks,\forall m\in Modes,m.taskId=t.id\end{array}$$

(3)

$$\begin{array}{c}\hfill \mathit{endBeforeStart}(tas{k}_{t1},tas{k}_{t2})\forall t1\in Tasks,t2.id\in t1.succs\end{array}$$

(4)

$$\begin{array}{c}\hfill \mathit{startAtStart}(tas{k}_{t1},tas{k}_{t2})\forall t1\in T^{\prime },t_2\in TaskId{s}_{t1}\end{array}$$

(5)

The objective function in Equation (1) is used to minimize the makespan $C_{max}$, which is calculated by the expression $ma{x}_{t\in Tasks}$endOf($tas{k}_t$) that corresponds to the task that finishes last. Constraints in Equation (2) guarantee that renewable resource availability is respected. In the context of the MRFJSSP, this ensures that each renewable resource (machine) does not perform multiple tasks simultaneously (or equivalently, there is only one machine of each type r, that is, $A_r=1\forall r\in R$.). Constraints in Equation (3) ensure that each task is executed in exactly one mode. Equations (4) impose the precedence relationships between operations (route conditions). Finally, the constraints in Equation (5) ensure that every dummy task must start at the same time as its associated original task.

3.3. Instance Representation

In the following, we illustrate our procedure for building an instance data-file for the CP formulation proposed in Section 3.2. Specifically, we use the MRFJSSP instance named $mjs37$ from [12], which considers linear precedence constraints. This instance consists of $n=10$ jobs, $NR=20$ resources, and $o=10$ operations (original tasks) per job. For brevity, we focus our example only on the ten tasks belonging to Job 1 (see Figure 1).

Figure 1. A graphical representation of the precedence constraints for the operations of Job 1 in instance $mjs37$ with early resource release. Nodes with a subscript greater than or equal to 101 highlighted in light blue (e.g., $tas{k}_{101}$ and $tas{k}_{102}$) represent dummy activities added to model early resource release. The number in parentheses within each node represents the number of modes in which the activity can be executed.

As established in Section 3.1 and Section 3.2, to model the MRFJSSP with early resource release, we turn to dummy operations. In this context, the first operation of Job 1 ($t=1$) requires two resources ($R_1=2$). Therefore, we split it into two tasks (it is worth noting that when $R_t=1$ (for example, $t=4,5,6,9$ in Figure 1), $TaskId{s}_t=\left\{⌀\right\}$, and consequently, no dummy tasks are required for these operations.): $tas{k}_1$ (the original task) and $tas{k}_{101}$ (a dummy task) so that $TaskId{s}_1=\left\{tas{k}_{101}\right\}$. Note that we start listing the subscript of dummy tasks from $(n\times o)+1$ because, in the benchmark dataset, each job has the same number of operations. Additionally, $tas{k}_1$ and $tas{k}_{101}$ must start simultaneously, that is, $\mathit{startAtStart}(tas{k}_1,tas{k}_{101})$. In other words, the distance between the start of interval variable $tas{k}_1$ and the start of interval variable $tas{k}_{101}$ must be equal to zero. Lastly, $tas{k}_1$ and $tas{k}_{101}$ share the same set of predecessors (none, as they are the first operations of Job 1) and successors ($tas{k}_2$ and $tas{k}_{102}$).

Next, the $tas{k}_1$ must be performed by one of 8 possible resources ($m_1=8$) while $tas{k}_{101}$ must be performed by one of 5 possible resources ($m_{101}=5$) (the characteristic of resource flexibility in the problem, that is, $m_t\ge 1\forall t\in Tasks$). Specifically, for $tas{k}_1$ exactly one machine must be selected from the following set as the first resource: $R_{tas{k}_1}=\{17,20,13,15,1,3,4,7\}$ with the subsequent processing times (sorted in the same order as the machines): $p{t}_{tas{k}_1}=\{19,35,33,37,19,37,43,24\}$. Thus, $tas{k}_1$ can be processed on machine 17 with a processing time of 19, or on machine 20 with a processing time of 35, and so on. Analogously, for $tas{k}_{101}$ exactly one machine must be selected from the following set as the second resource: $R_{tas{k}_{101}}=\{2,20,8,10,5\}$ with the subsequent processing times (sorted in the same order as the machines): $p{t}_{tas{k}_{101}}=\{29,12,18,12,48\}$.

Note that the sets from which a resource can be selected to operate an original task and their associated dummy tasks are not necessarily disjoint (i.e., a resource may belong to several sets). In our example, machine 20 belongs to both set $R_{tas{k}_1}$ and $R_{tas{k}_{101}}$, that is, $R_{tas{k}_1}\cap R_{tas{k}_{101}}=\left\{20\right\}$, hence $R_{tas{k}_1}$ and $R_{tas{k}_{101}}$ are not disjoint sets. Recall that a resource cannot be assigned to more than one operation at the same time, which is enforced by constraint (2). Therefore, if machine 20 is selected to perform $tas{k}_1$, it cannot simultaneously be assigned to $tas{k}_{101}$ (and vice versa). It is also possible that machine 20 is not selected for either of these two tasks.

In Figure 1, the dashed arcs represent the precedence relations between the operations of Job 1. In this way, as soon as $tas{k}_1$ and $tas{k}_{101}$ are finished, $tas{k}_2$ (the second original operation) and $tas{k}_{102}$ (their respective dummy operation) can start their executions. Then, $tas{k}_2$ can be performed by one of the 4 ($m_2=4$) possible resources and $tas{k}_{102}$ can be performed by one of the 5 ($m_{102}=5$) resources. In fact, since $R_2=2$, it is only necessary to add one dummy task ($tas{k}_{102}$) related to the original task ($tas{k}_2$). Again, a resource r can process at most one operation at a time, because $A_r=1\forall r\in R$.

All in all, considering the example of Figure 1 with $TaskId{s}_1=\left\{tas{k}_{101}\right\}$, $TaskId{s}_2=\left\{tas{k}_{102}\right\}$, $TaskId{s}_3=\left\{tas{k}_{103}\right\}$, $TaskId{s}_7=\left\{tas{k}_{104}\right\}$, $TaskId{s}_8=\left\{tas{k}_{105}\right\}$, and $TaskId{s}_{10}=\left\{tas{k}_{106}\right\}$, the constraint set (5) produces the following conditions enforcing the simultaneous start of the operations in Job 1:

• $\mathit{startAtStart}(tas{k}_1,tas{k}_{101})$;
• $\mathit{startAtStart}(tas{k}_2,tas{k}_{102})$;
• $\mathit{startAtStart}(tas{k}_3,tas{k}_{103})$;
• $\mathit{startAtStart}(tas{k}_7,tas{k}_{104})$;
• $\mathit{startAtStart}(tas{k}_8,tas{k}_{105})$;
• $\mathit{startAtStart}(tas{k}_{10},tas{k}_{106})$.

3.4. CP Formulation for the MRFJSSP with Simultaneous Occupation

The CP formulation proposed in Section 3.2 for the MRFJSSP with early resource release can be adapted to tackle the problem under the simultaneous occupation policy. Although this particular variant of the problem is not the primary focus of the present work, we consider its incorporation necessary to test the behavior of the makespan under both scenarios (simultaneous occupancy and early release), in addition to providing a point of comparison for the results previously reported by [3] for the MRFJSSP with simultaneous occupation.

For this purpose, we detail below the necessary modifications to our original CP formulation, complementing its presentation with the support of the illustrative example presented in Figure 1:

• Additional Decision Variables:

• $au{x}_{t\in T^{\prime }}$: Auxiliary interval variable associated with the original $tas{k}_t$ if $TaskId{s}_t\ne \left\{⌀\right\}$;
• $lockO{R}_{t\in T^{\prime },m\in Modes}$: Optional interval variable associated with $au{x}_t$. $lockO{R}_{tm}$ indicates if $tas{k}_t$ is performed under $mod{e}_m$;
• $lockD{U}_{t\in T^{\prime }|TaskId{s}_t\ne \left\{⌀\right\},m\in Modes}$: Optional interval variables associated with $tas{k}_t$ and their respective elements in set $TaskId{s}_t$. $lockD{U}_{tm}$ indicates if $task{s}_t\in TaskId{s}_t$ is performed under $mod{e}_m$.

These additional decision variables allow us to model the full interval duration of each operation that requires two or more resources simultaneously, thereby keeping resources that are released early blocked until the completion of the process in the selected resource with the largest processing time.

Back to our example in Figure 1, an auxiliary variable $au{x}_t$ is added for each $tas{k}_t$ to the extent that $TaskId{s}_t\ne \left\{⌀\right\}$. Consequently we define $au{x}_1$, $au{x}_2$, $au{x}_3$, $au{x}_7$, $au{x}_8$, and $au{x}_{10}$ for the operations of Job 1 in instance mjs37. These auxiliary variables model the entire duration of the intervals associated with $tas{k}_1$–$tas{k}_{101}$ ($au{x}_1$), $tas{k}_2$–$tas{k}_{102}$ ($au{x}_2$), $tas{k}_3$–$tas{k}_{103}$ ($au{x}_3$), $tas{k}_7$–$tas{k}_{104}$ ($au{x}_7$), $tas{k}_8$–$tas{k}_{105}$ ($au{x}_8$), and $tas{k}_{10}$–$tas{k}_{106}$ ($au{x}_{10}$). Additionally, we added the optional interval variables $lockO{R}_{1m}$, $lockO{R}_{2m}$, $lockO{R}_{3m}$, $lockO{R}_{7m}$, $lockO{R}_{8m}$, and $lockO{R}_{10m}$, to represent the time interval that elapses between the end of $tas{k}_1$ and the end of $au{x}_1$ ($lockO{R}_1$), the end of $tas{k}_2$ and the end of $au{x}_2$ ($lockO{R}_2$), and so on. Finally, $lockD{U}_{tm}$ variables are defined to model the time that elapses between the completion of a dummy task and its respective auxiliary variable. Thus, $lockD{U}_{101m}$ represents the time interval between the completion of $tas{k}_{101}$ and the end of $au{x}_1$, $lockD{U}_{102m}$ for the time interval between the completion of $tas{k}_{102}$ and the end of $au{x}_2$, and so on.

• Additional Constraints:

Below, we present the additional sets of constraints that allow modeling the simultaneous occupation variant of the MRFJSSP.

$$\begin{array}{c}\hfill \mathit{span}(au{x}_t,all\{t\mid t\in TaskId{s}_t\})\forall t\in Tasks\mid TaskId{s}_t\ne \left\{⌀\right\}\end{array}$$

(6)

Constraint (6) imposes that the auxiliary variable $au{x}_t$ spans all present intervals in the set of dummy tasks that belong to the non-empty set $TaskId{s}_t$. Hence, at least one of these intervals must support the start (respectively, the end) of $au{x}_t$. In the example of Figure 1, for $t=1$, $au{x}_1$ spans $tas{k}_1$ and $tas{k}_{101}$, i.e., $span(au{x}_1,tas{k}_1,tas{k}_{101})$.

$$\begin{array}{c}\hfill \mathit{startAtEnd}(lockO{R}_{tm},tas{k}_t)\forall t\in Tasks\mid TaskId{s}_t\ne \left\{⌀\right\},\forall m\in Modes\end{array}$$

(7)

$$\begin{array}{c}\hfill \mathit{endAtEnd}(lockO{R}_{tm},au{x}_t)\forall t\in Tasks\mid TaskId{s}_t\ne \left\{⌀\right\},\forall m\in Modes\end{array}$$

(8)

Constraint (7) requires that the start time of $lockO{R}_{tm}$ matches the end time of their related original $tas{k}_t$ (e.g., $startAtEnd(lockO{R}_{1m},tas{k}_1)$ and $startAtEnd(lockO{R}_{2m},tas{k}_2)$). Additionally, Constraint (8) requires that $lockO{R}_{tm}$ and $au{x}_t$ end at the same time (e.g., $endAtEnd(lockO{R}_{1m},au{x}_1)$ and $endAtEnd(lockO{R}_{2m},au{x}_2)$).

$$\begin{array}{c}\hfill \mathit{startAtEnd}(lockD{U}_{tm},tas{k}_t)\forall t\in Tasks\mid TaskId{s}_t\ne \left\{⌀\right\},\forall m\in Modes\end{array}$$

(9)

$$\begin{array}{c}\hfill \mathit{endAtEnd}(lockD{U}_{tm},au{x}_t)\forall t\in Tasks\mid TaskId{s}_t\ne \left\{⌀\right\},\forall m\in Modes\end{array}$$

(10)

Similarly, Constraint (9) enforces that $lockD{U}_{tm}$ begins immediately after the dummy $tas{k}_t$ ends (i.e., $startAtEnd(lockD{U}_{101m},tas{k}_{101})$), while Constraint (10) ensures that $lockD{U}_{tm}$ and $au{x}_t$ conclude simultaneously (i.e., $endAtEnd(lockD{U}_{101m},au{x}_1)$).

In Figure 2, we illustrate how these constraints interact for an excerpt of the schedule of Job 1 in the $mjs37$ instance shown in Figure 1. For example, assume that $tas{k}_1$ begins on machine 17 at time zero (using mode 1), with a processing time of 19 time units. Simultaneously, $tas{k}_{101}$ starts processing on machine 10 (due to Constraint (5)) using mode 4, with a processing time of 12 time units. In this case, the interval variable $lockD{U}_{101}$ is active since the completion of $tas{k}_{101}$ (Constraint (9)) until the completion of $au{x}_1$ (Constraint (10)), effectively locking machine 10 during these 7 time units. For the same example, if the selected modes for $tas{k}_1$ and $tas{k}_{101}$ complete simultaneously, both $lockO{R}_1$ and $lockD{U}_{101}$ have zero duration. Conversely, if $tas{k}_1$ completes before $tas{k}_{101}$, $lockO{R}_1$ is activated, and the interval representing $lockD{U}_{101}$ has zero length.

$$\begin{array}{cc}\hfill \mathit{presenceOf}\left(mod{e}_{m\mid m.taskId=t}\right)=& \mathit{presenceOf}\left(lockO{R}_{tm}\right)\hfill \\ \hfill & \forall t\in Tasks\mid TaskId{s}_t\ne \left\{⌀\right\},\forall m\in Modes\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathit{presenceOf}\left(mod{e}_{m\mid m.taskId=t}\right)=& \mathit{presenceOf}\left(lockD{U}_{tm}\right)\hfill \\ \hfill & \forall t\in Tasks\mid TaskId{s}_t\ne \left\{⌀\right\},\forall m\in Modes\hfill \end{array}$$

(12)

Figure 2. An example illustrating the activation of the optional interval variable $lockD{U}_{101}$, which occurs because $tas{k}_{101}$ completes before $tas{k}_1$. Observe that, in this case, the interval variable $lockO{R}_1$ has zero duration.

Constraint (11) ensures that the interval variable $lockO{R}_{tm}$ uses the same mode m (and hence the same resource) as the one selected for the original $tas{k}_t$. Similarly, Constraint (12) ensures that $lockD{U}_{tm}$ is assigned the same mode m as its corresponding dummy $tas{k}_t$.

• Cumul Function Expression:

Finally, the definition of the cumul function expression from Section 3.2 must be updated to account for machine usage during the execution of the optional interval variables $lockO{R}_{tm}$ and $lockD{U}_{tm}$ when they have positive values. This effect is illustrated by the shaded bars in the Gantt Chart shown in Figure 3.

$$\begin{array}{cc}\hfill Usag{e}_r=\sum _{m\in Modes\mid m.rsrcId=r}pulse(mod{e}_m,1)& \\ \hfill & +pulse(lockO{R}_{tm},1)+pulse(lockD{U}_{tm},1)\hfill \end{array}$$

Figure 3. A Gantt Chart for an optimal solution of instance $mjs37$ with “simultaneous occupation” and $C_{max}=202$. The shaded boxes represent time intervals during which a machine cannot be occupied until another resource, selected to perform the job operation, has completed its task.

4. Computational Results

This section provides and examines the results of the computational experiments conducted in this paper.

4.1. Experimental Setup

All experiments were performed on an AMD Ryzen 7-7730U CPU at 2.0 GHz with 8 cores (16 workers in parallel) and 16 GB of RAM, running at 3200 MHz. IBM ILOG CP Optimizer 22.1.1 was used to solve the CP formulations presented in Section 3.2 and Section 3.4 under the default Auto-search strategy, that is, the decision of choosing the search type for a worker is automatically made by the solver.

Problem instances are well-known case studies from the literature. In all these cases, the time each resource is required for each operation is explicitly defined. We have considered that these times are required at the start of the operation, and hence, resources can be released once the time has passed.

Each instance is solved once. Finally, we established a time limit of three hours (10,800 s) to solve each instance, the same as [3]. These results are presented in Section 4.2. Nevertheless, in Section 4.3, we analyze the performance of our CP approach for different time limits, in order to evaluate whether significant improvements in results are achieved as computational time increases. Finally, in Section 4.4 we implemented the method proposed by [7] to evaluate whether it is possible to increase the lower bound values obtained in our experiments and thereby obtain a more accurate estimate of the optimality gap.

4.2. Results

Table 2 summarizes the numerical results for 65 instances of the MRFJSSP with linear precedence constraints (it is worth noting that the original set comprised 70 instances. However, five of them ($mjs00$, $mjs55$, $mjs57$, $mjs58$, and $mjs69$) were excluded from the experiments due to data consistency issues. See [3].). These test instances can be obtained from the work of [12]. The first multicolumn contains the characteristics of each instance: name, number of original operations, number of resources, the maximum number of resources required per operation (MR), the maximum number of possible resources that can be selected as a required resource (MF), and whether the sets of resources are disjoint or not. The second multicolumn provides the results of [3] under the assumption of simultaneous occupation and includes the lower bound (LB), the $C_{max}$ (or equivalently, the upper bound (UB)), the optimality gap ($Gap={\displaystyle \frac{UB-LB}{UB}}$), and the total computational time in seconds. The third multicolumn presents the same information as the previous one, but for our implementation of the MRFJSSP under simultaneous occupation, as formulated in Section 3.4. Finally, the last multicolumn presents the results of the CP formulation described in Section 3.2, which assumes early resource release. All instances have 10 jobs, and the number of operations is divided equally among these jobs (i.e., 10 operations per job for instances with 100 operations and 15 operations per job for instances with 150 operations). Finally, the optimal values of $C_{max}$ are highlighted in bold, whereas the new valid best lower bounds and upper bounds for the MRFJSSP with simultaneous occupation are highlighted in red and blue, respectively.

Table 2. Results on the MRFJSSP benchmark set of [12] with linear precedence constraints for both constraint policies: simultaneous occupation (SO) and early resource release (ERR).

Name	Instance Characteristics					MRFJSSP with SO Reported in [3]				Our Implementation of MRFJSSP with SO				MRFJSSP with Early Resource Release
	Oper.	Res.	MR	MF	Disj.	LB	$C_{\mathit{max}}$	Gap %	Time [s]	LB	$C_{\mathit{max}}$	Gap %	Time [s]	LB	$C_{\mathit{max}}$	Gap %	Time [s]
mjs01	100	20	2	2	Yes	361	361	0.00	17	361	361	0.00	5.82	347	347$\downarrow$(3.88%)	0.00	1.16
mjs02	100	20	2	2	Yes	381	381	0.00	25	381	381	0.00	11.77	360	360$\downarrow$(5.51%)	0.00	1.81
mjs03	100	20	2	2	Yes	376	376	0.00	51	376	376	0.00	12.26	355	355$\downarrow$(5.59%)	0.00	2.97
mjs04	100	20	2	2	Yes	391	391	0.00	34	391	391	0.00	10.28	355	355$\downarrow$(9.21%)	0.00	2.12
mjs05	150	20	2	2	Yes	623	623	0.00	659	623	623	0.00	140.45	595	595$\downarrow$(4.49%)	0.00	8.83
mjs06	150	20	2	2	Yes	547	547	0.00	1171	547	547	0.00	717.59	515	515$\downarrow$(5.85%)	0.00	13.02
mjs07	150	20	2	2	Yes	610	610	0.00	10,553	610	610	0.00	3910.32	563	563$\downarrow$(7.70%)	0.00	88.62
mjs08	150	20	2	2	Yes	552	552	0.00	1424	552	552	0.00	479.01	518	518$\downarrow$(6.16%)	0.00	22.60
mjs09	150	20	2	2	Yes	563	563	0.00	104	563	563	0.00	36.18	540	540$\downarrow$(4.09%)	0.00	4.98
mjs10	100	20	5	2	No	444	828	46.38	10,800	453$\uparrow$(2.03%)	818$\downarrow$(1.21%)	44.62	10,800	453$\uparrow$(2.03%)	619	26.82	10,800
mjs11	100	20	5	2	No	487	901	45.95	10,800	501$\uparrow$(2.87%)	892$\downarrow$(1.00%)	43.83	10,800	494	683	27.67	10,800
mjs12	100	20	5	2	No	446	790	43.54	10,800	448$\uparrow$(0.45%)	781$\downarrow$(1.14%)	42.64	10,800	447	615	27.32	10,800
mjs13	100	20	5	2	No	434	791	45.13	10,800	433	801	45.94	10,800	432	593	27.15	10,800
mjs14	100	20	5	2	No	552	910	39.34	10,800	552	887$\downarrow$(2.53%)	37.77	10,800	533	664	19.73	10,800
mjs15	150	20	5	2	No	655	1292	49.30	10,800	654	1291$\downarrow$(0.08%)	49.34	10,800	659$\uparrow$(0.61%)	972	32.20	10,800
mjs16	150	20	5	2	No	581	1198	51.50	10,800	581	1174$\downarrow$(2.00%)	50.51	10,800	575	927	37.97	10,800
mjs17	150	20	5	2	No	647	1407	54.02	10,800	657$\uparrow$(1.55%)	1355$\downarrow$(3.70%)	51.51	10,800	645	979	34.12	10,800
mjs18	150	20	5	2	No	668	1396	52.15	10,800	664	1374$\downarrow$(1.58%)	51.67	10,800	668	1005	33.53	10,800
mjs19	150	20	5	2	No	674	1321	48.98	10,800	674	1307$\downarrow$(1.06%)	48.43	10,800	663	1015	34.68	10,800
mjs20	100	20	5	2	Yes	500	902	44.57	10,800	500	912	45.18	10,800	491	685	28.32	10,800
mjs21	100	20	5	2	Yes	438	836	47.61	10,800	438	832$\downarrow$(0.48%)	47.36	10,800	436	599	27.21	10,800
mjs22	100	20	5	2	Yes	467	865	46.01	10,800	474$\uparrow$(1.50%)	866	45.27	10,800	460	632	27.22	10,800
mjs23	100	20	5	2	Yes	475	849	44.05	10,800	477$\uparrow$(0.42%)	877	45.61	10,800	463	662	30.06	10,800
mjs24	100	20	5	2	Yes	419	790	46.96	10,800	420	819	48.72	10,800	422$\uparrow$(0.72%)	613	31.16	10,800
mjs25	150	20	5	2	Yes	653	1315	50.34	10,800	653	1335	51.09	10,800	653	1036	36.97	10,800
mjs26	150	20	5	2	Yes	620	1203	48.46	10,800	623	1190$\downarrow$(1.08%)	47.65	10,800	625$\uparrow$(0.81%)	902	30.71	10,800
mjs27	150	20	5	2	Yes	633	1327	52.30	10,800	637$\uparrow$(0.63%)	1306$\downarrow$(1.58%)	51.23	10,800	637$\uparrow$(0.63%)	979	34.93	10,800
mjs28	150	20	5	2	Yes	610	1325	53.96	10,800	606	1268$\downarrow$(4.30%)	52.21	10,800	610	907	32.75	10,800
mjs29	150	20	5	2	Yes	690	1215	43.21	10,800	690	1195$\downarrow$(1.65%)	42.26	10,800	677	892	24.10	10,800
mjs30	100	20	2	10	No	216	216	0.00	50	216	216	0.00	48.20	216	216	0.00	2.07
mjs31	100	20	2	10	No	218	218	0.00	18	218	218	0.00	23.83	218	218	0.00	1.81
mjs32	100	20	2	10	No	216	216	0.00	429	216	216	0.00	287.28	210	210$\downarrow$(2.78%)	0.00	7.36
mjs33	100	20	2	10	No	217	217	0.00	99	217	217	0.00	80.12	214	214$\downarrow$(1.38%)	0.00	2.89
mjs34	100	20	2	10	No	213	213	0.00	8	213	213	0.00	6.56	213	213	0.00	1.67
mjs35	100	20	2	10	No	265	265	0.00	4	265	265	0.00	2.64	265	265	0.00	0.70
mjs36	100	20	2	10	No	223	223	0.00	27	223	223	0.00	29.46	220	220$\downarrow$(1.35%)	0.00	2.17
mjs37	100	20	2	10	No	202	202	0.00	64	202	202	0.00	82.02	197	197$\downarrow$(2.48%)	0.00	7.08
mjs38	100	20	2	10	No	241	241	0.00	5	241	241	0.00	7.40	241	241	0.00	1.13
mjs39	100	20	2	10	No	210	210	0.00	56	210	210	0.00	19.21	210	210	0.00	1.65
mjs40	100	20	2	10	Yes	241	241	0.00	3	241	241	0.00	1.54	241	241	0.00	0.86
mjs41	100	20	2	10	Yes	210	210	0.00	680	210	210	0.00	1957.41	205	205$\downarrow$(2.38%)	0.00	6.99
mjs42	100	20	2	10	Yes	250	250	0.00	2	250	250	0.00	2.47	250	250	0.00	0.85
mjs43	100	20	2	10	Yes	219	219	0.00	6	219	219	0.00	2.62	219	219	0.00	0.77
mjs44	100	20	2	10	Yes	252	252	0.00	8	252	252	0.00	7.40	252	252	0.00	0.84
mjs45	150	20	2	10	Yes	294	294	0.00	73	294	294	0.00	103.07	294	294	0.00	3.34
mjs46	150	20	2	10	Yes	296	296	0.00	556	296	296	0.00	270.97	295	295$\downarrow$(0.34%)	0.00	6.64
mjs47	150	20	2	10	Yes	330	330	0.00	109	330	330	0.00	81.18	330	330	0.00	3.18
mjs48	150	20	2	10	Yes	315	315	0.00	164	315	315	0.00	256.23	311	311$\downarrow$(1.27%)	0.00	32.56
mjs49	150	20	2	10	Yes	356	356	0.00	8	356	356	0.00	10.36	351	351$\downarrow$(1.40%)	0.00	2.55
mjs50	100	30	5	10	No	279	326	14.42	10,800	279	324$\downarrow$(0.61%)	13.89	10,800	279	279	0.00	24.41
mjs51	100	30	5	10	No	289	367	21.25	10,800	291$\uparrow$(0.69%)	367	20.71	10,800	284	284	0.00	210.74
mjs52	100	30	5	10	No	286	317	9.78	10,800	286	310$\downarrow$(2.21%)	7.74	10,800	286	286	0.00	7.71
mjs53	100	30	5	10	No	267	353	24.36	10,800	268$\uparrow$(0.37%)	341$\downarrow$(3.40%)	21.41	10,800	265	265	0.00	55.53
mjs54	100	30	5	10	No	241	299	19.40	10,800	240	307	21.82	10,800	245$\uparrow$(1.66%)	245	0.00	866.59
mjs56	150	30	5	10	No	380	534	28.84	10,800	384	484$\downarrow$(9.36%)	20.66	10,800	387$\uparrow$(1.84%)	387	0.00	7100.14
mjs59	150	30	5	10	No	346	476	27.31	10,800	346	465$\downarrow$(2.31%)	25.59	10,800	347$\uparrow$(0.29%)	363	4.41	10,800
mjs60	100	50	5	10	Yes	246	246	0.00	301	246	246	0.00	409.40	246	246	0.00	7.30
mjs61	100	50	5	10	Yes	301	301	0.00	101	301	301	0.00	12.08	301	301	0.00	0.19
mjs62	100	50	5	10	Yes	284	284	0.00	63	284	284	0.00	13.66	284	284	0.00	3.09
mjs63	100	50	5	10	Yes	286	286	0.00	44	286	286	0.00	138.29	286	286	0.00	3.54
mjs64	100	50	5	10	Yes	240	240	0.00	61	240	240	0.00	174.69	239	239$\downarrow$(0.42%)	0.00	2.68
mjs65	150	50	5	10	Yes	375	375	0.00	357	375	375	0.00	646.02	375	375	0.00	9.01
mjs66	150	50	5	10	Yes	423	423	0.00	796	423	423	0.00	74.23	423	423	0.00	6.93
mjs67	150	50	5	10	Yes	400	400	0.00	1000	400	400	0.00	1399.92	399	399$\downarrow$(0.25%)	0.00	9.55
mjs68	150	50	5	10	Yes	382	382	0.00	1189	382	382	0.00	1008.19	381	381$\downarrow$(0.26%)	0.00	8.83

Before analyzing the results in detail, it is essential to note that our findings for the MRFJSSP with early resource release cannot be directly compared with those reported by [3]. This is not only because different software and hardware conditions were used (and, importantly, the CP formulations differ), but also because different assumptions were made about resource release. However, since the MRFJSSP with early resource release is a more relaxed version of the same problem under “simultaneous occupation”, any LB obtained in our experiments is also valid for this latter case. Additionally, we find it helpful to compare the results between the two scenarios to understand how much the makespan can be reduced by relaxing the simultaneous occupation assumption. That said, we now proceed to analyze the results, together with discussing some related managerial insights:

• It appears that the most challenging instances in the simultaneous occupation scenario remain difficult even under the assumption of early resource release. However, in our experiments (last set of columns), we successfully solved instances ranging from $mjs50$ to $mjs54$ and $mjs56$ to optimality.
• For the 38 instances where both approaches reach an optimal solution ($mjs01$ to $mjs09$, $mjs30$ to $mjs49$, and $mjs60$ to $mjs68$), the average $C_{max}$ was reduced by 1.76%. Notably, the optimal value is identical in 18 of these instances.
• The reduction in $C_{max}$ was more significant for instances $mjs01$ to $mjs09$, averaging 5.83%. In contrast, instances $mjs60$ to $mjs68$ showed the smallest reduction in $C_{max}$, averaging just 0.10%. In light of the results, it appears that as the maximum number of possible resources that can be selected as required resources increases (e.g., $MF=10$), the percentage difference in $C_{max}$ between the two approaches decreases. In our view, this represents an important managerial insight, as increasing the scheduling possibilities of an operation should lead to smaller differences (in terms of the achieved makespan) between the two approaches. Similarly, as flexibility decreases, early resource release is expected to have a greater effect on reducing the schedule makespan.
• For the fifteen instances in which we provided new lower bounds under the “simultaneous occupation” policy (here indicated in red), the average increase in the lower bound was approximately 1.10%.
• Our implementation of the MRFJSSP with simultaneous occupation proves competitive with the results reported in [3], contributing 9 new LB and 19 new UB to the literature, with an average makespan reduction of 2.17% in these instances.
• The number of operations per instance ($Oper$ equal to 100 or 150) does not seem to have a major influence on the results. Moreover, the characteristics of the instances, specifically whether the sets of necessary resources are disjoint, do not appear to have a significant influence either.
• The hardest instances in terms of the average optimality gap (from $mjs10$ to $mjs29$) share three common characteristics: the number of resources ($Res=20$), the maximum number of resources required per operation ($MR=5$), and the maximum number of possible resources that can be selected as a required resource ($MF=2$). However, in our experiments, we solved all instances with $MF=10$ to optimality, except for $mjs59$, which, in any case, has a relatively low optimality gap. This finding suggests that increasing the $Res$ and/or $MF$ does not necessarily imply that the instance becomes more complex. Furthermore, a low quotient q between $Res$ and $MR$ appears to be a reasonable predictor of the hardness of an MRFJSSP instance ($q={\displaystyle \frac{Res}{MR}}=4$ or 6). We believe that this last result is also relevant for decision-making, for example, having a low q ratio should be a strong incentive both for the early release of resources (as an operational policy that aims to decrease the length of the schedule) and for seeking the incorporation of new multipurpose renewable resources that can be used in alternative ways to process the operations of the works.

Finally, we are interested in examining the Gantt chart for two optimal solutions (one for each scenario: simultaneous occupation and early resource release) for the instance previously used as an example in Figure 1. In this context, Figure 3 provides a Gantt chart for an optimal solution of instance $mjs37$ with simultaneous occupation constraints and $C_{max}=202$. The shaded boxes represent time intervals during which a resource (machine) cannot be occupied until another resource, selected to perform the job operation, finishes. For example, the first operation of Job 1 ($J_1$) (see Figure 1), needs two resources: $tas{k}_1$ and $tas{k}_{101}$. The $tas{k}_1$ operation is performed in mode 1 (on machine 17 with a processing time of 19), while the $tas{k}_{101}$ operation is performed in mode 4 (on machine 10 with a processing time of 12). Both operations start at time zero, but machine 10 cannot be occupied immediately after completing $tas{k}_{101}$ (it must wait until $tas{k}_1$ is finished). See Figure 2.

In contrast, Figure 4 provides a Gantt chart for an optimal solution of instance $mjs37$ with early resource release constraints and $C_{max}=197$. It can be seen that $tas{k}_1$ and $tas{k}_{101}$ are performed in the same modes as in Figure 3, and, as before, both operations start simultaneously at time zero. However, as soon as machine 10 finishes $tas{k}_{101}$, the resource is released and can begin a new operation (in this case, the first operation of Job 5).

Figure 4. A Gantt Chart for an optimal solution of instance $mjs37$ with early resource release constraints and $C_{max}=197$. Here, a job operation begins simultaneously on all allocated resources, although these resources may not be occupied for the same duration.

4.3. Sensitivity Analysis for Different Time Limits

According to the analysis in Section 4, 21 out of 65 instances could not be solved to optimality within the 10,800-s time limit, with an average optimality gap of 29.00%. In this section, we analyze the impact of extending the time limit to 21,600 and 32,400 s, which correspond to twice and three times the original limit, respectively. This sensitivity analysis is motivated primarily by the optimality gaps observed in some instances, which may initially appear relatively large. The results of these experiments are detailed in Table 3. In bold, we highlight the changes in both lower and upper bounds for the makespan.

Table 3. Results on the MRFJSSP benchmark set of [12] with linear precedence constraints and early resource release for different time limits. The makespans of instances for which increasing the computational time led to improved values are highlighted in bold.

Name	Instance Characteristics					t = 10,800 [s]			t = 21,600 [s]			t = 32,400 [s]
	Oper.	Res.	MR	MF	Disj.	LB	$C_{\mathit{max}}$	Gap %	LB	$C_{\mathit{max}}$	Gap %	LB	$C_{\mathit{max}}$	Gap %
mjs10	100	20	5	2	No	453	619	26.82	453	619	26.82	453	619	26.82
mjs11	100	20	5	2	No	494	683	27.67	494	683	27.67	494	683	27.67
mjs12	100	20	5	2	No	447	615	27.32	447	608	26.48	447	608	26.48
mjs13	100	20	5	2	No	432	593	27.15	432	593	27.15	432	587	26.41
mjs14	100	20	5	2	No	533	664	19.73	533	663	19.61	533	663	19.61
mjs15	150	20	5	2	No	659	972	32.20	659	972	32.20	659	972	32.20
mjs16	150	20	5	2	No	575	927	37.97	575	927	37.97	575	925	37.84
mjs17	150	20	5	2	No	645	979	34.12	645	979	34.12	645	979	34.12
mjs18	150	20	5	2	No	668	1005	33.53	668	1005	33.53	668	1005	33.53
mjs19	150	20	5	2	No	663	1015	34.68	663	1015	34.68	663	1015	34.68
mjs20	100	20	5	2	Yes	491	685	28.32	491	685	28.32	491	684	28.22
mjs21	100	20	5	2	Yes	436	599	27.21	436	583	25.21	436	583	25.21
mjs22	100	20	5	2	Yes	460	632	27.22	460	625	26.40	460	625	26.40
mjs23	100	20	5	2	Yes	463	662	30.06	463	662	30.06	463	662	30.06
mjs24	100	20	5	2	Yes	422	613	31.16	422	613	31.16	422	613	31.16
mjs25	150	20	5	2	Yes	653	1036	36.97	653	1036	36.97	653	1030	36.60
mjs26	150	20	5	2	Yes	625	902	30.71	625	902	30.71	625	902	30.71
mjs27	150	20	5	2	Yes	637	979	34.93	637	979	34.93	637	979	34.93
mjs28	150	20	5	2	Yes	610	907	32.75	610	907	32.75	610	907	32.75
mjs29	150	20	5	2	Yes	677	892	24.10	677	887	23.68	677	887	23.68
mjs59	150	30	5	10	No	347	363	4.41	347	361	3.88	347	360	3.61

In this context, the improvements resulting from increased computational time are modest. The lower bounds remain unchanged, and the average optimality gap is reduced to 28.78% and 28.70% when the time limit is extended to 21,600 s and 32,400 s, respectively. A reduction in the upper bound was achieved in only 10 instances. These results suggest that further improvements in upper bounds are likely to be relatively marginal, and that the most significant opportunity for reducing the optimality gap lies in improving the current lower bounds. In this regard, a lower-bound augmentation procedure such as the one proposed by [7] for the JSSP under makespan minimization could represent a promising direction for future work.

4.4. Lower Bound Analysis

In this section, we consider the 21 instances of the MRFJSSP with early resource release for which, according to Table 2, optimality could not be demonstrated within the execution time limit. The average optimality gap for these instances is 29.00%. Notably, as shown in Section 4.3, allocating additional computational time yields only modest improvements, reducing the average gap to 28.70% when the time limit is tripled. Consequently, it is reasonable to assume that further reductions in the optimality gap for these instances primarily require increasing the current lower bound. To this end, we implemented the method proposed by [7] for the JSSP. This method consists of two phases: in the first phase, a relaxation of the original problem is solved; in the second phase, this relaxation is iteratively tightened until a time limit is reached or no further improvements are obtained. For these computational tests, we allocated 10,800 s per instance. The results are reported in Table 4.

Table 4. Results of the lower-bounding procedure proposed by [7] for the 21 MRFJSSP instances with early resource release for which, according to Table 2, optimality could not be demonstrated.

Name	LB	${\mathit{C}}_{\mathbf{max}}$	GAP %	LB Phase 2	GAP %
mjs10	453	619	26.82	561	9.37
mjs11	494	683	27.67	572	16.25
mjs12	447	615	27.32	526	14.47
mjs13	432	593	27.15	527	11.13
mjs14	533	664	19.73	610	8.13
mjs15	659	972	32.2	744	23.46
mjs16	575	927	37.97	643	30.64
mjs17	645	979	34.12	684	30.13
mjs18	668	1005	33.53	744	25.97
mjs19	663	1015	34.68	735	27.59
mjs20	491	685	28.32	586	14.45
mjs21	436	599	27.21	520	13.19
mjs22	460	632	27.22	534	15.51
mjs23	463	662	30.06	534	19.34
mjs24	422	613	31.16	507	17.29
mjs25	653	1036	36.97	716	30.89
mjs26	625	902	30.71	706	21.73
mjs27	637	979	34.93	718	26.66
mjs28	610	907	32.75	670	26.13
mjs29	677	892	24.1	752	15.70
mjs59	347	363	4.41	351	3.31
AVG			29.00		19.11

According to Table 4, the lower bound increased significantly for all evaluated instances. Based on these results, the effective optimality gap for this subset of instances is reduced to 19.11%. Naturally, these results can still be improved as computational time increases. This indicates that the makespan achieved in our experiments is competitive and that the primary challenge in reducing the optimality gap is to increase the lower bound.

5. Conclusions

In this paper, we propose a constraint programming (CP) formulation for the multi-mode resource-constrained project scheduling problem (MMRCPSP) to address the multiresource flexible job-shop scheduling problem (MRFJSSP), in which the operations of a given job follow a linear precedence order. Our focus is on relaxing the simultaneous occupation policy, which assumes that a job operation starts only when all its assigned resources are available and releases them simultaneously. Instead, we assume that a job operation begins simultaneously across all allocated resources, even if these resources may not be occupied for the same duration. The results validated our CP approach, as we achieved optimality in 44 of 65 instances, with an average relative optimality gap of 9.37%.

Additionally, we extended our formulation to the case of simultaneous resource occupation to compare the results with those previously reported by [3] for this scenario. In this context, we observed that there are significant incentives to implement the early resource release policy, especially in industrial applications where flexibility is relatively low, that is, when there is a relatively small number of renewable resources available to process the operations of a job.

Several extensions of this study could be explored in future research, motivated by the limitations of this article. For example, the MRFJSSP with early resource release and arbitrary precedence graphs is a natural extension for future study. Additionally, it is worth studying the standard extension of replicating identical machines in parallel to increase overall production capacity, balance stage capacities, and/or reduce the impact of bottlenecks on shop floor operations (see [30]). Furthermore, extending the computational experiments to include metaheuristic approaches represents a promising avenue for future research, given their demonstrated competitiveness in solving various scheduling problems. Naturally, these extensions are not exclusive and can be studied together.