IAES Inter national J our nal of Articial Intelligence (IJ-AI) V ol. 15, No. 2, April 2026, pp. 1261 ∼ 1274 ISSN: 2252-8938, DOI: 10.11591/ijai.v15.i2.pp1261-1274 ❒ 1261 Genetic algorithm f or generalized time-windo w assignment pr oblem Ali Kansou 1 , Bilal Kanso 1 , Houssein W ehbe 1,2,3 , Haydar Bazzi 1 , Ali Mcheik 1 1 Department of Computer Science, F aculty of Sciences, Lebanese Uni v ersity , Beirut, Lebanon 2 School of Arts and Sciences, Lebanese International Uni v ersity , Nabatieh, Lebanon 3 Colle ge of Arts and Sciences, American Uni v ersity of Iraq-Baghdad, Baghdad, Iraq Article Inf o Article history: Recei v ed Aug 1, 2025 Re vised Jan 3, 2026 Accepted Jan 22, 2026 K eyw ords: Assignment with time windo ws Constructi v e heuristic Genetic algorithms Local search procedure Multi-resource scheduling ABSTRA CT This paper presents a h ybrid genetic algorithm (GA) for the generalized time-windo w assignment pr oblem (GTW AP), a comple x articial intelligence (AI) scheduling challenge that in v olv es assigning agents to resources under strict temporal and capacity constraints . Our method inte grates a problem- specic heuristics a nd a repair mechanism to generate feasible and high- quality solutions. W e pro vide a mathematica l formulation for GTW AP and introduce a ne w public benchmark set, using CPLEX to obtain e xact solutions. Computational e xperiments demonstrate that our GA is highly competiti v e with CPLEX, often matching its performance. This ef fecti v eness mak es our method a practical and scalable AI-dri v en tool for comple x scheduling in domains lik e logistics and healthcare. This is an open access article under the CC BY -SA license . Corresponding A uthor: Bilal Kanso Department of Computer Science, F aculty of Sciences, Lebanese Uni v ersity Beirut, Lebanon Email: bilal kanso@hotmail.com 1. INTR ODUCTION Assignment and scheduling problems ha v e become increasingly comple x in recent years due to the introduction of temporal constraints, posing signicant challenges for articial intelligence (AI) and operations research. A common challenge in man y real-w orld appli cations is matching agents (e.g., patients, w ork ers, or v oters) to appropriate f acilities (e.g., medical centers, job sites, and polling stations) while respecting f acility capacities and operational time constraints. T raditional assignment models, such as the f acility location problem or the classical linear assignment problem, often f ail to capture the k e y aspects such as agents’ temporal a v ailability , spatial dispersion, and resource heterogeneity . T o bridge this g ap, we propose the generalized time-windo w assignment problem (GTW AP), a e xible and unied assignment frame w ork capable of representing a wide range of real-w orld situations. Unlik e domain-specic approaches, such as staf f scheduling or v ehicle routing with tim e windo ws, GTW AP inte grates multiple f actors, including eligibility assignment rules, f aci lity capacities and time windo w constraints, into a single model. This allo ws the de v elopment of scalable and ef cient algorithms that can be used in di v erse elds such as public administration, healthcare, maintenance, and w orkforce management. In GTW AP , f acilities (e.g., polling stations, v accination centers, or w orkstations) operate under stri ct capacity limits and specic opening hours, each with dif ferent service durations. Agents, in turn, ha v e their o wn a v ailability and preferences re g arding which f acilities the y can access. The objecti v e is to assign agents to suitable f acilities while satisfying all constraints, making GTW AP a po werful tool for solving man y assignment J ournal homepage: http://ijai.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
1262 ❒ ISSN: 2252-8938 problems used in dif ferent areas (healthcare, logistics, education, and w orkforce coordination). T o address GTW AP , genetic algorithm (GA) that goes be yond standard implementations is proposed. Initial e xperiments sho wed that st andard GA operators struggle under the problem’ s tight constraints, ofte n generating infeasible solutions. T o o v ercome this, our approach uses a structured chromosome representation aligned with decision v ariables, together with a specialized repair mechanism and constrai nt-handling strate gies (bes t allocation procedure (B AP)). This mechanism ensures feasibility and is inte grated directly into e v olutionary process. This paper mak es se v eral k e y contrib utions. First, we introduce a no v el inte ger linear programming formulation for GTW AP that captures the essential constraints and objecti v es of a wide range of real-w orld scenarios. Second, we propose a constructi v e assignment heuristic to generate high-quality initial solutions. Third, we implement tw o metaheuristic approaches: a GA specically designed for GTW AP , and a general v ariable neighborhood search (GVNS) used here mainly to e v aluate the GA performance. F or further assessment, we use ILOG CPLEX Optimizer to obtain high-quality solutions and lo wer bounds. F ourth, since no public benchmark e xists for GTW AP , we de v elop a dedicated and di v erse benchmark set that reects realistic and v aried assignment conte xts. Computational e xperiments demonstrate that the GA consistently impro v es upon the initial heuristic, is competiti v e with CPLEX and sho ws a slight performance adv antage o v er GVNS on lar ger instances. Its con v er gence is further v alidated using statistical test s, conrming its potential as a practical tool for lar ge-scale scheduling problems. The remainder of this paper is or g anized as follo ws. Se ction 2 int roduces problem denition. Section 3 presents the mathematical formulation. Section 4 describes the GA-based solution and GVNS approach to solv e GTW AP . Section 5 presents and discusses the computational results. Finally , section 6 presents the conclusion. Assignment and scheduling problems with temporal and capacity constraints ha v e been widely studied across v arious domains. These problems of ten inte grate time windo ws, resource a v ailability , and spatial dispersion of agents and f acilities. Recent w ork has increasingly focused on assigning agents to service windo ws or tasks under tight temporal and capacity constraints. F or e xample, P aradiso et al. [1] proposes a stochastic look-ahead method for dynamic windo w assignment under uncertainty . Ca v aliere et al . [2] de v elops tw o-stage models for time-windo w assignment in tr a v eli ng salesperson problem (TSP) settings with stochastic tra v el times. Demirbilek [3] in v estig ates optional and mandatory assignment strate gies in dynamic v ehicle routing under hard time windo ws. C ˆ ot ´ e et al . [4] addresses multi-acti vity w orkforce scheduling by inte grating sequencing and time-windo w constraints within a unied optimization frame w ork, whereas Gozali [5] designs a modied GA with local search to handle time-constrained w ork er assignment in manuf acturing en vironments. These contrib utions sho w ho w assignment problems become increasi n gl y comple x when time windo ws, agent a v ailability , and capacity const raints interact. At the same time, earlier studies ha v e e xplored assignment and scheduling problems, of fering k e y modeling frame w orks and solution strate gies. F or e xample, Schmidt et al. [6] formulates an inte ger programming model to consolidate polling places while minimizing v oter tra v el and ensuring acceptable w ait times, whereas Dur ´ an et al. [7] de v elops an analytical optimization model to redesign v oter assignments considering both geographic distance and queue delays. Similar modeling perspecti v es appear in w orkforce scheduling [8], course assignment [9], and health resource allocation [10] where time-dependent a v ailability and heterogeneity complicate the decision process. Other w orks focus on generalized v ariants of the assignment problem [11], dynamic supply chains under uncertainty [12], or spatio- temporal task allocation in cro wdsourcing and mobile sensing [13], [14]. These contrib utions demonstrate the di v ersity of application settings b ut typically emphasize model construction and e xact optimization, which often f ace serious scalability limitations as problem size increases. T o address these challenges, metaheuristic and h ybrid approaches ha v e become central to lar ge-scale, time-constrained assignment problems. Early studies applied pure GAs with inte ger or random-k e y representations [15]–[17], sho wing that e v en standard operators can produce competiti v e solutions compared to e xact solv ers. Ho we v er , these classical GAs often required long runtimes to con v er ge, or con v er ged quickly at the e xpense of di v ersity and rob ustness. T o address this, heuristic augmentations ha v e been proposed. F or instance, K otw al and Dhope [18] inte grated domain-specic heuristics into GA decoding, while Y ounas et al. [19] compared GA with particle sw arm optimization (PSO) and dif ferential e v olution (DE), demonstrating superior solution quality b ut higher computational o v erhead. More recently , h ybridization trends ha v e intensied. Reinforcement learning has been coupled with GAs for adapti v e task assignment [20], and machine learning has been used to enhance GA-based order picking under f atigue ef fects [21]. Such approaches conrm the benets of embedding learning or heuristic guidance into e v olutionary search to balance e xploration and e xploitat ion. Be yond single-objecti v e GAs, adv ances in multi-objecti v e optimization ha v e demonstrated the ef fecti v eness of e v olutionary algorithms for Int J Artif Intell, V ol. 15, No. 2, April 2026: 1261–1274 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 1263 comple x combinatorial problems. Enhanced NSGA-II v ari ants [22], [23] and MOEA/D approaches [24], [25] ha v e sho wn strong performance and e xcellent con v er gence in logistics and urban systems, while h yper -heuristics [26], [27] of fer adapti v e stra te gies to dynamically balance e xplorat ion and e xploitation. Ne v ertheless, despite their success, applications of these methods to assignment problems with strict time windo ws and heterogeneous resource constraints remain limited. This g ap calls for specialized algorithms, such as GA for GTW AP , that can directly handle feasibility challenges while maintaining computational ef cienc y . 2. PR OBLEM DESCRIPTION Consider a set V of N agents and a set F of M f acilities. Each agent i ∈ V has a subset of preferred f acilities P i ⊆ F . The service time windo w for agent i at f acility f ∈ P i is denoted by T f i = [ a f i , b f i ] , where a f i and b f i represent the earliest and latest possible starting times, respecti v ely . The actual starting time is denoted by S T f i and must f all within both the time windo ws T f i and [ A f , B f ] associated with the f acility f . Each f acility f has a daily capacity M f max representing the maximum number of agents it can serv e. As in real-w orld, agents may require dif ferent service durations S i . T w o asymmetric distances: d if (from agent i to f acility f ) and d f i (return distance) are considered. The total round-trip distance for agent i to f acility f is denoted by D f i and equal to d if + d f i . Agents cannot arri v e late; if the y arri v e early to the f acility , the y must w ait until their scheduled time. W aiting times are therefore not considered in the cost function. The objecti v e of our problem is to assign each agent i ∈ V to e xactly one of their preferred f acilities and determine a feasible service start time such that all constraints are satised. The total distances tra v eled by all agents must be minimized, while respecting indi vidual service durations, f acility time windo ws, agent time windo ws, agent preference f acilities list and f acility capacity limits. T able 1 illustrates an instance with N = 8 agents and M = 3 f acilities. W e assume that f acilities operate from 8:00AM to 10:00AM (i.e. A f = 8 , B f = 10 for all f ∈ F ), and each f acility can serv e up to three agents per day (i.e. M f max = 3 ). Each agent has tw o preferred f acilities. The table lists service duration S i , preferred f acilities P i , total tra v el distance D f i , and time windo ws T f i . A feasible solution L consists of M ordered lists L f , one for each f acility f ∈ F , where L f represents a sequence of agents assigned to f satisfying all the problem constraints. Figure 1 illustrates the problem and its solution. Specically , Figure 1(a) presents an instance of the problem, while Figure 1(b) sho ws a corresponding feasible solution composed of three ordered lists. The feasible proposed solution is denoted by L = S f ∈ F L f . The total tra v el cost of the solution is computed as cost ( L ) = X f ∈ F cost ( L f ) = cost ( L 1 ) + cost ( L 2 ) + cost ( L 3 ) = 170 + 170 + 120 = 460 . T able 1. Instance e xample with 8 agents and 3 f acilities Agents v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S i ( min ) 5 10 5 5 10 5 5 5 P i f 1 , f 2 f 2 , f 3 f 1 , f 3 f 2 , f 3 f 1 , f 2 f 1 , f 3 f 2 , f 3 f 1 , f 2 D f 1 i 60 ∞ 60 ∞ 50 60 ∞ 50 D f 2 i 50 50 ∞ 60 60 ∞ 70 80 D f 3 i ∞ 50 80 70 ∞ 70 70 ∞ T f 1 i [9-10] [0-0] [8h30-9h30] [0-0] [9-10] [8-9] [0-0] [8-8h30] T f 2 i [8-9] [8-9] [0-0] [8h40-10] [8h30-9h30] [0-0] [8-9] [9-10] T f 3 i [0-0] [8-9] [9-10] [9-10] [0-0] [9-10] [8-9] [0-0] f 1 v 8 v 6 v 3 v 5 v 4 v 7 v 1 f 2 v 2 f 3 (a) f 1 v 8 v 6 v 3 f 2 v 1 v 5 v 4 f 3 v 2 v 7 (b) Figure 1. Problem instance and its solution: (a) problem solution instance and (b) solution-lists representation Genetic algorithm for g ener alized time-window assignment pr oblem (Ali Kansou) Evaluation Warning : The document was created with Spire.PDF for Python.
1264 ❒ ISSN: 2252-8938 3. MA THEMA TICAL MODEL This problem introduces no v el constraints particularly the preferred f acility requirement, that distinguish it from classical scheduling and assignment problems in the literature. T o formalize our problem, we adapt the mathematical models of the v ehic le routing problem with time windo ws (VRPTW) [28]. W e consider the follo wing tw o decision v ariables: i) x f ij = 1 if agent j is serv ed immediately after agent i at f acility f , and 0 otherwise; ii) S T f i is the actual starting time of agent i if assigned to f acility f . W e assign a ctitious starting agent denoted by 0 to each f acility f ∈ F with T f 0 = [ A f , B f ] and D f 0 = 0 . The problem is formulated as the follo wing inte ger linear programming model: (1) is the objecti v e function, which minimizes the total tra v el distance for all agents. Constraint (2) ensures that each agent is assigned to e xactly one preferred f acility f ∈ P i . Constraint (3) denes the start and end of service at each f acility . Constraint (4) ensures the connecti vity constraints between agents. Constraints (5) to (7) ensure the respect of the time windo ws of both agents and f acilities. Constraint (8) ensures that the limit of the number of agents per f acility f does not e xceed M f max . Constraints (9) and (10) ensure that e v ery agent is serv ed e xactly once. Constraint (11) states that each agent can only be assigned to f acilities in their preferred set P i . Constraints (12) and (13) dene the v ariable domains for all used decision v ariables. Minimize: X f ∈ F X i ∈ V ∪{ 0 } X j ∈ V ∪{ 0 } x f ij D f j (1) Subject to: X f ∈ P i X j ∈ V ∪{ 0 } ,j ̸ = i x f ij = 1 ∀ i ∈ V (2) X j ∈ V x f 0 j = X i ∈ V x f i 0 = 1 ∀ f ∈ F (3) X j ∈ V ,j ̸ = i x f ij = X j ∈ V ,j ̸ = i x f j i ∀ f ∈ F , ∀ i ∈ V ∪ { 0 } (4) S T f i + S i × x f ij ≤ S T f j + [1 − x f ij ] b f i ∀ i, j ∈ V ∪ { 0 } , ∀ f ∈ F (5) a f i X j ∈ V ∪{ 0 } ,j ̸ = i x f ij ≤ S T f i ≤ b f i X j ∈ V ∪{ 0 } ,j ̸ = i x f ij ∀ i ∈ V , ∀ f ∈ F (6) a f i ≤ S T f i ≤ b f i ∀ f ∈ F , ∀ i ∈ V ∪ { 0 } (7) X i ∈ V ∪{ 0 } X j ∈ V ∪{ 0 } x f ij ≤ M f max + 1 ∀ f ∈ F (8) x f ii = 0 ∀ f ∈ F , ∀ i ∈ V ∪ { 0 } (9) x f ij + x f j i ≤ 1 ∀ f ∈ F , ∀ i ∈ V ∪ { 0 } , ∀ j ∈ V ∪ { 0 } (10) X j ∈ V ∪{ 0 } x f ij = X k ∈ V ∪{ 0 } x f k i = 0 ∀ i ∈ V , ∀ f / ∈ P i (11) x f ij ∈ { 0 , 1 } ∀ i, j ∈ V ∪ { 0 } , ∀ f ∈ F (12) S T f i ∈ R + ∀ i ∈ V ∪ { 0 } , ∀ f ∈ F (13) Int J Artif Intell, V ol. 15, No. 2, April 2026: 1261–1274 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 1265 4. SOLUTION METHOD This section presents tw o meta-heuristic approaches de v eloped to solv e studied problem. The rst i s GA specically designed to handle the comple x constraints of GTW AP . The second is GVNS method, which serv es both as alternati v e solution approach and as means of v alidating the performance of primary method. 4.1. Genetic algorithm f or the assignment pr oblem GAs, a class of metaheuristics inspired by the principles of natural selecti on and genetics, were r st proposed in [29]. F or the GTW AP , the critical step lies in designing a suitable chromosome representation that encodes a feasible solution. W e adopt an appropriate representation scheme using an ( N + M ) -dimensional v ector composed of M -lists of agents, as illustrate d in Figure 1. Our GA implementation follo ws the frame w ork of [30] and is structured as follo ws: i) construct an initial population of N pop feasible candidate solutions; ii) rank the solutions of the current population in ascending order according to the ’tness’ cost, dened as the total distance tra v eled by all agents; i ii) select tw o parent solutions for reproducti on using an elitist selection scheme; i v) apply a crosso v er operator to the selected parents to generate children solutions; v) impro v e di v ersity by applying a mutation procedure; vi) form a ne w population by replacing solutions with lo w tness cost with impro v ed ones; vii) check the stopping criterion, dened by reaching the predened maximum number of iterations ( N iter ). In the follo wing subsections, we pro vide a detailed e xplanation of each component of our proposed GA. 4.1.1. Initial population The initial population contains N pop feasible solutions and is b uilt as follo ws. One of these solutions is generated using a dedicated heuristic called the best allocation agent (B AA) method, while the rest of the solutions are generated via a random allocation method. The B AA method b uilds a solution iterati v ely by assigning agents to f acilities. At each step, it selects the best unassigned agent and assigns it to its most preferred f acility , i.e., the f acility that minimizes its allocation cost. If no agents remain unassigned, the feasible solution is found, and the process terminates. If no feasible allocation is possible, a repair procedure called the B AP is applied. This procedure consists of randomly destro ying 2 or 3 lists in the current (non-feasible) solution and then trying to complete it by randomly reassigning agents to their best possible f acilities. This procedure is repeated up to 5 times. If a feasible solution is obtained within these attempts, it is accepted. Otherwise, the B AA method is restarted. W e apply this method 100 times, and the best feasible solution will be considered as part of the initial population. On the other hand, the random solutions are generated using a similar process, b ut with one k e y dif ference: the agent to be assigned at each iteration is selected randomly rather than optimally . 4.1.2. Selection mechanism In this w ork, we use a tness-based selection method to guide the e v oluti on of the po pul ation. This step plays a critical role in the GA by selecting the ttest i n di viduals from the current population to reproduce and create the ne xt generation. Our selection process relies on tw o k e y principles: i) the best indi viduals of the population are directly preserv ed in the ne xt population without modication, and ii) this ensures that the solutions with the highest tness cost are ne v er lost during reproduction. 4.1.3. One-point cr osso v er operator Once tw o parents are selected from the current population using the selection method, we apply our three-step crosso v er operator to generate children, as illustrated in Figure 2. First, we select randomly tw o agents v and w from tw o dif ferent random lists L A in parent A and L B in parent B (see Figure 2(a)). In the second step, we remo v e all agents follo wing v in L A and all agents preceding w in L B , including v and w themselv es (see Figure 2(b)). This yields tw o intermediate infeasible children, each missing the agents remo v ed in this step. In the third step, we repair these infeasible solutions using the B AP , by reassigning miss ing agents to their most suitable f acilities while satisfying time windo w and capacity cons traints (see Figure 2(c)). The tw o resulting children are no w feasible and can be e v aluated further . Note that when this process f ails to assign all agent s to f acilities, we use the destruction-construction procedure (B AP) as part of the repair process for the children. Figure 2 illustrates the three steps of the crosso v er process and the resulting children. As sho wn, the tw o resulting children outperform their parents: the second parent has a total cost of 510 , while its child achie v es a better cost of 490 , repres enting a v ery good impro v ement of 20 . Finally , each child under goes a further impro v ement as mutation, which is detailed in the ne xt section. Genetic algorithm for g ener alized time-window assignment pr oblem (Ali Kansou) Evaluation Warning : The document was created with Spire.PDF for Python.
1266 ❒ ISSN: 2252-8938 4.1.4. Mutation strategy In this phase of GA, we enhance the feasible solutions generated by one-point crosso v er via a mutation procedure applied with a predened probability . The mutation consists of a sequence of local search procedures applied up to N M times and in the follo wing order: relocate, sw ap, then B AP . Figures 3 and 4 illustrate ho w the relocate and sw ap procedures w ork using the same e xample presented in sect ion 2 (T able 1). Figure 3(a) sho ws the initial solution, while Figure 3(b) presents the result after applying the rel ocate mo v e: agent v 4 is reassigned, yielding a cost impro v ement of 10 . Similarly , Figure 4(a) depicts the solution before the while sw ap mo v e, and Figure 4(b) sho ws the solution after sw apping agents v 1 and v 8 , resulting in a more signicant impro v ement of 40 . Note, that after applying the mutation, both the impro v ed children and their parent solutions are g athered into a single list, which is then arranged in ascending order based on the total solution cost. (a) (b) (c) Figure 2. Illustration of the three steps of the crosso v er process: (a) agent selection, (b) remo v al and infeasible child generation, and (c) repair via B AP to ensure feasibility (a) (b) Figure 3. Illustration of the relocate procedure (a) before relocate: v 4 assigned to L 3 with a total cost of 520 and (b) after relocate: v 4 mo v ed to L 2 , reducing the total cost from 10 to 510 (a) (b) Figure 4. Illustration of the sw ap procedure (a) before sw ap: v 1 assigned to L 1 and v 8 to L 2 , with a total cost of 520 and (b) after sw ap: v 1 and v 8 e xchange their f acilities, reducing the total cost from 520 to 480 Int J Artif Intell, V ol. 15, No. 2, April 2026: 1261–1274 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 1267 4.2. General v ariable neighborhood sear ch appr oach T o v alidate our GA and pro vide an alternati v e solution method, we de v eloped an algorithm based on a GVNS. GVNS is a po werful metaheuristic used for combinatorial optimization problems, that changes systematically neighborhood structures to escape local optima [31]. The idea of t his metaheuristic is to alternate between tw o phases at each iteration: stochastic shaking and local search. The shaking (or perturbation) phase is used to perturb the current solution by nding a neighborhood solution S ′ after applying in sequence one mo v e from the three mo v es (relocate, sw ap, and B AP). The local search then re nes the perturbed solution by e xploring impro v ements within its imm ediate neighborhood. Whene v er a neighborhood yields an impro v ement, GVNS updates the ne w current solution and resets the search with the smallest neighborhood. Otherwise, it proceeds to the ne xt neighborhood. This process is repeated for a predened number of iterations ( β max ) to allo w a rob ust e xploration of the solution space. The pseudo-code of the GVNS algorithm is e xplained in Algorithm 1. Algorithm 1: The pseudo code of GVNS algorithm Input : an initial solution S 0 , the maximum number of iterations β max = 2000 Initialization : iter = 1 , S = S 0 while iter ≤ β max do k=1 while k ≤ 3 do Apply k − th shaking move to obtain S ′ Apply local search phase to find a solution S ′ ∗ If cost ( S ′ ∗ ) < cost ( S ) then S ← S ′ ∗ k = 1 Else k=k+1 iter = iter + 1 5. EXPERIMENTS AND RESUL TS This section pre sents the results obtained using the IBM ILOG CPLEX CP Optimizer and our proposed GA and GVNS methods for solving the GTW AP . The mathematical model w as implemented in CPLEX with the CP Optimize r solv er , which w as used to compute either optimal solutions or v alid lo wer bounds. CP Optimizer [32] is a state-of-the-art constraint programming engine that e xtends classical CP with adv anced scheduling constructs. Its e xact search algorithm inte grates metaheuristics, such as self-adapti v e lar ge neighborhood search, enabling f ast disco v ery of high-quality solutions and, in man y cases, proofs of optimality . 5.1. Benchmark description T o e v aluate the performance of our methods, we de v eloped a dedicated GTW AP benchmark. This benchmark includes instances of dif ferent sizes and characteristics, designed to reect realistic and di v erse operational scenarios. The number of agents ranges from 20 to 1187, while the number of a v ailable f acilities v aries from 2 to 20. Agent time windo ws dif f er in width, ranging from tight to wide or e v en absent, allo wing e v aluation under dif ferent le v els of scheduling e xibility . Assignment durations depend on the f acility and are chosen to approximate the minimal g ap between time windo ws, with some controlled random v ariations. Each agent must be assigned e xactly once, and the total number of a v ailable resources for all f aciliti es matches the number of agents. F acility time windo ws are randomly distrib uted, and agent arri v al and service times are aligned with the o v erall problem parameters. These instances were randomly generated according to some criteria such as normal distrib ution and clustering. Service durations v ar y by f acility and roughly correspond to the minimum time windo w dif ferences, with additional random adjustments. The benchmark comprises 130 instances di vided into tw o subsets: 65 smaller to medium-sized instances with the number of agents ranges from 20 to 98, and f acilities from 2 to 8, and 65 lar ger and more comple x instances with 144 to 1187 agents and 4 to 20 f acilities. This set is used to test scalability and rob ustness of the methods. 5.2. Results T able 2 summarizes the results for the rst set. The columns correspond respecti v ely to: i) the i nstance name; ii) the number of agents; iii) the number of f acilities; i v) the lo wer bound (lb) computed by the CP Genetic algorithm for g ener alized time-window assignment pr oblem (Ali Kansou) Evaluation Warning : The document was created with Spire.PDF for Python.
1268 ❒ ISSN: 2252-8938 Optimizer solv er; v) the cost obtained during the creation of the initial population; vi) the cost obtained by our GA; vii) the percentage g ap between the GA cost and the lo wer bound (GAP1); viii) the percentage impro v ement of GA o v er the initial method (GAP2); and ix) the number of feasible solutions found by the CP solv er within the time limit. The results are presented as listed in the table. T able 2. Results on the rst set of instances Instance # Agents # F acilities LB INIT GA GAP1 (%) GAP2 (%) # Solutions gtw ap1 20 2 372 428 372 0 13.08 11 gtw ap2 20 2 450 482 450 0 6.64 12 gtw ap3 20 2 490 514 490 0 4.67 7 gtw ap4 20 2 430 446 430 0 3.59 8 gtw ap5 20 2 468 494 468 0 5.26 10 gtw ap6 20 2 480 512 480 0 6.25 10 gtw ap7 20 2 482 508 482 0 5.12 9 gtw ap8 20 2 514 530 514 0 3.02 10 gtw ap9 20 2 376 434 376 0 13.36 13 gtw ap10 20 2 506 532 506 0 4.89 7 gtw ap11 20 2 498 538 498 0 7.43 10 gtw ap12 20 2 462 510 462 0 9.41 15 gtw ap13 20 2 498 546 498 0 8.79 7 gtw ap14 20 2 478 512 478 0 6.64 9 gtw ap15 20 2 492 512 492 0 3.91 5 gtw ap16 20 2 606 652 606 0 7.06 10 gtw ap17 20 2 468 492 468 0 4.88 19 gtw ap18 20 2 582 610 582 0 4.59 7 gtw ap19 20 2 480 516 480 0 6.98 8 gtw ap20 20 2 538 580 538 0 7.24 10 gtw ap21 20 2 396 418 396 0 5.26 7 gtw ap22 20 2 352 380 352 0 7.37 7 gtw ap23 20 2 388 432 388 0 10.19 13 gtw ap24 20 2 412 444 412 0 7.21 8 gtw ap25 20 2 336 380 336 0 11.58 13 gtw ap26 48 4 2470 2658 2470 0 7.07 47 gtw ap27 50 5 928 1010 928 0 8.12 29 gtw ap28 50 5 1084 1144 1084 0 5.24 39 gtw ap29 50 5 976 1064 976 0 8.27 34 gtw ap30 50 5 974 1048 974 0 7.06 27 gtw ap31 50 5 1072 1152 1072 0 6.94 35 gtw ap32 50 5 868 958 868 0 9.39 24 gtw ap33 50 5 826 928 826 0 10.99 32 gtw ap34 50 5 904 962 904 0 6.03 37 gtw ap35 50 5 900 950 900 0 5.26 36 gtw ap36 50 5 926 1016 926 0 8.86 35 gtw ap37 50 5 558 640 558 0 12.81 17 gtw ap38 50 5 546 624 546 0 12.5 31 gtw ap39 50 5 462 534 462 0 13.48 34 gtw ap40 50 5 468 520 468 0 10 39 gtw ap41 50 5 442 526 442 0 15.97 31 gtw ap42 72 6 3314 3694 3314 0 10.29 71 gtw ap43 80 8 1322 1436 1322 0 7.94 52 gtw ap44 80 8 1306 1430 1306 0 8.67 62 gtw ap45 80 8 1396 1522 1396 0 8.28 80 gtw ap46 80 8 1384 1508 1384 0 8.22 75 gtw ap47 80 8 1466 1570 1466 0 6.62 81 gtw ap48 80 8 998 1078 998 0 7.42 45 gtw ap49 80 8 1072 1200 1072 0 10.67 76 gtw ap50 80 8 1092 1186 1092 0 7.93 80 gtw ap51 80 8 1058 1180 1058 0 10.34 69 gtw ap52 80 8 1006 1116 1006 0 9.86 74 gtw ap53 86 8 1334 1402 1332 -0.15 4.99 18 gtw ap54 86 8 1437 1466 1437 0 1.98 23 gtw ap55 86 8 1418 1465 1422 0.28 2.94 44 gtw ap56 86 8 1297 1301 1297 0 0.31 22 gtw ap57 90 7 1525 1604 1530 0.33 4.61 11 gtw ap58 90 7 1550 1599 1556 0.39 2.69 32 gtw ap59 96 4 4968 5166 4962 -0.12 3.95 93 gtw ap60 96 4 4978 5220 4962 -0.32 4.94 105 gtw ap61 98 7 1756 1781 1752 -0.23 1.63 34 gtw ap62 98 7 1647 1649 1647 0 0.12 65 gtw ap63 98 7 1688 1702 1688 0 0.82 23 gtw ap64 98 7 1691 1709 1684 -0.42 1.46 12 gtw ap65 98 7 1778 1801 1789 0.61 0.67 31 Int J Artif Intell, V ol. 15, No. 2, April 2026: 1261–1274 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 1269 Solutions pro v en to be optimal within the time limit are mark ed in bol d. Note that for this set, GVNS and GA obtained the same results, thus, only GA results are reported. T able 3 reports the results for all instances in the second set. The columns correspond respecti v ely to: instance name, number of agents, number of f acilities, INIT cost, GA cost, GVNS cost, percentage g ap of GA o v er INIT and percentage g ap of GVNS o v er INIT . Gaps are computed as: GAP = INIT - X INIT × 100 where X ∈ { GA , GVNS } . T able 3. Results on the second set of instances Instance #Agents #F acilities INIT GA GVNS GAP1 (%) GAP2 (%) gtw ap66 144 4 8312 8168 8168 1.732 1.732 gtw ap67 144 6 7532 7248 7388 3.771 1.912 gtw ap68 144 4 8458 8168 8168 3.429 3.429 gtw ap69 144 6 7494 7248 7248 3.283 3.283 gtw ap70 148 4 2766 2470 2479 10.701 10.376 gtw ap71 192 4 11016 10604 10604 3.740 3.740 gtw ap72 192 4 10844 10604 10604 2.213 2.213 gtw ap73 216 6 10454 10174 10189 2.678 2.535 gtw ap74 216 6 10552 10174 10174 3.582 3.582 gtw ap75 230 6 4129 3997 4001 3.197 3.1 gtw ap76 238 6 4146 4130 4130 0.386 0.386 gtw ap77 240 4 12748 12628 12628 0.941 0.941 gtw ap78 240 4 12806 12628 12628 1.390 1.390 gtw ap79 270 6 14928 14925 14925 0.020 0.020 gtw ap80 274 6 4942 4933 4933 0.182 0.182 gtw ap81 275 6 15321 14874 14874 2.918 2.918 gtw ap82 288 4 15182 14648 14648 3.517 3.517 gtw ap83 288 6 15230 15004 15045 1.484 1.215 gtw ap84 288 4 14690 14562 14562 0.871 0.871 gtw ap85 288 4 3738 3314 3333 11.343 11.835 gtw ap86 288 6 16070 15016 15016 6.559 6.559 gtw ap87 305 6 15961 15650 15650 1.948 1.948 gtw ap88 305 6 5565 5520 5520 0.809 0.809 gtw ap89 305 6 15438 15412 15412 0.168 0.168 gtw ap90 355 6 16851 16327 16368 3.110 2.866 gtw ap91 355 6 36392 36278 36278 0.313 0.313 gtw ap92 360 4 46840 46390 46390 0.961 0.961 gtw ap93 360 4 41490 40806 40806 1.649 1.649 gtw ap94 360 6 43500 42268 42290 2.832 2.782 gtw ap95 360 6 40388 38868 38868 3.763 3.763 gtw ap96 400 6 6986 6971 6971 0.215 0.215 gtw ap97 420 12 40458 39044 39051 3.495 3.478 gtw ap98 420 12 40592 39050 39074 3.799 3.74 gtw ap99 480 4 61712 61348 61348 0.590 0.590 gtw ap100 480 4 54426 54100 54100 0.599 0.599 gtw ap101 520 6 62196 61382 61382 1.309 1.309 gtw ap102 520 6 56628 55412 55412 2.147 2.147 gtw ap103 600 4 76516 76268 76268 0.324 0.324 gtw ap104 600 4 69814 68696 68696 1.601 1.601 gtw ap105 600 12 59002 57094 57094 3.234 3.234 gtw ap106 600 12 60180 57238 57249 4.889 4.87 gtw ap107 700 6 86586 83320 83320 3.772 3.772 gtw ap108 700 6 79388 75364 7541 5.069 4.864 gtw ap109 720 4 92182 91236 91236 1.026 1.026 gtw ap110 720 4 82722 82344 82344 0.457 0.457 gtw ap111 780 12 80570 77180 77186 4.208 4.2 gtw ap112 780 12 80586 76868 76873 4.614 4.608 gtw ap113 800 12 11109 10966 10966 1.287 1.287 gtw ap114 800 12 20483 20260 20260 1.089 1.089 gtw ap115 800 12 10761 10713 10713 0.446 0.446 gtw ap116 800 12 11097 10989 10989 0.973 0.973 gtw ap117 800 12 10593 10452 10452 1.331 1.331 gtw ap118 829 15 10310 10230 10230 0.776 0.776 gtw ap119 840 4 106144 105816 105816 0.309 0.309 gtw ap120 840 4 97504 96904 96904 0.615 0.615 gtw ap121 850 15 10215 10134 10134 0.793 0.793 gtw ap122 880 6 95676 94388 94388 1.346 1.346 gtw ap123 880 6 96172 94532 94541 1.705 1.696 gtw ap124 960 4 120192 119910 119910 0.235 0.235 gtw ap125 960 4 111514 110934 110934 0.520 0.520 gtw ap126 960 12 94840 93516 93519 1.396 1.393 gtw ap127 960 12 97080 93193 93220 4.004 3.976 gtw ap128 1015 18 11596 11556 11556 0.345 0.345 gtw ap129 1063 20 12221 12146 12146 0.614 0.614 gtw ap130 1115 16 33873 33773 33773 0.295 0.295 Genetic algorithm for g ener alized time-window assignment pr oblem (Ali Kansou) Evaluation Warning : The document was created with Spire.PDF for Python.
1270 ❒ ISSN: 2252-8938 5.3. Analysis and discussion The INIT method w as run for 1500 iterations, and each instance w as solv ed v e times; the reported v alues correspond to the a v erage. F or the rst set, the CP Optimizer computed lo wer bounds with a time limit of three hours. In some cases, it found optimal solutions in under one minute. GA outperformed the CP Optimizer in 5 instances, while the CP outperformed GA in 4 instances. F or the remaining 56 instances, both performed equally . The a v erage g ap between GA and CP Optimizer w as only 0.005692%, and both methods solv ed 26 instances to optimality . This is e xpected, as the search space for small instances is relati v ely limited. The a v erage number of feasible solutions per instances in the rst set w as approximately 31, making e xhausti v e e xploration feasible within short computation times. Compared to INIT , GA impro v ed results by an a v erage of 6.919% in Set 1 and 2.2% in Set 2, leading to an o v erall a v erage g ain of 4.56%. The e x ecution time of the INIT method is v ery short, a v eraging less than 2 seconds for Set 1 and under 30 sec on ds for Set 2. As a result, INIT e x ecution times were not reported in the tables. GA also ran ef ciently , with a v erage e x ecution times of 10 seconds for Set 1 and 77 seconds for Set 2. GA and GVNS ha v e the same performance on all 65 small instances. F or the lar ger and more challenging instances in Set 2, GA demonstrated slightly better , outperforming GVNS in 16 instances. The g aps compared to INIT are 2.198% for GA and 2.139% for GVNS. Figure 5 graphically illustrates the results for INIT , GA and GVNS. Runtime for both methods are comparable as sho wn in Figure 6, making GA the preferred choice due to its mar ginally higher solution quality . Figure 5. INIT vs GA vs GVNS Figure 6. T ime GA vs GVNS T o assess the ef fecti v eness of our proposed approach, we applied the W ilcoxon signed-rank test, a widely used non-parametric statistical method to compare pai red results between tw o dif ferent methods [33]. This test w as applied to statistically e v aluate the performance of the GA method relati v e to the INIT method across the tw o benchmark sets. The analysis, performed separately for each set, yielded p-v alues of 2 . 3 × 10 − 12 Int J Artif Intell, V ol. 15, No. 2, April 2026: 1261–1274 Evaluation Warning : The document was created with Spire.PDF for Python.