1. INTRODUCTION
Manpower planning plays a key role in the success of almost every organization especially production and service industries. It increases their efficiency and helps them to achieve targets. The major reasons for its need are organizational goals, workers preferences, work regulations, collective bargaining, and legal constraints.
Flexible working is an efficient way of managing manpower by providing better employee satisfaction. It increases workers productivity and engagement and provides flexible retirement, attraction, and retention of senior executives etc. (See, Direction, 2008;Report, 2012;Wheatley, 2016;RARC Report, 2015;Qin et al., 2015).
Annualized hours policy is one of the methods of flexible working that helps to schedule and calculate working time throughout the year. Also, it helps the organization to efficiently match workforce demand and it’s availability over the year. The variation in demand may be due to seasonal effects whereas the variation in workforce availability is due to illness, training, vacation etc. As high demand leads to hectic overtime and less demand leads to underutilization of the workforce, there is a need for proper management. AH policy provides effective plans to manage better worklife balance, less overtime, improved working flexibility and minimizing capacity shortages during peak requirements (Hung, 1997;RARC Report, 2015;Bell and Hart, 2003;Ryan and Wallace, 2016).
Earlier applications of AH were prevalent among several French, German and Scandinavian companies (Teriet, 1977;Lynch, 1995;Gall, 1996;Tucker et al., 2001). Later, it is as used with different labor legislation and collective bargaining of countries like France (Grabot and Letouzey, 2000), Britain (Rodriguez, 2003) and Kouzis and Kretsos (2003). A brief list of wide implementation and labor legislation of AH can be found in Hung (1997), Kouzis and Kretsos (2003), Bell and Hart (2003). It gives greater flexibility by distributing working hours of the hired workforce over the year to fulfill the demand fluctuation. The greater advantage of Annualized hours policy is that it leads to a reduction in overtime, underutilization of workforce, total cost, parttime workers and gives improved service level. It gives fruitful results in laborintensive industries for eg. Filho and Marçola (2001) reduces overtime by 94% and the use of temporary workers by 53%. Reduction in temporary workers gives improved service level in Thomas Sanderson (Workforce Logistics, 2016) productivity increased by 25% and labor cost reduced by previous year costs. Also, McMeekin (1995) in Tesco Distribution showed that the stock levels had reduced to a large extent on the implementation of AH policy.
The layout of this paper is as follows; Section 2 discusses related literature; In Section 3 presents problem motivation and contribution; Section 4 describes the problem and its mathematical programming formulation; Section 5 is about the experimental study for proposed models of Annualized hours planning problem (AHPP). Finally, section 6 is the conclusion.
2. LITERATURE REVIEW
Manpower planning and scheduling in an AH environment is well studied by many authors. Some of them are Hung (1999a, 1999b), Filho and Marçola (2001), Azmat and Widmer (2004), Azmat et al. (2004), Corominas et al. (2002), Corominas et al. (2007b), Lusa et al. (2008a, 2008b), Hertz et al. (2010), Corominas and Pastor (2010), Lusa and Pastor (2011), Van der Veen et al. (2012), Sureshkumar and Pillai (2012, 2013), Van der Veen et al. (2014) and Hasan and Hasan (2017). Detailed Characterization and classification of the annualized working hours planning problems are presented by Corominas et al. (2004a).
Hung (1999a, 199b) proposes heuristics for determining minimal manpower and their scheduling in single and multiple shifts over the year. Azmat and Widmer (2004) present a threestep heuristic approach for manpower scheduling with minimal staff determination, overtime estimation, and holiday consideration. Most of the models in AH planning and scheduling problems are based on mathematical programming.
A brief Summary of mathematical models is presented through table.
3. PROBLEM MOTIVATION AND CONTRIBUTION
Very limited work is present in literature for Planning and scheduling problems in an Annualized hours environment. This work is an extension of previously proposed mathematical programming models for multiple tasks by Corominas et al. (2007c), Sureshkumar and Pillai (2012, 2013) and for the single task by Lusa et al. (2008a, 2008c). In these models, the problem objective for single task problem is to minimize the sum of maximum and average of relative capacity shortages. While for multiple task problem the objective is to minimize the sum of maximum relative capacity shortage among all the tasks and an average of relative capacity shortages of all tasks.
The objective function for Single Task:
$\text{Min:}\hspace{0.33em}\hspace{0.33em}Z=\alpha \times S+(1\alpha )\times \frac{{{\displaystyle \sum}}_{t\in T}\text{}\frac{{S}_{t}}{{D}_{t}}}{\text{total}\hspace{0.33em}\text{weeks}}$
The objective function for Multiple Tasks:
$\text{Min:}Z=\alpha \times {S}^{\text{'}}+(1\alpha )\times \frac{{{\displaystyle \sum}}_{j\in J}\text{}{{\displaystyle \sum}}_{t\in T}\text{}\frac{{S}_{jt}}{{D}_{jt}}}{(\text{total}\hspace{0.33em}\text{weeks})\times (\text{total}\hspace{0.33em}\text{tasks})}$
where S and S′ are maximum relative capacity shortage for single and among multiple tasks with weight α.
Proposed Objective function:
$\text{Min:}Z={\beta}_{j}\times {Z}_{j}$
where,
$\sum}_{j\in J}\text{}{\beta}_{j}=1$
and
${Z}_{j}={\alpha}_{j}\times {S}_{j}+(1{\alpha}_{j})\times \frac{{{\displaystyle \sum}}_{t\in T}\text{}\frac{{S}_{jt}}{{D}_{jt}}}{\text{total}\hspace{0.33em}\text{weeks}}$
where S_{j} is maximum relative capacity shortage for task j with weight α_{j}. The proposed objective helps for better decision making with respect to the preference of minimizing capacity shortages for a particular task. It is the weighted sum of relative capacity shortages for multiple tasks.
In addition to the new objective, the problem is analyzed by introducing workers willingness to work on multiple tasks and weeks. The performance of workers is determined by many factors such as Age, experience, intelligence, mental and physical strength, working conditions, worklife balance, attitude, job type etc. (see, Blumberg and Pringle, 1982;Riggs, 1981;Waldman and Spangler, 1989;Sjöberg, 1994). Recently the concept of workers willingness or readiness is introduced in the planning and scheduling of workers by Othman et al. (2012) and Yang et al. (2016). They used it as a job satisfaction factor of workers. Both Othman and Yang considers workers willingness to work on different machines in a particular time period t. Here we introduced this concept for working in different weeks. It provides greater flexibility to the workers. Previous work in AH planning only considers the relative efficiency of workers on each task. Here, we considered the willingness in two different forms i.e. willingness to work on different tasks and to work in each week. The willingness is determined over a scale of 1, if willingness is 0.3 then it means that the worker is 30% ready to work.
The objectives of the proposed models are:

(1) Minimize relative capacity shortage with respect to demand as per the preference of Decision Maker for each task.

(2) To determine the scheduled working hours for each employee.

(3) To analyze the model by providing more flexibility to the workers through willingness and annualized hours policy.
4. PROBLEM FORMULATION
In the present work, we are working on a basic model of AH planning problem and optimizing the service level by taking into account workers working efficiency and willingness. No temporary workers are considered. Overtime is allowed with an upper bound. It is assumed that there are multiple tasks and that the company forecasts the demand and establishes the capacity requirement. As overtime is bounded, so the capacity shortage is possible in certain weeks. To make the model more real, crosstrained workers with willingness to work on each task is considered with different relative efficiency (RE) (scale of 1, for eg. a value of 0.6 means that such a worker needs to work 1/0.6 h to perform a task that a worker with a RE equal to 1 would perform in 1 hour) and willingness (scale of 1) associated to them. The number of weekly working hours fall within an interval with upper and lower bound. Also, the total working hours for the planning period lies within an interval. Four Mathematical programming formulations M1, M2, M3, and M4 are presented. Where M1 considers new objective and relative efficiency, M2 considers workers willingness to work in each week in addition of new objective and relative efficiency, M3 considers workers willingness to work on each task with new objective and relative efficiency and M4 in addition of M3 takes account of workers willingness to work in each week of the year.
The problem studied and modeled in this paper is based upon the minimization of the relative capacity shortage with respect to demand of workforce. Since, high relative capacity shortage leads to worse service level, hence the objective of the problem is to minimize.
The characteristics of the problem are as follows:

(1) Planning period of the allocation of the workforce is taken as 52 weeks (i.e 1 year).

(2) Holiday weeks are predetermined by workers.

(3) Working hours for each week and year is lower and upper bounded.

(4) Multiple tasks with forecasted demand are taken with the assumption that the staff is multitasking having different relative efficiency associated to them.

(5) Workers willingness for each task is considered.

(6) Overtime is permitted with an upper bound.

(7) Hiring temporary workers is not allowed.

(8) The average number of working hours for a group of 12 consecutive weeks cannot be larger than 44 hours per week.

(9) A Utility function is to be optimized.
The list of notations which will be used throughout the paper is given below:
Decision variables and Model parameters

I set of workers indexed by i.

J set of tasks indexed by j.

T set of week indexed by t.

D_{jt} demand for task j in week t.

L_{it}, U_{it} minimum and a maximum number of allowed working hours for worker i in week t.

L_{i}, U_{i} minimum and a maximum number of working hours allowed for worker i in entire.
planning horizon T i.e. a year.

RE_{ij} relative efficiency of worker i on task j.

WT_{ij} worker i willingness to work on task j.

WW_{it} worker i willingness to work in week t.

A_{ijt} an effective number of working hours by worker i on task j during week t.

A_{it} an effective number of working hours by worker i during week t.
i.e. $\{{A}_{it}={\displaystyle {\sum}_{j\in J}{A}_{ijt},\hspace{0.33em}\hspace{0.33em}i\in I,\hspace{0.33em}\hspace{0.33em}t\in T}\}$

A_{i} the number of effective working hours by worker i during planning period T.
i.e. $\{{A}_{i}={\displaystyle {\sum}_{j\in J}\hspace{0.33em}{\displaystyle {\sum}_{t\in T}\text{}{A}_{ijt},\hspace{0.33em}i\in I}}\text{}\}$

Y_{ijt} the number of working hours by worker i with full efficiency on task j.
during week t.

Y_{it} the number of working hours by worker i with full efficiency during week t.
i.e. $\{{Y}_{it}={\displaystyle {\sum}_{j\in J}{Y}_{ijt},\hspace{0.33em}i\in I,\hspace{0.33em}t\in T}\}$

S_{j} maximum relative capacity shortage of workers with respect to demand D_{j} for task j

S_{jt} capacity shortage of workers for task j in week t.

α_{j} the weight associated with the maximum relative capacity shortage of task j.

β_{j} the weight associated to the sum of the maximum and average relative capacity shortage of each task j.

Z the weighted sum of relative capacity shortages Z_{j} for all tasks.
4.1 Mathematical Programming Models
The mathematical programming models of above explained problem are as follows:
M1 model:
This model minimizes the new objective considering the relative efficiency of workers on each task.
Objective function:
$\text{Min}:Z={\beta}_{j}\times {Z}_{j}$
where,
$\sum}_{j\in J}\text{}{\beta}_{j}=1$
and
${Z}_{j}={\alpha}_{j}\times {S}_{j}+(1{\alpha}_{j})\times \frac{{{\displaystyle \sum}}_{t\in T}\text{}\frac{{S}_{jt}}{{D}_{jt}}}{\text{total}\hspace{0.33em}\text{weeks}}$
Subject to,
Maximum relative capacity shortage:
The maximum relative capacity shortage is greater than the weekly relative capacity shortages.
Annual working hour constraint:
The Annual Sum of weekly working hours to be allotted to each worker with full efficiency is bounded by the minimum number of working hours and the maximum number of working hours which includes overtime hours.
Weekly working hour constraint:
Weekly working hours to be allotted to each worker with full efficiency is bounded by the minimum number of working hours and the maximum number of working hours.
Demand constraint:
The number of working hours allotted to the workers for each task and for each week added to the shortage hours must be greater and equal to the forecasted demand.
Effective hours constraint:
A_{ijt} is the number of effective working hours by the worker i for each task j and week t after considering their relative efficiency.
Average working hour constraint:
It imposes an upper bound on the average weekly working hours for any consecutive L working weeks.
Nonnegativity restrictions:
and
The rest of the three models M2, M3, and M4 are same as M1, except the change in demand and effective hours constraint.
M2 model:
Demand constraint:
The number of working hours allotted to the workers, after including their efficiency and willingness to work in each week added to the shortage hours, must be greater and equal to the forecasted demand.
Effective hours constraint:
${A}_{ijt}$ is the number of effective working hours by the worker i for each task j and week t after considering relative efficiency and willingness to work in each week.
M3 model:
Demand constraint:
The number of working hours by the workers, after including their efficiency and willingness to work on each task added to the shortage hours, must be greater and equal to the forecasted demand.
Effective hours constraint:
A_{jt} is the number of effective working hours by the worker i for each task j and week t after considering relative efficiency and willingness to work on each task.
M4 model:
Demand constraint:
The number of working hours allotted to the workers, after including their efficiency, willingness to work on each task and week added to the shortage hours, must be greater and equal to the forecasted demand.
A_{ijt} is the number of effective working hours by the worker i for each task j and week t after considering relative efficiency, willingness to work on each task and week.
5. EXPERIMENTAL STUDY
To show the applicability and decision aspects of our proposed models, we consider a hypothetical manpower planning problem. Suppose a production firm needed workers for two tasks (Task1 and Task2) for a period of a year. The demand for each task in terms of working hours is forecasted with different patterns and presented in Figure 1. Demand for Task1 and Task2 follows different patterns. A staff of 13 workers is considered to fulfill the total demand. These workers are crosstrained with the different relative efficiency of working on each task. To provide flexibility to the staff, the firm asks the workers for their willingness to work on each task and in each week. The firm restricts the workers by allowing a maximum of 3 weeks for 0.3 (30%) and 4 weeks for 0.8 (80%) willingness without holidays. Two consecutive holiday weeks are allowed in summer and winter season only without willingness to work in particular week. The firm needs the following results for better manpower planning in an AH environment:

(1) To schedule workers throughout the year.

(2) The firm wants to minimize more the relative capacity shortage of Task2 as compare to Task1.

(3) To determine the relative capacity shortages, effective hours in each week, annual effective working hours, overtime and undertime by the employees.
To analyze these results four cases are considered:

Case1: It works on model M1 and takes account of only relative efficiency of workers.

Case2: It considers relative efficiency and willingness of workers to work in each week and uses model M2.

Case3: It considers relative efficiency as well as the willingness of workers to work on each task and uses model M3.

Case4: It utilizes model M4 and in addition of relative efficiency, workers willingness on each task, it considers workers willingness of working in each week of the year.
The data for the problem is presented below:

Number of Workers (I) 13 workers
Number of Tasks (J) 2 Tasks
Planning horizon (T) 52 weeks (approx. a year)
Minimum and maximum no. of weekly working hours 35, 50 hours
i.e. ${L}_{it},\text{\hspace{0.17em}\hspace{0.17em}}{U}_{it}\forall i\in I,\text{\hspace{0.17em}\hspace{0.17em}}t\in T$
Minimum and maximum no. of annual working hours 1920, 2100 hours
i.e. ${L}_{i},\text{\hspace{0.17em}\hspace{0.17em}}{U}_{i}\forall i\in I$
${\alpha}_{j}\forall j\in J$ 0.99, 0.99
${\beta}_{j}\forall j\in J$ 0.1, 0.90
Relative efficiency (RE) See Table 1
Workers willingness to work on each task (WT) See Table 1
Workers holiday weeks See Table 1
Workers willingness to work in each week (WW) See Table 2
${h}_{L}$ 44 hours
$L$ 12 weeks
The relative efficiency, holidays and willingness to work on each task with respect to the workers is presented in Table 1 and willingness to work in each week is in Table 2.
These models M1, M2, M3 and M4 are modeled in AMPL language and solved using BARON solver using NEOS server online facility provided by Wisconsin Institutes for Discovery at the University of Wisconsin in Madison for solving Optimization problem (see, Czyzyk et al., 1998;Dolan, 2001;Gropp and Moré, 1997;NEOS Server, 2016).
5.1 Results
The Proposed models are solved and the results are obtained. The proposed models minimize the demand and supply gap with preference to a particular task as compared to other tasks in an AH environment considering flexible working. The computation time by solvers is found acceptable and gives optimal results within seconds for all models. Since the decrease in the sum of relative capacity shortage and maximum relative capacity shortage gives an increase in service level. Here, for this problem, the decision maker is interested to minimize shortages of task2 as compared to task1. Since the data of results is too much, The results are presented through figures. The demand, supply, and shortage in each week for task1 and 2 is presented in Figure 2. Task1 shows high shortages while task2 have no shortages in all cases. As we move from case1 to case4, the capacity shortage in different weeks for task1 increases, while for task2 is zero. In case1 during some weeks, the supply of working hours is higher than the demand. Total annual scheduled allocated working hours, Overtime and undertime by each employee is given in Figure 3. The negative values are the undertime hours by the workers, i.e. below contracted 1920 hours. As flexibility increases effective hours decreases by increasing the undertime i.e. lesser than 1920 hours. Figure 4 shows the capacity shortage in each week for all cases. The maximum variation in capacity shortage can be seen in case4 while minimum variation lies in case1. In some weeks capacity shortage of task1 is high for case1 as compared case2 and case3 as compared to case4, although the total annual capacity shortage increases from case1 to case4. The values of maximum relative capacity shortages and the annual sum of relative capacity shortages is presented in Figure 5. It shows that as the flexibility to working increases from case1 to case4, there is an increase in maximum relative capacity shortages and total relative shortages over the year. In Figure 6, there is a fewer increase in total annual capacity shortages from case1 to case2 but a high increase from case2 to case3 and case4. It shows that willingness to work in weeks do not affect the shortages more as compared to the willingness of task. There is a higher increase when we consider the willingness of task (case3). The scheduled working hours on each task for case1, 2, 3 and 4 are presented in Appendix I, II, III and IV.
where, $Max\text{\hspace{0.17em}}S1={S}_{1}$, $Max\text{\hspace{0.17em}}S2={S}_{2}$, $RESUM1={\displaystyle {\sum}_{t\in T}\text{}\frac{{S}_{1t}}{{D}_{1t}}}$, $RESUM2={\displaystyle {\sum}_{t\in T}\text{}\frac{{S}_{2t}}{{D}_{2t}}}$
6. CONCLUSION AND DISCUSSION
The contribution of this paper is to propose mathematical programming models of Annualized hours planning problem with a new objective and the integration of workers willingness to work on each task and week. This paper considers a new flexibility factor of willingness to work in weeks. Four mathematical programming formulations (M1, M2, M3, and M4) are presented in section 4. These proposed models are the best fit when there is a need for minimizing capacity shortage of multiple tasks with preference weights for different tasks. These models will help the decision makers to plan and schedule their workers in an AH environment with a lot of results of overtime, undertime, annual and weekly effective working hours by each employee, shortages in each week, to manage the flexible working of willingness etc.
Also, a hypothetical experimental study has been done to show the effectiveness of the proposed model. For better understanding, an analysis has been done on a problem of two task demand scenario through four cases. As the data is huge with respect to paper writing, the summary of the optimal solution obtained after solving the models are presented by figures and tables. Results of the problem show that as flexibility increases the relative capacity shortage increases from case1 to case4. In some weeks the total capacity shortage for case2 is high than case1 and case3 is high than case4. Only Task1 shows shortages, task2 do not show any shortage to all cases. Surely, flexibility helps employees to better manage their worklife balance but high shortages cost to employers. So, there is a need for proper control and balance of flexibility. Model1 performs best as do not considers the flexibility concept of willingness to work on tasks and in weeks. This work can be extended by considering stochastic or fuzzy uncertainties. Since uncertainty always lies in real life problems, so there is a need for planning and scheduling study in AH environment.