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Abstract

This research is the first to introduce a mathematical model for hybrid workforce scheduling with a focus on enhancing in-person interaction. By integrating organizational requirements and individual preferences, it provides a framework that can be tailored to different institutional needs, offering a valuable tool to managers aiming to design hybrid work schedules. A mixed-integer linear programming model is developed to optimize weekly hybrid work schedules by maximizing in-office interactions and minimizing conflicts with employees’ remote work preferences. Numerical experiments, based on randomly generated instances, explore trade-offs between interaction and satisfaction under varying parameter values, including minimum office attendance, fixed office days, and penalties for violating employee preferences. The findings provide practical insights for managers: (1) Increasing structured office attendance substantially enhances interaction but sharply decreases employee satisfaction, particularly when fixed office days remove individual choice. (2) Skewed remote work preferences further complicate scheduling, potentially increasing dissatisfaction and reducing flexibility. (3) However, measures such as moderate enforcement of in-office presence achieves a balance, maintaining relatively high interaction with limited dissatisfaction. The proposed model can serve as a practical decision-support tool, adaptable to diverse organizational contexts and policies, offering a customized approach to equitable hybrid workforce planning.

Keywords

face-to-face interactions hybrid work scheduling integer programming operations management optimization

Introduction

There has been a significant increase in the adoption of hybrid work models by companies in recent years. The COVID-19 pandemic was a key driver for this trend, as companies around the world were forced to quickly switch to remote work to maintain their business practices. As the effects of the pandemic decreased, many companies understood that reverting to a totally office-based model was no longer a business requirement or an employee preference. Rather, a mixed strategy that incorporates the adaptability of remote work combined with the benefits of in-person cooperation in an office setup has become the preferred choice, providing a balance between autonomy and interaction among people. In recent years, many companies have adopted hybrid work models in which employees divide their time between a traditional workplace and a place other than the office, typically home, to meet the needs of their workforce while ensuring operational efficiency.¹ Despite the growing appeal of hybrid work models, several organizational difficulties can emerge in such environments. An important difficulty is maintaining a cohesive team of workers when they are not literally together. The absence of spontaneous in-person communications can impede team bonding, decrease informal expertise sharing, and make it more difficult to cultivate a sense of community. Another difficulty is ensuring a sense of equality between remote and in-office staff members, as there is a threat that those working from another location may really feel excluded from important meetings, decisions, or social gatherings, possibly resulting in a two-tier workforce,² and also leading to concerns about missing out on opportunities for learning and career advancement.³ In addition, the complexity of coordinating hybrid teams can disrupt workflows and project timelines, creating communication and collaboration barriers.

Most likely because of the above-mentioned organizational difficulties, there is a significant skepticism and backlash towards remote and hybrid work: A recent NY Times news article⁴ mentions how CEOs of companies like Amazon, JPMorgan Chase, Goldman Sachs, and Dell expressed their dislike towards hybrid work. The main argument CEOs and many managers have against hybrid work is that related work models decrease interaction among employees. In a memo informing employees that they decided to go back to the office, Amazon CEO Andy Jessy stated that “the advantages of being together in the office are significant”.⁴ Even little remote work can have a big impact on the frequency of face-to- face interactions in the workplace; spending an average of 3 days a week in the office is estimated to limit potential encounters between any two workers by 64% compared with pre-pandemic norms, increasing to 84% with 2 days a week.⁵ Despite skepticism and backlash, a survey by Barrero, Bloom and Davis⁶ shows that as of January 2025, one-third of the paid working days of Americans were from home. The same survey also shows that businesses that can work remotely plan an average of 2.5 working days from home per week. A new study⁷ found that working from home improves retention without damaging performance. Considering the advantages both office and remote work offer, one could assume that hybrid work models are here to stay. Therefore, providing the right tools, such as advanced collaboration software and flexible scheduling systems, can help bridge communication gaps, increase face-to-face interactions, and improve workflow coordination between hybrid teams.

We can categorize the existing hybrid work models into four: split-week, adaptable hybrid work, remote-first, and office-occasional models. One widely used model is the “split-week” model, where employees alternate between remote work and office based on an established routine, usually splitting the week between both. One other version is the “adaptable” hybrid work model, where workers have the autonomy to select which days they work from another location and which days they are in the office, depending upon their personal preferences or team needs. Some firms opt for the “remote-first” model, where employees mostly function from a remote location but are needed to come into the office for conferences, occasions, or task milestones. Another hybrid work model is the “office-occasional” model, which allows remote work as the default, with occasional workplace visits when definitely needed. These variations in hybrid work allow businesses to tailor their work strategies based on business needs and the preferences of the workforce. Traditionally, regardless of the hybrid work model type, hybrid work schedules are developed by team leaders or department heads ad hoc, resulting in a sense of exclusion, unfair treatment, and a two-tier workforce.⁸

In this article, we develop a mathematical model that attempts to maintain a cohesive team of workers and avoid the formation of a two-tier workforce by emphasizing office interaction in a hybrid work environment. As rightly pointed out in Alexander, De Smet and Mysore,² face-to-face interactions are much superior to virtual interactions in creating trust, a collaborative culture, and innovation. To the best of our knowledge, this is the first work that utilizes mathematical modeling, specifically integer programming, to address work schedule planning focused on increased interaction in a hybrid work environment. Besides its academic value, such a mathematical modelling approach would also benefit organizations, considering that manual scheduling becomes impractical when the number of employees increases. The article is organized as follows. In the next section, we review the literature related to hybrid work, and then, we develop an integer programming model that maximizes the interactions between employees. After presenting our computational results the article is concluded.

Literature review

Academic work on workforce planning typically focuses on planning personnel operations so that customer demands are satisfied while observing other work-related constraints. Several good reviews of such research are available.^9–12 Recently, the COVID-19 pandemic triggered research on scheduling personnel by allowing remote work under several constraints. Zucchi, Iori and Subramanian¹³ schedule the personnel shifts in a pharmaceutical distribution warehouse for contract workers to minimize the contagion risk by employing integer programming. Guerriero and Guido¹⁴ develop and solve optimization models that take into account demand requirements, employees’ personal and family responsibilities, and anti-COVID-19 measures at the same time at an Italian university. Melkonian¹⁵ uses integer programming to decide when employees should work in a hybrid environment. He assumes that there are some known savings due to remote work, and his models maximize the total savings while following constraints such as minimum and maximum numbers of days to work remotely. Moallemi and Patange¹⁶ describe the hybrid scheduling system they implemented at Columbia Business School during the COVID-19 pandemic that allows some students to attend in-person classes with social distancing, while their peers attend online, and schedules vary by day.

The research problem posed in this work also deals with hybrid work but it is different: We approach the problem from a top-managerial perspective and provide a tool to plan the workdays of the personnel to primarily maximize the physical interaction among the employees. At the same time, our model aims to observe the employees’ desired remote working days as much as possible. To the best of our knowledge, there is no prior work concentrating on this specific issue. However, there are numerous reports from consulting companies and articles from academics on the effects of hybrid work and on employee and employer preferences, and how it may affect the work environment.

In a recent review on remote and hybrid work, Mustajab¹⁷ systematically categorizes the literature on remote and hybrid work in relation to productivity, job satisfaction, work-life balance, leadership strategies, and organizational culture. His review indicates that remote work enhances productivity by reducing workplace distractions and commuting time while increasing job satisfaction due to greater autonomy and flexibility. However, it also identifies some challenges, such as social isolation and blurred work-life boundaries, which can negatively impact employee well-being. To address these issues, he concludes that a structured hybrid work model, supported by proactive leadership and strong digital infrastructure, offers the most effective solution for optimizing workforce productivity and engagement in modern organizational settings. In another study, Roy¹⁸ provides a critical examination of hybrid workspaces in the public sector, identifying the challenges, opportunities, and strategic considerations for governments adapting to post-pandemic workforce expectations and analyzes the impact of COVID-19 on employee preferences, managerial concerns, and policy adaptations. He suggests that the successful implementation of hybrid work in the public sector depends on three key design principles: (1) differentiation, the need for flexible policies tailored to various job roles, employee preferences, and organizational structures; (2) engagement, the need for maintaining employee motivation, collaboration, and trust through digital tools and structured workforce interactions; (3) inclusion, the need for expanding diversity and accessibility. He stresses that governments must approach hybrid work as a long-term strategic shift, investing in data-driven policies and research to ensure workforce effectiveness and equity in digital-era governance. Gratton⁸ provides a strategic framework for designing effective hybrid work models, emphasizing the need to balance institutional objectives with individual employee needs. Her approach builds on existing research on workplace flexibility, productivity, and digital transformation, reinforcing findings that hybrid models enhance work-life balance and efficiency when implemented thoughtfully.^7,19 Gratton’s emphasis on employee preferences and fairness echoes self-determination theory²⁰ and research on workplace equity,²¹ highlighting the risks of uneven access to flexibility. Gratton’s work synthesizes key insights on hybrid work, offering a practical guide for organizations navigating post-pandemic workforce transformations, while also pointing to the need for further research on long-term implications across industries. Krajcık, Schmidt and Barath²² conducted a survey analyzing the employee preferences in Slovenia and Kuwait for work settings in post-COVID-19. Their study reveals that most employees in both countries prefer a hybrid approach over fully remote or fully office-based work even though employees preferring some form of hybrid work is much higher in Slovenia than the ones in Kuwait. The survey results show that employees think that structured hybrid models, where employees split their time between home and office, offer the best balance between productivity, collaboration, and employee well-being. However, challenges such as maintaining team cohesion, ensuring equitable career opportunities, and addressing digital security concerns remain significant. The study suggests that organizations must design tailored hybrid policies that align with employee preferences and operational needs, ensuring an inclusive hybrid work environment. In a similar study, Angreni and Mahyuni²³ explore the impact of hybrid work, work-life balance, and work engagement on employee performance, particularly among Generation Z employees in Indonesia. Their survey of 170 respondents indicates that hybrid work has the strongest positive effect on employee performance, followed by work engagement and work-life balance. Their insights suggest that organizations should prioritize hybrid work models and employee engagement strategies to improve performance and job satisfaction. Hopkins and Bardoel²⁴ examine the rise of hybrid work in post-pandemic Australia, identifying five distinct hybrid work models that balance flexibility and productivity. Through interviews with senior human resources managers, they outline five key pillars for successful hybrid work: operations, culture, communication, well-being, and future skills. The study highlights that hybrid work improves work-life balance, employee satisfaction, and sustainability goals, but also presents challenges such as social isolation, coordination difficulties, and proximity bias. They suggest that organizations must carefully design policies, invest in supportive infrastructure, and adopt technologies like collaboration tools and cybersecurity measures to ensure the success of hybrid work. Eng, Tjernberg and Champoux-Larsson²⁵ explore the factors that contribute to effectiveness, work engagement, work-life balance, and well-being in hybrid work environments. Using reflexive thematic analysis, the study examines the experiences of 33 hybrid workers, identifying key success factors including workplace flexibility, structured communication, and the integration of digital tools. The findings emphasize that hybrid work enhances productivity when employees can optimize their tasks based on location suitability, but it also requires intentional strategies to maintain collaboration and social engagement.

Table 1 summarizes the previous research we included in our literature review. Each entry in Table 1 lists the authors and year of the study, its main objective, methodological approach, and principal findings.

Table 1.

Summary of previous studies on workforce scheduling and hybrid work models (newest to oldest).

Authors (year)	Objective	Methodology/Approach	Key findings
Barrero, Bloom and Davis (2025)⁶	Examine long-term trends in remote and hybrid work adoption	Empirical analysis using the survey on work arrangements	Hybrid work stabilized at 25–30% of workdays; employees prefer hybrid flexibility, prompting employer adaptation via digital investment
Angreni and Mahyuni (2024)²³	Analyze hybrid work, work–life balance, and engagement on Gen Z performance in Indonesia	Empirical survey (n = 170) analyzed via PLS-SEM (SmartPLS 4.0)	Work–life balance, engagement, and hybrid work all improve performance; engagement mediates work–life balance effects
Bloom, Han and Liang (2024)⁷	Assess productivity and management outcomes under hybrid work	Randomized controlled field experiments and firm-level productivity data	Hybrid work improves satisfaction and retention without reducing productivity; coordination and fairness are key
Eng, Tjernberg and Champoux-Larsson (2024)²⁵	Explore employee well-being and hybrid work experiences	Qualitative thematic analysis of employee interviews	Well-being shaped by flexibility, socialization, ICT maturity, and empowerment; structured communication is vital
Moallemi and Patange (2024)¹⁶	Develop a hybrid work scheduling system for business school	Mixed-integer programming optimization model	Efficiently scheduled 2500+ students; minimized conflicts and maximized fairness and interaction opportunities
Mustajab (2024)¹⁷	Evaluate effectiveness of hybrid work on productivity, satisfaction, and leadership	Empirical systematic literature review	Hybrid work enhances productivity but may increase isolation; leadership support and digital infrastructure are key success factors
Hopkins and Bardoel (2023)²⁴	Identify sustainable hybrid work models in Australia	Qualitative study of HR leaders using semi-structured interviews	Defined “Five Pillars of Hybrid Work”: operations, culture, communication, well-being, and skills; trust and flexibility critical
Krajcık, Schmidt and Barath (2023)²²	Examine cultural differences in hybrid work preferences	Empirical cross-country survey (Slovenia, Kuwait)	Structured hybrid models enhance satisfaction and performance; adoption varies by cultural context
Melkonian (2023)¹⁵	Develop hybrid scheduling models balancing flexibility and presence	Mixed-integer linear programming model	Proposed three models allowing flexible remote options; achieved time-efficient solutions and cost savings through relaxed in-office requirements
Smite, Moe, Hildrum et al. (2023)¹	Examine post-pandemic WFH and hybrid work policies	Multiple-case study of 17 software-intensive firms in Sweden and Norway	Employees show diverse WFH preferences shaped by role, age, and commute; firms adopt varied hybrid policies balancing flexibility, equity, and operational needs
Vij, Souza, Barrie et al. (2023)³	Assess worker preferences and wage trade-offs for remote work in Australia	Survey (1113 workers) with discrete-choice modeling plus employer interviews	Average workers would give up 4–8% salary for remote options, but preferences vary widely; attitudes toward learning, collaboration, and career influence valuations
Guerriero and Guido (2022)¹⁴	Optimize academic staff schedules under COVID-19 constraints	Mixed-integer programming model	Balanced safety, demand, and staff constraints; demonstrated feasibility of hybrid scheduling optimization
Roy (2022)¹⁸	Assess hybrid work implications for public sector governance in Canada	Qualitative synthesis of surveys (26,000+) and elite interviews	Proposed governance principles of differentiation, engagement, and inclusion; emphasized readiness and inclusion challenges
Yang et al. (2022)²⁶	Assess the causal effects of firm-wide remote work on collaboration and communication	Natural experiment using digital trace data from 61,182 microsoft employees in the U.S.	Remote work made collaboration networks more siloed, reduced cross-team ties, and shifted work from synchronous to asynchronous communication.
Gratton (2021)⁸	Develop framework for human-centered hybrid work design	Conceptual synthesis of 100+ organizational cases	Introduced “place–time matrix” linking tasks, preferences, and workflows; advocated fairness and structured flexibility
Xu and Hall (2021)¹²	Review operational implications of hybrid work for firms	Conceptual analytical synthesis	Highlighted coordination complexity and adaptive scheduling requirements under hybrid conditions
Zucchi, Iori and Subramanian (2021)¹³	Model workforce scheduling with contagion risk during COVID-19	Mixed-integer linear programming using CBC and Gurobi solvers	Improved manual schedules by 37.3% while reducing contagion risk (R from 3.7–5.7 to 1.9–2.5)
Alexander, De Smet and Mysore (2020)²	Identify post-pandemic hybrid work trends and productivity outcomes	Practitioner survey and McKinsey case analyses	Reported that 80% of workers prefer hybrid options; emphasized collaboration tools and workplace redesign
Weizenegger et al. (2020)	Examine challenges and opportunities of remote work during the COVID-19 transition	Qualitative case study using interviews and internal document analysis	Remote work improved flexibility and efficiency but increased coordination complexity and communication load
De Bruecker, Van den Bergh, Belien et al. (2015)¹¹	Review workforce planning models incorporating employee skills	Review combining operations research and management perspectives	Identified lack of integration between optimization and human factors; proposed hybrid models considering skills and flexibility
Rockmann and Pratt (2015)²¹	Explore meaning and identity in hybrid and remote work	Conceptual theoretical framework	Found that hybrid work reshapes professional identity and belonging; emphasized need for connection to sustain engagement
Van den Bergh, Belien, De Bruecker et al. (2013)¹⁰	Review advances in personnel scheduling models	Systematic review of optimization and heuristic methods	Classified mathematical and heuristic models; noted increasing focus on worker preferences
Ernst, Jiang, Krishnamoorthy et al. (2004)⁹	Review staff scheduling and rostering models across industries	Quantitative taxonomy of mathematical and heuristic approaches	Defined six core rostering modules: Demand modeling, days-off scheduling, shift scheduling, task assignment, and staff allocation
Bailey and Kurland (2002)¹⁹	Review early telecommuting literature	Conceptual synthesis of telework research	Highlighted flexibility and technology as drivers; noted challenges of isolation and productivity monitoring
Deci and Ryan (2000)²⁰	Develop self-determination theory for motivation in flexible work settings	Conceptual theoretical development	Identified autonomy, competence, and relatedness as key motivational needs; foundational to hybrid work engagement research

As summarized in Table 1, the existing body of research on hybrid work and workforce scheduling consists primarily of empirical studies, with only a few adopting mathematical modeling approaches. Empirical research on hybrid work emphasizes the role of leadership support, communication quality, flexibility, and digital readiness to increase employee satisfaction, engagement, and productivity.^8,24,25 In contrast, the mathematical models developed in prior studies focus mainly on workforce scheduling, space allocation, and contagion-risk mitigation (e.g., Guerriero and Guido¹⁴; Moallemi and Patange¹⁶; Zucchi, Iori and Subramanian¹³). While these models provide computational efficiency and operational insights, they typically overlook the interactions and collaboration patterns among employees that empirical research identifies as critical for team performance and organizational cohesion. The following section, therefore, develops a mixed-integer linear programming (MILP) model that explicitly incorporates employee interaction and implicitly accounts for collaboration through weighted relationships among employees.

Mathematical model

We start this section by first giving the notation used in the mathematical formulation of the problem. Our model is about day-of-week planning since we are focusing on maximizing potential face-to-face interactions by being in the office on the same day while observing the days employees prefer working from home.

Index sets

D = index set of departments.

E = index set of employees.

E_d = index set of employees in department d.

T = index set of days in the planning horizon. The planning horizon would be five business days for weekly planning.

UT_e = index set of unwanted office days for employee e.

Parameters

M_e = a large penalty coefficient incurred when employee e has to work on a day with remote work preference.

nDays = number of working days in the planning horizon.

nMin_d = minimum number of employees who should be present in the office on each day in department d.

nOffc_e = minimum number of days employee e should be in the office during the planning horizon. Could be the same for all employees, or for all employees of a department.

w_e,f = weight assigned to the interaction between employees e and f.

Decision variables

h_e,t = 1 if employee e works from home on day t; 0 otherwise.

o_e,t = 1 if employee e works in the office on day t; 0 otherwise.

u_e = number of preferred remote work days not satisfied for employee e.

y_e,f,t = 1 if both employees e and f are in the office on day t; 0 otherwise.

Formulation

The mixed-integer linear programming formulation of the problem is:

\max Z = \sum_{e \in E} \sum_{f \in E, f > e} \sum_{t \in T} w_{e, f} \cdot y_{e, f, t} - \sum_{e \in E} M_{e} \cdot u_{e}

(1)

subject to:

\sum_{t \in T} o_{e, t} \geq n O f f c_{e} \forall e \in E

(2)

\sum_{t \in T} (o_{e, t} + h_{e, t}) = n D a y s \forall e \in E

(3)

o_{e, t} + h_{e, t} = 1 \forall e \in E, \forall t \in T

(4)

y_{e, f, t} \leq o_{e, t} \forall e \in E, \forall f \in E, f > e, \forall t \in T

(5)

y_{e, f, t} \leq o_{f, t} \forall e \in E, \forall f \in E, f > e, \forall t \in T

(6)

o_{e, t} + o_{f, t} \leq 1 + y_{e, f, t} \forall e \in E, \forall f \in E, f > e, \forall t \in T

(7)

\sum_{t \in U T_{e}} o_{e, t} \leq u_{e} \forall e \in E

(8)

\sum_{e \in E_{d}} o_{e, t} \geq n M i n_{d} \forall d \in D, \forall t \in T

(9)

h_{e, t}, o_{e, t} \in {0, 1} \forall e \in E, \forall t \in T

(10)

y_{e, f, t} \in {0, 1} \forall e \in E, \forall f \in E f > e, \forall t \in T

(11)

u_{e} \geq 0 \forall e \in E

(12)

The objective function maximizes the weighted office interaction among employees. If two employees are both present in the office on a given day, this counts as an interaction. When the associated weight coefficients are all the same, say one, the interactions have the same importance. Recent literature suggests that not all interactions are equal in organizational value (e.g., Gratton,⁸ Yang, Holtz, Jaffe et al.²⁶). Our model allows interaction weights to be adjusted to reflect this heterogeneity, enabling future implementations to differentiate between strategic meetings and informal collisions (Rockman et al.²¹, Waizenegger et al.²⁷). However, if certain interactions are deemed more important, such interactions can be assigned a larger weight. For example, if it is important for two specific departments to interact more, the weights among employees of those departments can be given larger values. Moreover, the objective function penalizes the total number of preferred remote work days specified by the employees that are not satisfied. Minimizing the number of undesired office days serves the purpose of increasing employee satisfaction. Note that hybrid work dynamics–and therefore employee interaction and satisfaction–are also influenced by many external factors, such as technological infrastructure, economic conditions, leadership support, and workplace safety concerns. For the purposes of mathematical modelling, however, our definitions of employee interaction and satisfaction are intentionally limited in scope to be quantified in the scheduling objective.

By choosing different penalty coefficients (M_e), the employees’ wishes can also be prioritized depending on their role in the company. Constraints (2) state that each employee has to be at least nOffc_e days (determined by the company) in the office. The number of office days can differ by employee, for example, based on their department. Constraints (3) let each employee work exactly nDays, e.g., in the case of a typical work week, five total days. Constraints (4) make sure that each employee is either working at home or in the office on any given day. Constraints (5), (6), and (7) serve the purpose of capturing the interactions among the employees. Constraints (8) find the number of unwanted days an employee has to work in the office rather than at home. Constraints (9) make sure that there are at least nMin_d employees from each department in the office on each day. This type of restriction may be needed to prevent everybody from not coming to the office on the same day when some office presence is required for the department to function properly on any given day. Constraints (10), (11) and (12) determine the types of the decision variables. Note that, the number of pairwise interaction opportunities grows quadratically since the number of potential interactions on any given day is equal to n·(n−1)/2 where n is the total number of employees.

Computational results and business insights

In this section, we present the results of numerical experiments run under different scenarios. Numerical experiments were executed on an Intel Core i7-1165G7 CPU 2.8 GHz computer with 16 GB of RAM using Gurobi 9.5.1 as a solver and Python as a programming tool.

Since different combinations of parameter settings would lead to an innumerable number of experiments, we chose to fix certain parameter values. The importance of employee interactions is controlled via weight coefficients. While the model allows those weights to be set differently depending on how important particular interactions are, in the absence of such information, the weights of the interactions among employees were set to be equal. Furthermore, we assumed that there is only one department with 100 employees. The choice of 100 employees was for simplicity because we are reporting percentages in experiments. Moreover, the model allows for having more than one department. When there are no interactions among departments, their optimizations can be handled independently. The case of interdepartmental interactions, on the other hand, is not much different than having one department since the importance of interactions is controlled by the weight coefficients among specific employees who can also be from different departments. Unless stated otherwise, we set the minimum number of employees to be present in the office on each day equal to 10% of the personnel–for lack of information, thinking that some minimum number of employees need to be present. Our experiments, however, provide a sensitivity analysis regarding the model parameters including this one. The planning horizon was taken to be a week consisting of five business days, Monday through Friday.

In our model, the penalty coefficient M_e serves as a critical balancing parameter that governs the trade-off between maximizing percent interactions and minimizing percent unsatisfied employees. A higher value of M_e places greater emphasis on respecting employee preferences, encouraging the model to schedule office days in alignment with individual desires—potentially at the cost of reduced interaction. In contrast, a lower M_e allows the model to increase in-office overlap and promote collaboration, even if that means overriding some employee preferences. Depending on the organizational priorities, M_e values may be calibrated. Table 2 shows how M_e values affect interaction and satisfaction among employees when the number of preferred days working remotely for each employee (nRemote) is set to three, the minimum number of office days for each employee nOffc_e to two and when a uniform distribution of preferred remote work days is assumed. In Table 2, the second column, Interactions (%), reports the total percentage of weekly interactions. This value is obtained by summing the daily interaction percentages across all five workdays. The daily interaction percentages, reported in the columns labeled M, T, W, Th, and F are the ratios of the number of interactions realized by the model on that day to the total number of possible interactions among all employees. Note that, with 100 employees, there are 100 · 99/2 = 4, 950 daily and 5 · 4, 950 = 24, 750 weekly potential interactions. The Unwanted days (%) column reports the percentage of days on which employees are scheduled to be in the office despite having indicated a preference to work remotely on that day. When employees specify more than one preferred remote work day then it is possible that their wishes are partially fulfilled. Therefore, we also report the percentage of employees for whom at least one preferred remote work day could not be satisfied under the Unsatisfied emps. (%) column. Finally, the last column in the table reports the average optimization runtime of 100 randomly generated instances from an employee remote work preference distribution modeled as a discrete probability distribution with given preference probabilities for remote work days. The column structure in Table 2 repeats in all of the remaining tables in the article.

Table 2.

Calibration of M_e; nRemote = 3, nMin_d = 10, nOffc_e = 2, probabilities of preferred remote work days = [0.2, 0.2, 0.2, 0.2, 0.2].

M _e	Interactions (%)	M	T	W	Th	F	Unwanted days (%)	Unsatisfied emps. (%)	Ave. run time (secs.)
80	15.98	3.13	3.15	3.32	3.19	3.20	0.00	0.00	0.541
75	16.86	3.42	3.29	3.47	3.33	3.35	0.95	2.85	0.555
74	17.91	3.42	3.44	3.92	3.48	3.65	2.11	6.34	0.540
73	20.05	4.03	3.75	4.38	3.79	4.11	4.51	13.54	0.862
72	25.25	4.97	4.86	5.64	5.05	4.73	10.40	30.23	1.036
71	34.58	6.58	6.63	7.42	7.30	6.66	21.13	56.16	1.179
70	45.27	8.55	8.93	10.05	9.28	8.47	33.61	76.45	0.760
69	60.70	11.23	11.44	13.91	11.63	12.49	51.86	93.60	0.870
68	74.02	13.28	14.85	16.30	14.02	15.57	67.84	99.33	0.774
67	83.58	15.72	16.24	17.69	16.97	16.96	79.47	99.85	0.874
66	92.76	18.18	18.01	19.10	19.09	18.37	90.81	100.00	0.929
65	96.34	19.26	18.91	19.64	19.45	19.09	95.31	100.00	0.779
60	100.00	20.00	20.00	20.00	20.00	20.00	100.00	100.00	0.783

The trade-off between two competing objectives in our hybrid workforce scheduling model, maximizing in-office interactions and maintaining employee satisfaction, can be clearly seen in Figure 1. A low M_e value prioritizes interaction by allowing the model to override employee preferences, resulting in high levels of face-to-face collaboration but low satisfaction. Conversely, a high M_e prioritizes employee preferences, leading to high employee satisfaction but reduced opportunities for in-office interaction. The interaction percentage can be increased by decreasing the penalty. Results in Table 2 demonstrate this effect. While M_e is decreased from 80 to 60, the interaction percentages increase (nonlinearly). This, however, causes increasing dissatisfaction among the employees who prefer to have some remote work days. A company can choose any of these solutions to implement depending on targeted employee interactions and policies on remote work. By calibrating M_e values, the model allows the upper management to decide on its hybrid work priorities: Organizations seeking to enhance collaboration may opt for a lower M_e, while those aiming to preserve flexibility and employee well-being may prefer a higher M_e. In the rest of the numerical examples, M_e, was set to 100 for all employees. This high value ensures that the model prioritizes satisfying employees’ preferred remote work days as much as possible, while still aiming to maximize the number of in-person interactions.

Figure 1.

Trade-off between interaction percentage and employee satisfaction under varying values of the penalty coefficient M_e.

In Table 3 results, each employee is required to work in the office for 2 days per week (nOffc = 2), and the organization mandates that at least 10 employees (10% of the workforce) be present in the office on any given day (nMin_d = 10). The distribution of preferred remote work days is uniform across the five weekdays, and the number of preferred remote days per employee varies from one to three. Typically, the preferences for days to work remotely are not uniformly distributed. Mondays and Fridays are preferred more because they are after or before the weekend; Wednesdays and Thursdays, which split the week, are also the choice of some, although not as highly sought as Mondays and Fridays. Table 4 uses the same setting as in Table 3, except the distribution of remote day preferences is skewed toward Monday and Friday. Thus, this setting captures a more realistic distribution of remote work preferences, and allows us to numerically study the impact of skewed preferences for remote work days.

Table 3.

Effect of the number of preferred remote work days; nMin_d = 10, nOffc_e = 2, M_e = 100, probabilities of preferred remote work days = [0.2, 0.2, 0.2, 0.2, 0.2].

Remote days (#)	Interactions (%)	M	T	W	Th	F	Ave. run time (secs.)
1	63.99	12.91	12.82	12.84	12.69	12.74	0.487
2	35.98	7.12	7.13	7.27	7.19	7.27	0.551
3	15.98	3.13	3.15	3.32	3.19	3.20	0.567

Table 4.

Effect of the number of preferred remote work days; nMin_d = 10, nOffc_e = 2, M_e = 100, probabilities of preferred remote work days = [0.4, 0.0, 0.1, 0.1, 0.4].

Remote days (#)	Interactions (%)	M	T	W	Th	F	Unwanted days (%)	Unsatisfied emps. (%)	Ave. run time (secs.)
1	66.80	7.29	20.00	16.32	16.12	7.07	0.00	0.00	0.486
2	45.16	1.22	20.00	11.46	11.28	1.21	0.00	0.00	0.557
3	28.28	0.18	20.00	4.03	3.89	0.18	3.01	8.64	13.350
3 (nMin_d = 0)	28.03	0.07	20.00	4.03	3.89	0.05	0.00	0.00	0.755

In both tables, as the number of preferred remote work days increases from one to three, there is a notable decrease in the percentage of interactions. The daily interaction percentages in Table 3 are almost equal aligning with the uniformly distributed remote work day preferences. Compared to Table 3, the overall interaction percentages are higher for the same number of preferred remote days in Table 4. When employees have a propensity to work remotely on the same days (Mondays and Fridays in Table 4), the interactions are increased by about 10%. With two remote work days, the interaction percentage increases from 35.98 to 45.16; with three remote work days, from 15.98 to 28.28 in the case of skewed remote work preference distribution. However, most of this interaction actually occurs on Tuesdays when nobody minds working in the office. Daily interaction percentages for the most sought-after days (Mondays and Fridays) become almost non-existent, and more people end up coming to the office on the same days because remote work preferences are concentrated on similar days. Moreover, both the percentage of unwanted assignments and the percentage of unsatisfied employees become positive with three remote work days. This means that preference clustering around specific days reduces flexibility in scheduling and may worsen system-wide outcomes. The variance in interaction percentages between days also shows the influence of daily preferences for remote work. For instance, with skewed preferences, interactions on Wednesday remain lower throughout, due to the low probability of remote work on that day (Table 4). The penalty for unsatisfied preferred remote work days becomes more apparent when employees’ preferences are harder to satisfy, leading to an 8.64% (compared to 0% in the case of uniformly distributed preferred remote work days) unsatisfied employee rate when three preferred remote work days are requested, stressing the balancing act between employee preferences and team interactions. In most of the experiments, the minimum number of personnel who should be present in the office was kept relatively small at 10% of the total number of employees. Even such a small enforcement can actually have a positive effect on interactions. When this requirement is relaxed, the average percentage of interactions drops slightly from 28.28 to 28.03% by lowering the interaction percentages on Mondays and Fridays to observe more employees’ preferences, and leading to 0% unsatisfied employees (Table 4).

When n·(n − 1)/2 (the maximum number of possible daily interactions) is taken as pre-pandemic interaction levels, where working from the office was more of a norm, our experiments indicate that the interactions must have decreased dramatically in the presence of remote work after the pandemic. When employees have two remote work day preferences, the interactions are reduced by approximately 55%, and with 3 days by more than 70% (Table 4).

Table 5 examines the effect of keeping a daily minimum number of employees in the office on interactions and unsatisfied employees. As the minimum number of employees required to be in the office increases, the interaction percentage gradually increases. When the percentage of employees required to be in the office increases to 80%, interactions reach 66.48% but at the expense of not satisfying the preferred remote work days of 7.98% of the employees. Note that the number of face-to-face interactions will be maximized when all employees work in the office every day. Table 5 results indicate that a balance between the office presence (through minimum employee constraints) and individual preferences for remote work is needed for optimizing interactions. The experiments also show that a relatively high interaction level of more than 60% is achievable while enforcing more than half, even 60%, of your staff to come to the office, and by still observing the preferred remote work days. These findings suggest that hybrid scheduling policies should be implemented carefully: Modest in-office coordination goals may be sufficient to maintain collaboration, while overly strict rules risk widespread discontent with limited added benefit.

Table 5.

Effect of the daily minimum number of employees in the office; nOffc_e = 2, nRemote = 1, M_e = 100, probabilities of preferred remote work days = [0.2, 0.2, 0.2, 0.2, 0.2].

nMin _d	Interactions (%)	M	T	W	Th	F	Unwanted days (%)	Unsatisfied emps. (%)	Ave. run time (secs.)
20	63.99	12.91	12.82	12.84	12.69	12.74	0.00	0.00	0.672
40	63.99	12.91	12.82	12.84	12.69	12.74	0.00	0.00	0.479
60	63.99	12.91	12.82	12.84	12.69	12.74	0.00	0.00	0.520
80	66.48	13.36	13.28	13.29	13.29	13.25	7.98	7.98	0.871
100	100.00	20.00	20.00	20.00	20.00	20.00	100.00	100.00	0.065

Similar to Table 5, Table 6 also shows the effect of requiring more employees to be present in the office each day, albeit with a different remote work preference distribution. When the preferences are not equally distributed but concentrated on certain days, such as Mondays and Fridays, it becomes more difficult to fulfill those preferences (even if there is only 1 day employees are working remotely as in these experiments), resulting in a lot more employee dissatisfaction beyond a certain minimum office-presence level: 40.13% versus 7.98% in the case of 80% office presence but some dissatisfaction (3.84%) occurs even with 60% office presence. On the positive side, the interaction percentages increase.

Table 6.

Effect of the daily minimum number of employees in the office; nOffc_e = 2, nRemote = 1, M_e = 100, probabilities of preferred remote work days = [0.4, 0.0, 0.1, 0.1, 0.4].

nMin _d	Interactions (%)	M	T	W	Th	F	Unwanted days (%)	Unsatisfied emps. (%)	Ave. run time (secs.)
20	66.80	7.29	20.00	16.32	16.12	7.07	0.00	0.00	0.490
40	66.80	7.29	20.00	16.32	16.12	7.07	0.00	0.00	0.486
60	67.68	7.66	20.00	16.32	16.12	7.58	3.84	3.84	0.689
80	77.97	12.77	20.00	16.32	16.12	12.77	40.13	40.13	1.382
100	100.00	20.00	20.00	20.00	20.00	20.00	100.00	100.00	0.092

In Table 7, we ran experiments similar to the ones given in Table 5; however, this time with two preferred remote work days for each employee instead of one. The interaction percentage again increases with a higher percentage of employees in the office, but with a more significant penalty to employee satisfaction. At lower percentages for office presence, the interaction percentages drop compared to the case of one preferred remote work day, since the model tries to satisfy preferences for remote work days (when the penalty for not satisfying is high at M_e = 100). At 80% office presence, interactions rise to 63.83% (although they are slightly lower than the 66.48% with one preferred remote work day), but the percentage of unsatisfied employees spikes dramatically to 66.50% (up from 7.98% in the case of one preferred remote work day). A higher minimum presence in the office restricts the system’s flexibility, leading to increased unsatisfied employees when multiple preferred remote work days are specified. On the other hand, the mathematical model is still capable of keeping interaction percentages stable at close to 40% with a relatively high office presence percentage of 60%, while mostly observing preferred remote work days at the same time.

Table 7.

Effect of the daily minimum number of employees in the office; nOffc_e = 2, nRemote = 2, M_e = 100, probabilities of preferred remote work days = [0.2, 0.2, 0.2, 0.2, 0.2].

nMin _d	Interactions (%)	M	T	W	Th	F	Unwanted days (%)	Unsatisfied emps. (%)	Ave. run time (secs.)
20	35.98	7.12	7.13	7.27	7.19	7.27	0.00	0.00	0.551
40	35.98	7.12	7.13	7.27	7.19	7.27	0.00	0.00	0.512
60	38.18	7.64	7.58	7.63	7.63	7.71	4.82	9.08	1.569
80	63.83	12.77	12.77	12.77	12.77	12.77	50.00	66.50	3.233
100	100.00	20.00	20.00	20.00	20.00	20.00	100.00	100.00	0.086

Figure 2 compares Tables 5 and 7 visually to illustrate the effect of specifying one versus 2 days as remote. In Figure 2, we depicted two metrics: Interaction Percentage, and Percentage of Unwanted Days. Figure 2 shows that increasing the number of preferred remote days leads to a consistent drop in employee interaction unless stricter attendance requirements (high nMin_d) are enforced. Increasing employee flexibility by moving from one to two preferred remote days leads to a significant loss in workplace interaction. However, when nMin_d ≤ 40, both configurations yield full satisfaction and no unwanted days without a significant drop in interaction when minimum office presence requirements are reduced. When stricter attendance rules are enforced, interaction percentage increases but at the cost of widespread preference violations and employee satisfaction, especially in more flexible settings.

Figure 2.

Comparison of interaction percentage, percentage of unwanted days, and percentage of unsatisfied employees across varying minimum office attendance levels (nMin_d) for two hybrid work settings: one and two preferred remote days per employee.

These findings stress the value of our proposed model as a useful decision-support tool for human resources personnel tasked with designing hybrid work policies. By quantifying the trade-offs between employee preferences and in-person interaction, our model may help organizations to achieve both high satisfaction and effective collaboration, provided that they maintain a moderate level of required in-office presence.

Fixing office days as a company policy is expected to reduce the total flexibility of the system, leading to higher interaction percentages, but also higher unsatisfied employee percentages, especially when the number of fixed days increases. For achieving high interaction levels and low employee dissatisfaction, it is important that the office days are fixed so that they are aligned with the working-from-home preferences of the employees. Figure 3 (nOffc_e = 2, nMin_d = 10, nRemote = 2, M_e = 100, probabilities of preferred remote work days = [0.2, 0.2, 0.2, 0.2, 0.2]) illustrates the effect of fixing office days on overall interaction and employee satisfaction levels. As the number of fixed in-office days increases from only Monday (“Fix M”) to a full 4-day schedule (“Fix M,T,W,Th”), interaction percentage steadily rises from 48.90% to 87.27%. This increase reflects a more overlapping presence among employees, facilitating greater opportunities for face-to-face interaction. However, this improvement in interaction comes at a significant cost to satisfaction. The percentage of satisfied employees (computed as 100% minus the percentage of unsatisfied employees) drops sharply—from 59.67% in the “Fix M” case to 0% in the “Fix M,T,W,Th” configuration. This indicates that more rigid scheduling severely limits employees’ ability to follow their preferred remote work days, leading to high levels of dissatisfaction. The steep decline in satisfaction again highlights a trade-off between maximizing interaction and preserving individual workday flexibility in hybrid work design.

Figure 3.

Interaction and satisfaction levels under uniform remote work preferences as the number and combination of fixed in-office days increase.

Figure 4 (nOffc_e = 2, nMin_d = 10, nRemote = 2, M_e = 100, probabilities of preferred remote work days = [0.4, 0.0, 0.1, 0.1, 0.4]) shows how fixing in-office days affects interaction and employee satisfaction under skewed remote work preferences, where employees strongly favor Monday and Friday as their remote days. As more days are fixed, interaction improves—peaking at 91.28% for the “Fix M,W,F” policy while satisfaction declines sharply, dropping from 24.60% to 0% when three or more days are fixed. This reflects an insightful trade-off between coordination and flexibility: Even when the model tries to avoid preference violations, employees lose the ability to follow their preferred schedules as the number of fixed days increases. Interestingly, interaction drops from 91.28% (“Fix M,W,F”) to 81.20% in the “Fix M,T,W,Th” case, despite fixing more days. This happens because the policy requires all employees to be in the office from Monday through Thursday, leaving only Friday flexible. In this case, the model sacrifices some interaction to avoid a high level of unsatisfied employees since the related penalty is relatively high (M_e = 100).

Figure 4.

Interaction and satisfaction levels under skewed remote work preferences as the number and combination of fixed in-office days increase.

Thus, these results underscore the trade-off between ensuring sufficient in-office interaction and accommodating employee preferences for remote work. In general, there is a strong inverse relationship between the number of remote work days and the percentage of face-to-face interactions. The more employees’ preferences for remote work are observed, the more difficult it becomes to have office interactions, even with a model designed to maximize office interactions.

Conclusion

This study presents a mathematical optimization model for hybrid work scheduling that maximizes employee interactions over a planning horizon in a hybrid work environment. The model incorporates employee preferences, interaction importance, and minimum in-office requirements to generate weekly schedules that balance organizational interaction goals with individual employee flexibility. Through numerical experiments, we demonstrated how different scheduling strategies such as flexible versus fixed workdays and symmetric versus skewed preference distributions for working remotely affect interaction levels and employee satisfaction. The findings reveal that rigid scheduling policies can lead to significant dissatisfaction, even when aligned with stated preferences, while more adaptive strategies can maintain high interaction levels with considerably lower dissatisfaction.

While working from the office brings advantages in terms of increasing face-to-face interaction levels among employees, issues such as raising children, and long commutes to and from the workplace present other types of challenges where employees prefer to work remotely at times. Our work underlines the importance of finding a balance between meeting employee preferences for remote work to help increase employee satisfaction and ensuring sufficient in-office presence for effective team collaboration. Employee satisfaction is naturally important in the eyes of companies. However, a hybrid work format where each employee decides when to come and when not to come to the office solely according to their preferences and without any guidelines is not a good format to choose when interactions are valued. Uneven remote work day preference distribution prevalent in real-life makes it easier for companies to choose days for their employees to come to the office for face-to-face interactions and work from home as shown in our numerical experiments. Considering all the often contradicting parameters involved in such decisions, companies can benefit from having planning tools such as the model provided here when planning the office days of their personnel. These planning tools can be customized according to the specifications of a company and provide the best results for ensuring the desired interaction levels.

One future research direction is to incorporate stochasticity into the model, such as modeling uncertainty in employee attendance or variability in remote work preferences. This would reflect some real-world scheduling challenges and allow for the evaluation of the robustness of proposed policies. Another promising avenue is to introduce fairness constraints or penalty terms that ensure an equitable distribution of in-office days over time. Furthermore, it would be interesting to measure the impact of different work schedules on operational efficiency and employee productivity.

Footnotes

ORCID iDs

Tankut Atan

Ufuk Kula

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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