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 Strategic Direction, 2008;Future of work institute, 2012;Wheatley, 2016;RARC Report, 2015;Qin et al., 2015). Annualized hours (AH) 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 variations in workforce availability are due to illness, training, vacation etc. As high demand leads to hectic overtime and less demand leads to underutilization of workforce, there is a need for proper management. AH policy provides effective plans to manage better worklife balance, less overtime, improved working flexibility and minimising 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) and Britain (Rodriguez, 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 fulfil 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) reduced overtime by 94% and the use of temporary workers by 53%. Reduction in temporary workers gives improved service level. In Sanderson (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 an implementation of AH policy.
The layout of this paper is as follows, Section 2 discusses related literature and present contribution; Section 3 describes the problem and its mathematical programming formulation; Section 4 is the solution methodology; 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 AND CONTRIBUTION
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;Van der Veen et al., 2012, 2014;Sureshkumar and Pillai, 2012, 2013). Detailed Characterization and classification of the annualized working hours planning problems is presented by Corominas et al. (2004a).
Hung (1999a, 1999b) 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 literature summary of different solution methods along with consideration of uncertainty factor is presented in Table 1.
Zimmermann (1996) categorized uncertainty as stochastic and fuzzy uncertainty. Stochastic uncertainty occurs with the randomness of an event while fuzzy uncertainty appears when the information is vague, imprecise or ambiguous or when the information is not clearly defined. The concept of fuzzy set theory was first introduced by Zadeh (1965) and has been found extensive applications in different fields like operations research, control theory, management science and artificial intelligence etc. In 1970 Bellman and Zadeh (1970) developed a decision theory in fuzzy environment.
In mathematical programming, fuzzy mathematical programming (FMP) deals with fuzzy uncertainty. The fuzziness can appear in different forms i.e. with fuzzy inequalities, fuzzy objective function, both fuzzy inequalities and fuzzy objective function and fuzzy parameters. Fuzzy mathematical programming is further categorized as flexible programming, possibilistic programming and robust programming; see Cadenas and Verdegay (2006) and Tang et al. (2004). Flexible programming is applied when there is vagueness, the possibilistic approach is used when there is ambiguity and robust programming is applied when vagueness and ambiguity both occur simultaneously. Here in our Fuzzy Model of the AHPP we are using the flexible programming approach. Applications of fuzzy programming approach can be found in Lai and Hwang (1992), Pendharkar (1997), Rabbani et al. (2012), Selim and Ozkarahan (2008) and KhaliliDamghani et al. (2013a).
Fuzzy programming approaches are widely applied for solving multiobjective problems. Fuzzy approach measures the achievement degree of objective functions and helps the decision maker to decide the best efficient solution. Many approaches are available in literature Zimmermann (1978) developed the maxmin approach, Lai and Hwang (1993) proposed augmented maxmin approach and Selim and Ozkarahan (2008) presented modified Werner’s approach (1988).
Consequently, Torabi and Hassini (2008) (TH) proposes a better and efficient approach than previous approaches. It is the hybridization of Lai and Hwang (1993) and Selim and Ozkarahan (2008) methods. The Aköz and Petrovic (2007) method proposes a fuzzy goal programming method to solve multiobjective mathematical programming models, where the importance relations among goals is imprecise in linguistic terms, such as goal A is slightly or moderately or significantly more important than goal B. Previously, Aköz and Petrovic (2007) method was applied by Petrovic and Aköz (2008), Díaz et al. (2014), KhaliliDamghani et al. (2013b), KhaliliDamghani and Nezhad (2013), Tavana et al. (2013) and Khalili Damghani et al. (2014).
In this paper, we proposed a new modified fuzzy goal method with imprecise goal hierarchy and applied it on the multiobjective AHPP. This method is the hybridization of Torabi and Hassini (2008) and Aköz and Petrovic (2007) methods. The results are further compared with the Aköz and Petrovic (2007) method. We also extended the previous AHPP models to multiobjective model with fuzzy parameters and fuzzy goal hierarchies. Previously no work in the literature of AH planning models considered the fuzzy parameters and fuzzy goal hierarchies. Also, previous models in literature do not consider the proposed multiobjective model. This work fills this gap.
3. PROBLEM FORMULATION
In this paper, we proposed a multiobjective manpower planning model with annualized hours flexibility and optimizing the service level by taking into account workers working efficiency. 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 having different relative efficiency (RE) (scale of 1, for e.g. 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) associated to them are considered. The number of weekly working hours falls within an interval with upper and lower bound. Also, the total working hours for the planning period lies within an interval.
The characteristics of the problem are as follows:

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

2) Holiday weeks are considered 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) Overtime is permitted with an upper bound.

6) Hiring temporary workers is not allowed.

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

8) The Demand and overtime cost are considered fuzzy parameters.

9) The goals are assumed with fuzzy aspirations.

10) The priority between the goals is assumed as imprecise in linguistic terms.
The list of notations which will be used throughout the paper is given in Table 2.
The mathematical programming model of Multi Objective Annualized Hours Planning Model (MOAHPM) of the above explained problem is as follows:
MOAHPM:
Objective functions:
The objective functions ${Z}_{1j},\hspace{0.33em}j\in J$ minimizes the maximum and sum of relative capacity shortages for each task.
The objective function Z_{2} minimizes the maximum relative capacity shortage S, and sum of relative capacity shortages for all tasks.
Z_{3} minimizes the cost of overtime by each worker i.
Z_{4} minimizes the undertime (the number of hours worked below contracted hours) by each worker i∈I.
Subject to,
Maximum relative capacity shortage:
The maximum relative capacity shortage for each task j ∈ J is fcapacity shortages for each task.
The maximum relative capacity shortage for all the tasks j ∈ J is greater than the weekly relative capacity shortages for for each task.
Under and Overtime constraint:
The annual sum of working hours to be allotted to each worker with their relative efficiency must be equal to the minimum number of contracted working hours.
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
4. SOLUTION METHODOLOGY
The original multiobjective annualized hours problem model can be converted to APM and PMAPM by introducing the membership functions for the fuzzy goals and fuzzy relations. A defuzzification method is also introduced to determine the crisp value of fuzzy parameters.
4.1 Defuzzification of Fuzzy Parameters
The fuzzy parameters for our AH model are considered as fuzzy numbers. To address the defuzzification of fuzzy numbers we used the Yao and Wu (2000) signed distance method. The signed distance method for ranking fuzzy numbers can be explained as follows.
The signed distance of a triangular fuzzy number $\tilde{A}=({a}_{1},\hspace{0.33em}a,\hspace{0.33em}{a}_{2})$ and a trapezoidal fuzzy number $\tilde{B}=({b}_{1},\hspace{0.33em}b,\hspace{0.33em}{b}_{2},\hspace{0.33em}{b}_{3})$ is defined as,
Let $\tilde{A}$ and $\tilde{B}$ are two triangular or trapezoidal fuzzy numbers, their ranking relation is defined as $\tilde{A}\le \tilde{B}\hspace{0.33em}\iff d(\tilde{A},\hspace{0.33em}0)\le d(\tilde{B},\hspace{0.33em}0).$ The signed distance method for fuzzy numbers have similar properties to that of signed distance in real numbers.
4.2 Construction of Membership Functions
Fuzzy goal programming has the advantage that the decision maker is allowed to specify imprecise aspiration levels to the objectives. An objective with an imprecise aspiration level can be treated as a fuzzy goal. Let $Z=\{{Z}_{k}:k=1,\hspace{0.33em}\mathrm{...},\hspace{0.33em}K\}$ be a set of fuzzy goals, such that Z = BUC, where B and C represents maximization and minimization type fuzzy goals. The fuzzy goal programming model can be written as follows:
Here, OPT means to find an optimal solution x such that all fuzzy goals are satisfied. ⪆ and ⪅ implies essentially greater than or equal to and essentially less than or equal to repectively. a_{k} is the aspiration level for k^{th} goal Z_{k}. Ax ≤ b represents the constraints in vectors. Since, Our problem is multiobjective so we will apply the fuzzy goal programming as a solution approach. Fuzzy aspiration levels are either provided by the decision maker or determined from the payoff table. The payoff table is obtained by solving the defuzzified model separately for each objective and calculate other objectives for each solution.
Let us assume U_{k} and L_{k}, $k=1,\hspace{0.33em}\mathrm{...},\hspace{0.33em}K$ as the upper and lower limits for minimization and maximization type fuzzy goals respectively with a_{k} as their aspiration levels, then the membership function (μ_{k}) for objective the function Z_{k} can be modeled as:
For maximization type fuzzy goal we have:
For minimization type fuzzy goal we have:
When there lie fuzzy relations in linguistic terms like slightly or significantly or moderately more important then we employ the Aköz and Petrovic (2007) model which gives a fuzzy goal method with imprecise goal hierarchy. Basically three fuzzy relations ${\tilde{R}}_{1}(k,\hspace{0.33em}l),\hspace{0.33em}{\tilde{R}}_{2}(k,\hspace{0.33em}l)\hspace{0.33em}\text{and}\hspace{0.33em}{\tilde{R}}_{3}(k,\hspace{0.33em}l)$ for k^{th} goal is slightly more important than l^{th} goal, k^{th} goal is moderately more important than l^{th} goal and k^{th} goal is significantly more important than l^{th} goal are used with membership functions ${\mu}_{{\tilde{R}}_{1}(i,j)},\hspace{0.33em}{\mu}_{{\tilde{R}}_{2}(i,j)}\hspace{0.33em}\text{and}\hspace{0.33em}{\mu}_{{\tilde{R}}_{3}(i,j)}$ respectively.
4.3 Solution Models for Fuzzy Goal Programming with Imprecise Goal Hierarchies
Now, we will explain the AP and Modified AP model for imprecise goal hierarchy. The following fuzzy Goal programming model with fuzzy hierarchies as in Aköz and Petrovic (2007) is formulated as:
AP model (APM):
objective1: It maximizes the sum of achievement levels of all the fuzzy goals Z_{k}.
objective2: It maximizes the sum of membership grades for the linguistic preference relations.
A convex combination of the two objectives by applying the respective weights α and (1−α ) to obtain a compromise solution between the two objectives becomes:
subject to
Here in AP model constraint (30) and (31) represents the linear membership functions for maximization and minimization type fuzzy goals whereas, constraints (32) to (34) are for the achievement levels of fuzzy imprecise relations. The value of b_{kl} = 1, when a fuzzy relation exists otherwise its value is zero. In constraint (36) and (37), the non negativity restriction to variable x and bound to membership functions is presented.
Proposed Modified AP model (PMAPM)
The proposed model is obtained by hybridization of Torabi and Hassini (2008) approach with Aköz and Petrovic (2007) method. The objective 1 in AP model only maximizes the sum of satisfaction levels of objectives, it does not consider the maximization of minimum satisfaction level. By integrating the Torabi and Hassini (2008) approach in objective 1, we get more flexibility as it considers both the minimum satisfaction and sum of satisfaction of all the objectives.
objective1
objective2:
Convex combination of the two objectives becomes:
subject to
Rest of the constraints (30~37) are same as APM.
Here, $\lambda =\mathrm{min}\left\{{\mu}_{k}\right\}$ denote the minimum satisfaction level of objectives. γ is the weight associated to minimum achievement level (λ). This formulation has a new achievement function defined as a convex combination of the lower bound for satisfaction degree of objectives (λ ), and the sum of achievement degrees of objectives μ_{k} to ensure an adjustable balanced compromise solution. The PMAPM will become APM when γ = 0. Hence it gives wider applicability to the decision maker by considering extended objective function.
Also, γ controls the minimum satisfaction level of objectives as well as the compromise degree among the objectives implicitly. That is, the proposed formulation is capable of yielding both unbalanced and balanced compromised solutions for a given problem instance based on the decision maker’s preferences through adjusting the value of parameter γ . On the other side, the lower value for α means more attention is paid to obtain a solution with high satisfaction degree for fuzzy preference relations without any attention paid to the maximization of satisfaction degree of objectives (i.e. yielding unbalanced compromise solutions).
The solution procedure of the MOAHPM problem can be summarized as follows:

Step 1 At first defuzzify the fuzzy parameters to crisp form using signed distance method.

Step 2 Solve the MOAHPM for all objective functions as a singleobjective problem subject to the constraints.

Step 3 Construct the payoff table of all objectives for each solution obtained and determine the U_{k} and a_{k}k = 1, ⋯, K from it with consultation of the decision maker.

Step 4 Determine the membership functions for the objectives as well as the fuzzy relations between the objectives, for the obtained values of U_{k} and a_{k}.

Step 5 Now, Model the MOAHPM as APM and PMAPM.

Step 6 Solve the Problem.

Step 7 Stop, Solution is obtained.
5. EXPERIMENTAL STUDY
To show the applicability and decision aspects of our proposed models, we consider a hypothetical manpower planning problem. Because of the lack of availability of real data as per the model we consider hypothetical demand data. The data for the parameters may be changed as per the requirement. Suppose a production firm need to schedule 20 workers for three tasks (task1, task2 and task3) for a period of a year. The demand for each task in terms of working hours is forecasted and presented in Figure 1. Demand for task1, 2 and 3 follows different patterns. The fuzzy demand data is given in Table 5. The workers are assumed crosstrained with the different relative efficiency of working on each task. Two consecutive holiday weeks are allowed within six months. The firm needs to schedule workers throughout the year considering their holiday weeks and relative efficiency of working on each task for better manpower planning in an AH environment, also to determine the relative capacity shortages, effective hours in each week, annual effective working hours, overtime cost, overtime and undertime by the employees.
The firm provides imprecise relation in linguistic terms between the objectives. As we have three tasks then three objectives are of ${Z}_{1j}$ type (${Z}_{11},\hspace{0.33em}\hspace{0.33em}{Z}_{12},\hspace{0.33em}\hspace{0.33em}{Z}_{13}$), rest of the objectives are ${Z}_{2},\hspace{0.33em}\hspace{0.33em}{Z}_{3}\hspace{0.33em}\text{and}\hspace{0.33em}{Z}_{4}$. The firm provides the relation between the objectives linguistically i.e. ${Z}_{11}$ is slightly more important than ${Z}_{13},\hspace{0.33em}\hspace{0.33em}{Z}_{13}$ is moderately more important than ${Z}_{12}$ and ${Z}_{3}$ is significantly more important than ${Z}_{2}$. The demand for different tasks considered as trapezoidal fuzzy number ${\tilde{D}}_{jt}=({D}_{jt}^{a},\hspace{0.33em}{D}_{jt},\hspace{0.33em}{D}_{jt}^{b},\hspace{0.33em}{D}_{jt}^{c})$ is presented in Table 5. Overtime cost is assumed as triangular fuzzy number , relative efficiency and holiday weeks for each worker is presented in Table 4. The values of and , may be selected as the worst and best value for each objective determined from the payoff table or may be set by the decision maker. Now modeling the problem as MOAHPM and then to APM and PMAPM, we apply the solution algorithm to get the results. The data for the problem is presented in Table 3. The defuzzified values for fuzzy demand and overtime cost parameter becomes:
These models are modeled using AMPL language and solved by 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, 2016).
5.1 Results
This section presents the results obtained by applying the AP and Proposed AP model on the multiobjective annualized hours manpower planning model. The results are validated and evaluated through the experimental study. Table 6 presents the payoff between Z11, Z12, Z13, Z2, Z3, and Z4 objectives. Payoff table is determined by solving separately for each objective then values for all objectives are determined for the solution obtained. It gives the best and worst values of objectives to the decision maker. X_{k}, k = 11, 12, 13, 2, 3, 4 represents the solution when single objective problem model is solved for Z_{k} and value of other objectives are determined.
The values for all the objectives, membership functions and their sum, sum of overtime and under time, maximum and sum of relative capacity shortages for all the tasks and capacity shortages in hours for APM and PMAPM is presented in Table 7. The results are tabulated for different values of α for APM and different values of α and γ for PMAPM. The decision maker can change the values of these parameters and aspiration levels for the objectives on his/her preferences.
For better understanding and efficiency of solution methods, results are presented in Figures 25. Figure 2 display that the fuzzy parameter λ increases as the priority level γ increases. The maximum value (λ = 0.6891) is obtained for γ and λ. In Figure 5 we can see the trend for sum of satisfaction level for all the objectives and for fuzzy relations of APM to PMAPM. The fuzzy relations between the objectives are fully satisfied for α = 0.3 and then decreases from α = 0.60.99 for both APM and PMAPM methods. The maximum value 5.0606 of sum of achievement degrees of objectives is obtained in APM for α = 0.99 and minimum 5.0585 is by PMAPM for α = 0.99, γ = 0.99. In PMAPM, we are maximizing satisfaction level of all the objectives and minimum level of achievement while in APM it only aims at maximizing the sum of achievement levels of all objectives. As the priority level α increases, the sum of satisfaction level for fuzzy relations in PMAPM and APM decreases. The maximum relative capacity shortage increases while sum of relative capacity shortage and total shortage decreases from $\alpha =0.3$ in APM to $\alpha =0.3$ $\gamma =0.30.99$ for PMAPM. The trend of satisfaction level of objectives and fuzzy relations is presented in Figure 3 and 4.
On observing the APM objectives, for $\alpha =0.3$ we find that Z_{11}, Z_{13}, Z_{3} and Z_{4} gives their best value, whereas Z_{12} and Z_{2} reaches worst values of payoff matrix. For $\alpha =0.99$ , PM considers maximization of satisfaction level of all the objectives and provides a balanced solution. On comparing APM and PMAPM, PMAPM gives better balanced solution by considering all the objectives and minimum achievement level. For $\gamma =0$ , the PMAPM becomes APM model. It gives flexibility to the decision maker. For lower α and higher γ, results for APM and PMAPM are same but for high value of α and γ, the result in PMAPM changes as compared to AP model. This is due to the increase in weight associated to the minimum achievement level (λ). Also, the computation time by solvers is found acceptable and gives optimal results within seconds for all model. Since the data of results is too much, The results are presented in brief through figures and tables.
6. CONCLUSION
We have considered the problem of manpower planning with annualized hours flexibility. AH is a method of distributing working hours with respect to the demand over a year. The contribution of this work is to propose Multiobjective Manpower planning model with Annualized hours flexibility in a fuzzy environment. These models will help the decision makers to plan and schedule their workers in an AH environment. Demand and Overtime cost are considered as fuzzy parameters. The imprecise linguistic preference to fuzzy goals is modelled and two solution methods APM and PMAPM are applied. A new hybridized fuzzy goal programming method is presented and the results are compared with Akoz and Petrovic (2008) method. These models are the best fit when there is a need for minimizing capacity shortage of multiple tasks with imprecise linguistic preference hierarchies.
Finally, An experimental study is performed to validate the manpower planning model and analyze the effectiveness of the proposed hybridized method. The proposed method is the hybridization of Torabi and Hassini (2008) and Aköz and Petrovic (2007) method. It provides a better compromise solution for the Multiobjective AH planning problem. It has a new achievement function defined as a convex combination of the lower bound for satisfaction degree of objectives (λ), and the sum of achievement degrees of objectives μ_{k} to ensure an adjustable balanced compromise solution. The Hybridized model (PMAPM) will become AP model when γ=0. Hence, the proposed method provides more information and wider applicability to the decision maker for better and efficient decision making by considering extended objective function.
Previously no work in the literature of AH planning models considers fuzzy parameters and fuzzy goal hierarchies. Also, Previous models in literature do not consider the proposed multiobjective model. This paper fills this gap. This work may be further extended by considering multilevel fuzzy and stochastic goal hierarchies. Since uncertainty always lies in real life problems, so there is a need of studying AH planning and scheduling in an uncertain environment.