A MANUFACTURING STRATEGY APPROACH FOR SOLVING AGGREGATE PRODUCTION PLANNING PROBLEMS WITH MULTIPLE OBJECTIVES

ABSTRACT

Aggregate Production Planning (APP) aims at balancing the forecast demand for product families with the available production resources, thus allowing changes to be made in the system and the best action strategy to be found. APP decisions are strategic decisions for the organization, which take an intermediate time horizon which spans 12 months, in general. As different issues may influence business success, this kind of decision problem will take multiple objectives into account, when a strategic view is considered. Conflicts between objectives frequently increase the complexity of APP problems. In order to deal with this kind of problem, this paper proposes a multi-criteria decision model considering the five objectives related to manufacturing strategy, namely: quality, speed, dependability, flexibility, and cost. By applying a multi-objective optimizing model which considers the organization’s strategical objectives, and by using the NSGA-II algorithm, a subset of representative solutions of the Pareto Frontier is obtained, thereby enabling the application of the FITradeoff method. A multi-criteria method, such as the FITradeoff, assists Decision Makers (DMs) in choosing a production plan more suitable to their objectives. The results show that the FITradeoff method contributes to the evaluation of solutions by aiding the DM to consider a significant set of alternatives while at the same time reducing the cognitive effort that he/she needs to make.

Keywords:
aggregate production planning; manufacturing strategy; FITradeoff method; Multi-Criteria Decision-Making (MCDM); NSGA-II

1 INTRODUCTION

Aggregate Production Planning (APP) seeks to identify the best way to use production resources to cover fluctuations in demand (Slack 1992; Hill 1993; Slack et al., 2003). According to Wheelwright (1984), APP aims to balance the forecast demand using production resources that are available to the company. In this context, APP decisions try to change the production system to balance demand fluctuations.

Different strategies may be pursued depending on the company and the specificities of the environment. According to Nam & Logendran (1992), since 1950, APP problems have been discussed in the literature to evaluate different strategies to balance demand fluctuations. Buffa & Taubert (1972) present the basic strategies used, namely: adjust the production rate over or under time; adjust the workforce, use inventory, and allow lost sales.

Thus, generally, the most common strategies involve decisions regarding the workforce level in each period, considering production capacity in regular and overtime production. Another strategy considers using inventories over certain periods to cover seasonality and fluctuations in demand. In addition, subcontracting production resources is also considered to obtain the balance.

According to Yu et al. (2022), one or a combination of strategies can be adopted to respond to demand fluctuation. The authors also mentioned overtime time production rate adjustment, workforce size regulation, level production plus inventory policy, or sales shortfalls. In Jamalnia et al., (2019) several strategies have been also compared to evaluate the performance of different aggregate production planning (APP) problems.

All these strategies involve an intermediate horizon of time, which will remain in use for 6 to 12 months. APP decisions are taken in the medium term, connecting decisions concerning Capacity and Production Programming and Control (PPC) (Slack 1992; Hill 1993; Slack et al., 2003).

In general, companies always try to adopt strategies that minimize operational costs (Yu et al., 2022; Liu & Yang 2021; Jamalnia et al., 2019 Al-E-Hashem et al., 2011). Companies should reduce their operational costs since such an improvement in performance could result in a rise in profits and also the possibility of charging lower prices. Many companies seek to keep an advantage by charging low prices, which is especially true in the case of commodities, raw materials, and low-added-value products.

However, due to the dynamic nature of APP, various objectives are involved in these problems (Jamalnia et al., 2019). Considering only minimum-cost strategies is not sufficient to achieve sustained competitive advantage over time. It has been observed that customer satisfaction does not depend only on price but takes into account many factors.

Therefore, other objectives are also relevant to customer satisfaction. For example, objectives may be analyzed so as to determine order winners, which shall be maximized, and qualifiers, which shall be met at a defined level (Slack 1992; Hill 1993; Slack et al., 2003). This is the case for commodity producers who have been trying to add more value to their products, and therefore frequently associate different services with their products. In such scenarios, offering more than low prices is fundamental to attracting and keeping customers.

In this context, some techniques have been applied to support APP decisions considering multi-objective context (Yu et al., 2022; Liu & Yang 2021; Jamalnia et al., 2019, Rasmi et al., 2019; Al-E-Hashem et al., 2011; De Almeida Filho et al., 2006; Wang & Fang 2001). Nam & Logendran (1992) present a literature review about APP problems. In the study, several techniques are discussed to support this kind of problem. The authors classify them into optimal solution techniques and near-optimal solution techniques.

In the context of optimal solution techniques, optimization techniques are the major representant. In Al-E-Hashem et al. (2011), an optimization model has been developed which three objective functions to deal with an APP problem. In Rasmi et al., (2019) multi-objective APP model has been developed to analyse economic, social, environmental, and cultural pillars, an optimization technique for multi-objective mixed-integer linear programs (MOMILP) has been applied to solve this problem. Genetic Algorithm (GA) has been also applied to support APP decisions in a multi-objective context. In Liu & Yang (2021), a bi-objective model which includes cost versus workforce stability has been constructed and solved an APP problem using NSGA-II (Nondominated Sorting Genetic Algorithm II - Deb et al 2002). In addition, fuzzy multi-objective programming has been widely used to deal with APP problems (Jamalnia et al., 2019; Wang & Fang 2001, Tang et al., 2000; Satyadas & Chen 1992).

Another technique used is the Multi-Criteria Decision Making/Aiding (MCDM/A) approach. As different issues may influence business success, this kind of decision problem will take multiple objectives into account, when a strategic view is considered. Conflicts between objectives frequently increase the complexity of APP problems. Thus, the use of MCDM methods to support APP decisions can be very interesting for this field since MCDM methods permit the evaluation of strategies (or alternatives) considering multi-objectives, which are also present in the APP context. In addition, these methods seek to collect Decision-Makers’ (DMs) preferences and introduce them into mathematical models to conduct a rational decision process (Keeney & Raiffa 1976, Belton & Stewart 2002, de Almeida et al., 2015).

MCDM/A approach has been considered to support problems in the manufacturing context. For instance, Duvivier et al., (2013) and Pergher et al. (2020) applied MCDM/A methods to support industrial scheduling problems. Badri et al., (2014) used the TOPSIS method to solve the product mix problem. Uhde et al. (2015) performed a review study to investigate MCDM/A methods used in forest management planning. However, concerning to APP problems a few studies have been observed until now in the literature. Yu et al. (2022) considered the TOPSIS method (Hwang & Yoon, 1981) to solve a bi-objective problem. Jamalnia et al. (2019) also discuss different the use of MCDM/A methods to support APP problems.

One could claim that there is a possibility of improvement on those models considering that APP is part of the manufacturing strategy (Hill 1993; Slack et al., 2003) process. In manufacturing strategy, a set of strategic objectives are contemplated allowing an alignment with the top objectives of the production system.

In this context, this study applied an MCDM method to solve an APP problem considering five objectives from the manufacturing strategy: quality, speed, dependability, flexibility, and cost (Hill 1993; Slack et al., 2003). Therefore, the contribution of this study remains in using manufacturing strategy objectives, since previous studies do not consider these objectives, focusing on operational objectives of APP.

To solve the multi-criteria decision model proposed using the considering the five objectives of manufacturing strategy, the NSGA II (Deb et al 2002) has been applied to obtain the nondominated solutions. After that, the FITradeoff method (de Almeida et al., 2016; de Almeida et al., 2021), which is an MCDM/A method, has been applied to obtain the production plan based on the preferences expressed by the Decision-Maker.

As FITradeoff deals with discrete and stationary sets of alternatives, in the APP decision context what must be done first is to identify possible solutions. In addition, this study combines a multi-objective model with the NSGA II algorithm in order to obtain non-dominated solutions. After that, these non-dominated solutions have been evaluated considering the DM preferences in the FITradeoff DSS. As result, a production plan more suitable to DM objectives has been obtained.

This paper is organized as follows. First, section 2 discusses how the multi-objective model has been developed considering the five objectives of the manufacturing strategy. Section 3 introduces the conceptual formulations of the NSGA II the FITradeoff method which are the methods applied to solve the APP problem and its latest methodological improvements. Section 4 presents some aspects related to implementing NSGA II and the application of the proposed approach using the FITradeoff method for solving the APP problem. Finally, some conclusions are drawn in Section 5 and suggestions for future lines of research are made.

2 USING MANUFACTURING STRATEGY FOR A MULTI-CRITERIA AGGREGATE PRODUCTION MODEL

Since 1950, APP attracted considerable attention from DMs. As discussed in the introduction, different methodologies have been applied to deal with APP problems. Multi-criteria aggregate production planning has been considered in previous works (Yu et al., 2022; Liu & Yang 2021; Jamalnia et al., 2019 Al-E-Hashem et al., 2011), although not much of the manufacturing strategy has been applied to assign multiple objectives.

In Yu et al (2022), three objectives have been considered. The objectives are to maximize sales revenue, minimize total production costs, and minimize repair costs. In Liu & Yang (2021), a bi-objective model has been established to minimize total production costs and instability in the workforce. In Jamalnia et al., (2019), several objectives have been considered in the multi-objective decision-making model. These objectives are related to strategies to deal with the APP problem, such as maximizing total revenue, minimizing total production costs, minimizing total labour productivity costs, minimizing total costs of the changes in workforce level, and maximizing customer satisfaction. In Al-E-Hashem et al., (2011), also three objectives have been considered in the decision model. The first objective function is the sum of the expected total cost of the supply chain, the second one concerns to customer satisfaction approach, and the last objective involves maximizing workers’ productivity.

Therefore, in the context of manufacturing strategy, these studies consider only the objective of minimizing production costs. According to Rasmi et al., (2019), APP models traditionally aim to optimize monetary issues only. Among the commonly applied objectives, what the models and the approaches in the literature most frequently consider are costs (Slack 1992; Hill 1993; Slack et al., 2003).

The sum of production-related costs evaluates the total cost of production (costs of raw materials, workforce, energy supply, etc.). It is desirable to minimize it. The conceptual formulation for this objective function is illustrated in (1). Depending on the objective k, the alpha coefficients assume different meanings. In equation (1), r _t represents the regular time production, o _t represents the overtime production, and s _t represents the subcontracting production in period t. These parameters will be explored in Section 4.

$minOb{j}_k=\sum _{t=1}^n\left({\alpha }_r{r}_t+{\alpha }_o{o}_t+{\alpha }_s{s}_t\right)$ (1)

Therefore, objectives function in the format of (1) compose multi-objective aggregate production planning models (Yu et al., 2022; Liu & Yang 2021; Jamalnia et al., 2019 Al-E-Hashem et al., 2011).

The main contribution of this study has been to consider not only the cost, but the other four objectives related to manufacturing strategy, namely: quality, speed, dependability, and flexibility (Hill 1993; Slack et al., 2003). In de Almeida-Filho (2006) three out of five objectives related to manufacturing strategy have been considered.

Production management is a central factor for a company’s success since it assists companies to keep their products competitive in the market. Hence, production management decisions are related to the business strategy of all organizational levels.

APP problem should be integrated with the manufacturing strategy approach, in which strategic objectives are developed to fulfil this and other managerial decision problems. Corporative objectives should influence business objectives which will have been described in terms of operational objectives. Operational objectives are used to measure the management (industrial) system performance in terms of quality, flexibility, speed, dependability, and cost for any type of operation. These objectives aim to represent market requirements in operational statements, being considered as performance dimensions or strategic dimensions (Slack 1992; Hill 1993; Slack et al., 2003)

Therefore, the aim of this study has been to consider these objectives to solve an APP problem, since this kind of decision must be made by considering the company’s strategy. Figure 1 illustrates the aim of this study. Hence, as illustrated, the production plan variables are evaluated against several objectives, and as solution, the best Aggregate Planning Production (APP) is obtained.

Figure 1
Decision Model proposed in the study.

The decision model presented in Figure 1 has been worked out using the NSGA-II to obtain the Pareto Frontier solutions. After that, the FITradeoff method, which is an MCDM/A method, has been applied to do the preference modelling of the strategic objectives, which is essential for obtaining the production plan aligned with the strategies of the organization.

The next section presents the conceptual formulations of these methods. Then, in section 4, the basic formulation and application of this view for the APP problem, considering the manufacturing strategy is presented.

3 METHODS APPLIED

3.1 Genetic Algorithm to Find Pareto Optimal Solutions

As commented in the previous section, in this study, a subset of non-dominated alternatives was first obtained using the Nondominated Sorting Genetic Algorithm II (NSGA II - Deb et al 2002). This algorithm is considered an efficient way of obtaining multi-objective problem solutions.

While using GAs a few steps shall be executed:

• I. Generating the initial population: a population with n individuals, n being the desired number of solutions, is initially generated.

• In this study, the solutions were generated randomly but a restriction on the demand meeting was imposed. In short, solutions that did not have the demand fully met were not considered. In addition, the units of each production regime were illustrated in (2) (this restriction was considered due to aspects related to the company context). In equation (2), r _t represents the regular time production, ot represents the overtime production, and st represents the subcontracting production in period t. These parameters will be explored in Section 4.

$r_t>o_t>s_t$ (2)

• II. Fitness evaluation: as mentioned, the NSGA II was applied, so the evaluation is made based on non-dominated sorting (in which a ranking of the solutions is obtained) and crowding distances. In order to build the ranking of the solutions, a few domination rules shall be considered.

• III. Generating new sets of solutions based on genetic operators: the crossover and mutation operators to combine the solutions and create new populations must be defined. In this study, the simulated binary crossover operator and the alternate random mutation were applied, which modify a gene considering the vicinity of the original solution. A probability p is associated with the crossover operator, whereas (1-p) is associated with the mutation operator.

• The steps of the simulated binary crossover are as follows.

1. Select two parents I ₁ and I ₂;

2. Generate a number λ in the range [0;1) randomly;

3. Select the distribution index η_c , which in this case was defined as 1 to allow the children to be far from the parents;

4. Calculate β value;

$\beta \left\{\begin{array}{l}{\left(2\lambda \right)}^{\frac{1}{{\eta }_{c-1}}},\lambda \le 0.5\\ {\left(\frac{1}{2\left(1-\lambda \right)}\right)}^{\frac{1}{{\eta }_{c-1}}},\lambda >0.5\end{array}\right.$ (3)

• (e) Generating the children;

$C_1=0.5\left[\left(1+\beta \right)I_1+\left(1-\beta \right)I_2\right]$ (4)

$C_2=0.5\left[\left(1-\beta \right)I_1+\left(1+\beta \right)I_2\right]$ (5)

• IV. Selecting the survival population: at the end of each cycle a population of cardinality n shall be selected to continue the process. In this case, the selection is made considering the ranking levels. As suggested by Deb et al (2002), the algorithm selects the solutions level by level. If a level cannot be fully included in the population, the solutions with the bigger crowding distances shall be selected to preserve diversity.

Note that this is a constrained problem, which makes the process a bit more complex. Deb et al (2002) suggests considering a different concept of dominance when dealing with constrained problems. This concept admits that a solution si dominates a solution sj if one of the following relations is verified:

1. s_i is a feasible solution and s _j is infeasible;

2. should both of them be infeasible, s _i has a smaller constraint violation;

3. or if both of them are feasible solutions, s _i has better performances than s _j for all the considered objectives.

3.2 FITradeoff Method

The Flexible and Interactive Tradeoff (FITradeoff) method (de Almeida et al., 2016; de Almeida et al., 2021) is used to elicit the scaling constants in the context of Multi-Attribute Value Theory (MAVT - Keeney & Raiffa, 1976). Scaling constants are parameters of the additive model which are difficult to elicit, representing a challenging task in MCDM/A approach (Keeney & Raiffa, 1976).

In this context, the FITradeoff method was considered an important innovation since it keeps the axiomatic structure of the Tradeoff procedure (Keeney & Raiffa, 1976) allowing both linear and non-linear value functions and does not require that DMs defines the scaling constants directly, as has been done in methods which use the Swing procedure (Edwards & Barron, 1994). In addition, the method incorporates partial information concepts, thus DMs do not need to provide indifference points to obtain the solutions reducing DMs cognitive effort (de Almeida et al., 2016, da Silva et al., 2022).

Moreover, the FITradeoff method combines the two perspectives for preference modelling - elicitation by decomposition and holistic evaluations (de Almeida et al., 2021). In the former, consequences are compared in pairs. In the later, actions/alternatives are compared in pairs or in groups with more than two alternatives. Thus, DMs can combine both perspectives during the decision process, expressing preferences using those that feels more comfortable.

This feature provides flexibility for the FITradeoff method since DMs have different ways to interact and provide their preferences. In addition, each preference expressed during the process is included in the LPP model as a constraint. Thus, after each piece of information is provided, the method updates the partial results, which also allows DMs to decide if and how to continue the decision process.

The FITradeoff method was first proposed to solve choice problems which consists of selecting the best alternative from a set of candidates by considering multiple and possibly conflicting objectives (de Almeida et al., 2016). For instance, real decision problems have been solved using the FITradeoff method for choice problematic (Camilo, et al., 2020; Fossile et al., 2020; Carrillo et al., 2018). In addition, this study considers the FITradeoff method for choice problematic since it intends to obtain the best aggregate production plan by considering the DM’s preferences.

When dealing with choice problematic, the method looks for potentially optimal alternatives (POAs). An POA is an alternative that are at least as good as all the other ones for at least one scaling constants vector. The use of the potential optimality concept is necessary because, as the method does not require indifference points, it is not possible to specify a unique scaling constants vector, and so, throughout the elicitation procedure, there are several possible vectors to represent DM’s preferences. The existence of these vectors means that there are functions that can represent DM’s preferences and while he/she provides more information, the set of possible functions becomes narrower and narrower, and so it may occur that the most indicated solution for the problem becomes the same for all the set of possible aggregative functions, thus finalizing the decision problem.

FITradeoff implements a linear programming model to assess the possible aggregative functions. The linear programming model is solved at each interaction for each alternative that belongs to the set of POAs, thereby updating the partial results. At the very moment that the set of POAs contains a unique alternative, the decision process is finished, and the result is presented to the DM. This model is illustrated by equations (6) to (14) and it is discussed below.

$MaxV\left(A_i\right)=\sum _{j=1}^m{k}_j{v}_j\left(a_{ij}\right)$ (6)

$s/t{k}_1\ge k_2\ge ...\ge k_m$ (7)

$k_j{v}_j\left(x_j^{\text{'}}\right)>k_{j+1}$ (8)

$k_j{v}_j\left(x_j^{\text{'}\text{'}}\right)<k_{j+1}$ (9)

$\sum _{j=1}^m{k}_j{v}_j\left(a_{ij}\right)\ge \sum _{j=1}^m{k}_j{v}_j\left(a_{kj}\right)i\ne k$ (10)

$\sum _{j=1}^m{k}_j{v}_j\left(a_{zj}\right)>\sum _{j=1}^m{k}_j{v}_j\left(a_{ij}\right)z\ne l$ (11)

$k_j\ge 0$ (12)

$\sum _{j=1}^m{k}_j=1$ (13)

$\begin{array}{ccc}j=1,...,m& i=1,...,n& k=1,...,n\end{array}$ (14)

The objective function of the model is presented in (6), which consists of maximizing the alternative i overall value, in which a _ij represents the performance of the alternative, evaluated against the criterion j; v _j (a _ij ) is a value between 0 and 1 that represents the DM’s preference for the consequence a _ij using the interval scale, and is obtained by evaluating the marginal value function in the consequence. Finally, k _j is the scaling constant, also called the weight associated with criterion j.

Inequation (7) introduces into the model the scaling constants order previously defined by the DM at the beginning of the elicitation procedure. Inequations (8) and (9) introduce into the model the preference statements made by the DM while comparing consequences in the elicitation by decomposition.

At first sight, inequations (10) and (11) may seem to be repeated. However, their functions in the model are rather different. Inequations represented in (10) have to do with the constraints on potential optimality. They guarantee that the model will only present a feasible solution if and only if there is at least one scaling constants vector for which the alternative i, is at least as good as all the other alternatives. Inequations represented in (11), on the other hand, have to do with the constraints obtained by making holistic evaluations. Note that, in case an alternative z was said to be preferable to alternative i due to a holistic evaluation, there will be a constraint similar to (11). However, as for being a potentially optimal alternative, the alternative i must be at least as good as all the other ones (20), the model will not have any feasible solution, so the alternative i, in this case, would be eliminated.

Constraint (12) states the decision variables’ non-negativity and along with equation (13) guarantees that the values of the scaling constants belong to the range [0; 1]. Equations represented in (14) present the possible values of the indices.

The FITradeoff method has been also developed to support Ranking (Frej et al., 2019), Sorting (Kang et al., 2020) and Portfolio problematics (Frej et al., 2021; Marques et al., 2022). An extension of FITradeoff for dealing with multi-issues negotiation problems has also been proposed (Frej et al., 2022). In addition, the method has been implemented in a Decision Support System (DSS) that makes it possible to apply the method in an easy, flexible, interactive and accessible way. The DSS is available in web for free at www.fitradeoff.org.

In this context, the method has been applied to support several MCDM/A problems presented in the literature, in economic, environmental and social contexts (Pessoa et al., 2022; Dos Santos et al., 2022; Ribeiro et al., 2021; Rodriguez et al., 2021; Pergher et al., 2020; Camilo, et al., 2020; Fossile et al., 2020; Kang et al., 2018; de Macedo et al., 2018; Carrillo et al., 2018; Dell’Ovo et al., 2017). For instance, the FITradeoff has been applied to support the decisions regarding anti-fraud programs (Pessoa et al., 2022). For prioritizing alternatives to combat Aedes aegypti in a Brazilian city (Dos Santos et al., 2022). For solving a facility location decision problem which dealt with a shopping mall location in the Brazilian northeast countryside (Dell’Ovo et al., 2017). In the next section the basic formulation and application of the APP problem considering the manufacturing strategy objectives is presented.

4 SOLVING AN AGGREGATE PRODUCTION PLANNING PROBLEM WITH CONSIDER MANUFACTURING STRATEGY OBJECTIVES

4.1 Model Construction and Application of NSGA-II

To compose the multi-objective aggregate production planning model, five objectives function in the format of (1) have been considered. These objective functions are related to manufacturing strategy. The first one refers to the sum of production-related costs and it is desirable to minimize it. For other objectives, such as quality, it is desirable to maximize.

In order to illustrate the application of the proposed approach, a hypothetical APP problem regards a pharmaceutical industry is solved. The industry works with different families of products which are distributed to a large group of retailers. Although the model could easily be adapted for solving problems with multiple product families, this study deals with only one of them.

The motivation for studying this specific family of products relies on an increase in the demand that has been observed in the last years, so the company wishes to prepare itself to meet the forecasted demand in the best way possible.

The model can be described as follows and the notations used in the model are given.

• C_r =production cost of regular time production ($/unit)

• C_o =production cost of overtime production ($/unit)

• C_s =subcontracted production cost production ($/unit)

• C_i =inventory carrying cost for a period ($/unit)

• s_max =maximum outsourced production

• p_r =units produced per worker in regular time

• p_o =units produced per worker in overtime regime

• C_h =cost of hiring one worker

• C_f =cost of firing one worker

• w_max =maximum number of workers

• I_max =maximum production stored

• d_t =forecast demand for period t

• I_b =initial inventory level

• I_e =final inventory level

Decision variables:

• r_t =the unit of regular time production in period t

• o_t =the unit of overtime production in period t

• s_t =the unit of subcontracting production in period t

• I_t =inventory level in period t

• w_t =number of workers in period t

• h_t =number of hired workers in period t

• f_t =number of fired workers in period t

The programming problem is described by equations (15) to (30). Equations (15) to (19) are the objective functions while Eqs (20) to (30) represent the mathematical expression of constraints.

$Min\cos t\sum _{t=1}^n\left(C_r{r}_t+C_o{o}_t+C_s{s}_t+C_i{I}_t+C_h{h}_t+C_f{f}_t\right)$ (15)

$Maxquality\sum _{t=1}^n\left({\alpha }_r^q{r}_t+{\alpha }_o^q{o}_t+{\alpha }_s^q{s}_t\right)$ (16)

$Maxflexibility\sum _{t=1}^n\left({\alpha }_r^{fl}{r}_t+{\alpha }_o^{fl}{o}_t+{\alpha }_s^{fl}{s}_t\right)$ (17)

$Maxdependability\sum _{t=1}^n\left({\alpha }_r^{dp}{r}_t+{\alpha }_o^{dp}{o}_t+{\alpha }_s^{dp}{s}_t\right)$ (18)

$Maxspeed\sum _{t=1}^n\left({\alpha }_r^{sp}{r}_t+{\alpha }_o^{sp}{o}_t+{\alpha }_s^{sp}{s}_t\right)$ (19)

$\begin{array}{cc}s/tOb{j}_k\ge {\lambda }_k& k=1,...,4\end{array}$ (20)

$I_t\le I_{max}$ (21)

$r_t\le p_r\left(w_{t-1}+h_t-f_t\right)$ (22)

$o_t\le p_o\left(w_{t-1}+h_t-f_t\right)$ (23)

$s_t\le s_{max}$ (24)

$r_t+o_t+s_t+I_{t-1}\ge d_t$ (25)

$I_t=r_t+o_t+s_t+I_{t-1}-d_t$ (26)

$w_t=h_t-f_t$ (27)

$w_t\le w_{max}$ (28)

$I_l=I_b$ (29)

$I_n=I_e$ (30)

Depending on the objective k, the alpha coefficients assume different meanings. These parameters are used to evaluate the production performance. In general, the parameters will be a percentage score which evaluates the performance of the production regimes defined. The way of establishing such parameters may be based on the available historical data. Another possibility for defining the parameters is to build well-defined qualitative scales to evaluate the performance of the production regimes and then to apply a normalization procedure.

In (15), the objective function seeks to minimize the total cost, whereas in (16) to (19) it is desired to maximize the performance in the additional objective functions. Constraint (20) establishes a low boundary for each objective, which is a minimum acceptable value for each of them and is not mandatory. In case the objective is to be minimized, this boundary may represent the maximum acceptable value the DM aims at investing.

Constraints (21), (24) and (28) define respectively the maximum inventory level, the maximum subcontracted production (in units) and the maximum workforce. Constraints (22) and (23) define the maximum regular and overtime production (which depends on the workforce). Constraint (25) forces the solution to meet the demand (since it is not possible to maintain backorders). Constraint (26) states that all the products that are not absorbed by the demand shall be placed in the inventory. Finally, Constraints (27), (29) and (30) represent respectively the available workforce in each period (which depends on hiring and firing production line workers), and the initial and final inventory levels.

As for building the model, the company put together all the necessary data which consists of the forecasted demand for the products (table 1), the costs and rates associated with the available production regimes, besides the production-related policies established by them.

Table 1
Forecast demand.

Jan	Feb	Mar	Apr	May	Jun	Jul	Ago	Sep	Oct	Nov	Dec
1,700,317	1,852,013	1,580,973	1,085,060	966,924	1,000,222	1,671,220	1,098,395	1,015,707	1,455,420	1,123,001	1,615,317

The plant operates six days per week in a journey of seven hours per day. The products under study have most of their production process automatized, however, part of the process still requires the presence of operators. Currently, the production line has 15 workers.

As the products may affect people’s health, they should meet the ANVISA (Health Surveillance National Agency) requirements, so quality tests are made. Fundamentally, all the products dispatched are approved in two tests that are conducted by the operators.

The managers consider meeting the demand by acting over the number of items produced in regular and overtime journeys, it is also possible to outsource part of the production since it does not overcome 300.000 units per month, besides, the maximum number of items stored in the inventory should not surpass 500.000 items.

When it comes to the maximum number of hours in overtime journey, each operator may work a maximum of two hours per day. There are no items in the inventory at the beginning of the period and the managers wish to finish it with no pieces stored.

The coefficients of the objective functions have been derived from managerial data:

• Minimizing the cost: the unitary cost of producing internally and in regular time is smaller than producing it in overtime journey, which is smaller than the cost of outsourcing the production.

• Maximizing quality: the proportion of defective products is smaller when the production is outsourced, that is because the products are tested by the supplier and the defectives are eliminated. When it comes to intern production, the proportion is smaller in the regular period.

• Maximizing flexibility: flexibility is maximum in outsourced production once it is possible to count on the supplier’s installations and know-how, allowing to diversify the products. The minimum flexibility is observed in the overtime production.

• Maximizing dependability: dependability is higher when producing internally and during regular time. Historically, outsourced production also achieved a good performance in this objective, whereas overtime production demonstrated to contribute less to the achievement of this objective.

• Maximizing speed: the maximum speed is observed in the regular time production, followed by the production in overtime journeys. The outsourced production showed up the least attractive performance in this objective.

The time horizon of the planning is twelve months, and the costs of hiring and firing employees during the period have also been inputted into the model. The actual values applied for building the model are not presented due to confidentiality aspects.

Having built the model, the next step consists of finding the Pareto efficient alternatives, which is done by the application of NSGA-II. By doing so, a subset containing 100 efficient solutions has been obtained.

The problem solutions are, each of them, feasible plans. They allocate in different manners the production resources leading to distinct values for the objective functions. When the DM chooses a plan to implement, then he/she receives the outcomes of such a plan in all the objective functions. The challenge here is to select an alternative that best represents the DM’s preferences.

Figure 2 presents how the outcomes (on the left axis) of the solutions (represented by small squares) obtained are distributed in each of the objective functions. These figures provide an initial idea of the benefits the DM may obtain in this decision context. It is worth noting that the model can be applied to other objectives, it is not restricted to the objectives considered in this paper.

Figure 2
Solutions outcomes in each objective function.

From Figure 2 it is possible to visualize that the outcomes for the objective of cost minimization vary from 8.18 million reais to 10.51 million reais. In the same way, the outcomes for dependability maximization vary from 93% to 94% whereas the outcomes for quality maximization vary from 94.9% to 96% of items under specifications.

Finally, the outcomes for the objective of flexibility vary from 89,97% to 92,23% while the outcomes for speed maximization vary from 96% to 98%.

Regarding the variation of the outcomes, one may think that this is a smooth variation, and indeed it is. However, it is worth considering the fact that those percentages are going to affect a relatively large-scale production, so it is important to analyse them carefully. In the quality objective, for instance, a problem in a batch may lead to serious health problems, this is something that has to be considered while solving the problem.

It is worth mentioning that for plotting the graphs presented in Figure 2, the outcomes of the solutions were ordered according to each objective function, so then it would be possible to visualize clearly the variation range.

Once the subset of solutions has been obtained, it is possible to conduct the analysis through the FITradeoff method application. This step is presented in the following section.

4.2 Application of FITradeoff

The FITradeoff method has been applied to elicit the DM’s preferences concerning the APP problem.The consequence matrix was uploaded into the FITradeoff Decision Support System (DSS) which is an online tool that makes it easier to solve problematic situations. In this case, the choice problematic was considered since the objective is to select the best APP considering the DM’s preferences.

The first step of the method is to evaluate the intracriterion marginal value function (VF). Different from other methods, the axiomatic structure of FITradeoff allows the use of linear and non-linear value functions, besides, its DSS disposes of a module that conducts the elicitation of the VF with the DM.

In the literature, it is common that practitioners consider all the VF as linear, however, that may introduce errors to the model once in many cases the DM’s aspirations may be better represented by a non-linear VF. Figure 3 presents the VF associated with the criterion quality.

Figure 3
Value function associated with the criterion quality.

Continuing, the system requires the DM to rank the scaling constants of the criteria. The established order of the scaling constants is presented in (31).

$k_{\cos t}\ge k_{flexibility}\ge k_{quality}\ge k_{speed}\ge k_{dependability}$ (31)

Introducing only the order of the scaling constants, FITradeoff could already reduce the action space to six potentially optimal alternatives out of a hundred candidates. This observation highlights the efficiency of the method in eliminating non-efficient alternatives (Mendes et al., 2020).

At this point, the DM could choose how to proceed with the elicitation process. The DM decided to do a holistic evaluation pre-analysis, which consists of exploring the different kinds of visualizations (tabular visualization, bar, bubble, and spider graphics) by combining a different subset of alternatives to evaluate whether he/she feels confident to state a preference relation. Therefore, after combining some alternatives, the DM felt confident to state a preference while evaluating plans 48 and 3 by using the bar graph (Figure 4).

Figure 4
Comparison of two plans in the first holistic evaluation.

Some recommendations to support the decision of performing a holistic evaluation or continuing the elicitation by decomposition are made in De Almeida et al. (2021). In addition, in a previous behavioural study recommendations based on the mean and the standard deviation of the probability of success are discussed for each kind of visualization. Such recommendations may be very useful and shall be considered by analysts who provide support to the DM during the process Roselli et al. (2023).

It is worth mentioning that two distinct procedures can be undertaken in holistic evaluation, namely, selection and elimination procedures (De Almeida et al., 2021). In the former procedure, DMs shall compare the alternatives using the visualization that they feel more confident to state a dominance relation, i.e., to select the best alternative from the subset of POAs.

In the latter, however, the DM shall decide on which is the worst alternative from the subset analysed. DMs are free to choose which holistic procedure they wish to perform. If they identify a visualization that makes them confident to state their preferences in either procedure, they shall be encouraged by the analyst to proceed with the assessment.

Therefore, according to the DM’s preferences, plan 48 would be the best one. Although plan 3 had a better performance in the criterion cost, plan 48 had the best performance in the criteria flexibility and quality which, according to the DM’s preferences, could compensate for the advantage of plan 3.

When the DM states a preference relation during the decision process, a new inequality is included in the FITradeoff Linear Programming Problem (LPP), in this way, the space of scaling constants is updated, and the alternatives are once more evaluated. Sometimes DMs may not feel confident about performing a holistic assessment, thus they can continue the process by following the decomposition procedure.

With the information from the holistic evaluation, besides plan 3, another plan has been eliminated from the potentially optimal set of alternatives, which demonstrates the contribution of the holistic evaluation for accelerating preference elicitation.

After that, the DM continued through the elicitation by decomposition. In elicitation by decomposition, consequences are compared with the purpose of eliciting the DM’s preferences. This information is also included in the LPP, narrowing the scaling constants space.

An illustration of this procedure can be seen in Figure 5, the first consequence presents an intermediate performance of 95.66% in the criterion Quality (Consequence A) and the worst consequence in all the others. Such a consequence is compared to another one presenting the best performance in criterion speed (Consequence B) and the worst in all the others. Hence, the DM preferred Consequence A.

Figure 5
Elicitation by decomposition.

After the third question was answered the subset of POAs was reduced to four alternatives, so the DM decided to conduct another holistic pre-analysis. After analysing some possible combinations of POAs, the DM decided to evaluate holistically plan 1 and plan 88 (see Figure 6).

Figure 6
Comparison of two plans in the second holistic evaluation.

In this case, the DM considered plan 1 the most desirable. This consideration was done following the same idea as previously, to him, the advantage of plan 1 over plan 88 in the criteria of flexibility and quality was enough to establish a preference relation between the two alternatives.

The DM’s preferences may be justified due to the importance of following the ANVISA requirements when dealing with pharmaceutical products. If the specifications are not adequately verified, the batch would be withdrawn from the market, that not only would have a monetary impact but could also incur damages to the company’s image.

The DM continued the decision process throughout the elicitation by decomposition and after a few more questions, the optimal alternative has been found. Plan 1 turned out to be the most preferred one because of its outstanding performance in the criteria of flexibility and quality besides presenting a good performance in the criteria of cost, dependability and speed. The results are better discussed in the next section.

4.3 Discussion of Results

In this section, the solution obtained in the FITradeoff method has been discussed. Plan 1 has been selected as the best one following DM’s preferences. It is worth saying that when it comes to multicriteria problems, the best solution is the one that best fits each DM’s aspirations, that does not mean that the alternative has the best performance in all the criteria, if such an alternative were feasible, there would not exist a decision problem.

Table 2 presents the plan 1 outcome for each criterion. By following that plan, a mean of 1.047.047 products is produced in regular time and six operators shall be hired. When necessary, part of the production is outsourced, and no production in overtime journey has been recommended by the model solution.

Table 2
Plan 1 details.

	Cost (mi)	Dependability (%)	Quality (%)	Flexibility (%)	Speed (%)
Plan 1	$ 8.94	93.11	95.89	92.23	96.22

By evaluating the model, it is clear that overtime production has not been recommended due to its poor performance in the considered objectives, which is an attention point. The company should then, investigate the possible causes that might be reducing the performance of the overtime production and eliminate them, by doing so, they may save costs related to the outsourced production.

In order to enhance the analysis of the results, the FITradeoff DSS also allows for conducting the sensitivity analysis (SA). The SA intends to verify how changes in input can modify the output aiming to test the robustness of the model.

In this context, using the FITradeoff DSS, DMs shall select the criteria they wish to vary and the respective variation percentages. Thereafter, the system generates several hypothetical scenarios by varying the values of the consequences concerning the selected criteria and percentages.

At each simulation scenario, the FITradeoff LPP is evaluated and the frequency in which the alternatives are recommended as the optimal solution is stored and presented to DMs, allowing them to evaluate whether the solution obtained is robust or whether it is necessary to make any adjustment to the decision model inputs.

In this study, as for conducting the sensitivity analysis of the results, two different scenarios were created. In the first scenario, a variation of ±10% was applied to the criterion of cost. The second scenario consisted of varying the production quality by ±10%.

The SA showed that plan 1 was recommended in most of the simulation scenarios. Some other plans have also been included in the recommendation subset, however, the frequency in which they were included was so small that it could be neglected.

The criterion of quality presented a relatively higher impact on the solution, it indicates that the managers shall pay special attention to this aspect. This result reinforces the importance of enhancing the quality of their intern production not only to guarantee the attractiveness of the selected plan but also to eliminate costs.

As discussed, the sensitivity analysis in the FITradeoff DSS varies the consequences evaluated against each criterion, in the case of the APP problem, it represents a variation in the values of the objective functions. As the production levels in the plan do not change, this variation in the values of the objective functions could only be explained by a variation in the coefficients of the decision variables in the objective functions of the multi-objective LPP.

Analysing the effect of varying such parameters is important since there may be noises in their definition. Moreover, as suggested by studies in the literature, it would also be important to analyse factors related to changes in demand and other structural aspects. This would be a suggestion for future developments.

5 CONCLUSION

Aggregate production planning (APP) aims at balancing the forecasted demand for product families with the available production resources, thus allowing changes to be made in the system and the best action strategy to be found.

In this study, a FITradeoff-based approach has been considered to solve an APP problem which presents multiple objectives. By considering a multicriteria approach it is possible to evaluate more properly the alternatives and associate the decisions at this planning level with the business and manufacturing strategy.

For the solutions generation step, the NSGA II was applied, which is an efficient GA. For the selection of the best APP, on the other hand, the FITradeoff method, which conducts a flexible and interactive Tradeoff elicitation, has been used.

The FITradeoff, assists the Decision Maker (DM) in choosing a production plan more suitable to the company objectives. Hence, the use of partial information in a structured procedure may lead to more reliable solutions with significantly less effort.

It should be noticed that FITradeoff does not limit the number of criteria and alternatives, so the DM may consider as many objectives and Pareto’s frontier solutions as they wish.

In the case study, the flexibility of the FITradeoff method and DSS allowed the DM to choose the extent to which he felt confident to perform the preference elicitation procedure, leading not only to better solutions, but also enabling the multiple objectives to be evaluated in a learning process.

The sensitivity analysis of the results allowed assurance to be given that the recommended solution was indeed the best one even considering different scenarios. The case study also highlighted the contribution of the holistic evaluation as more than an alternative source of information, but also as a way of accelerating the elicitation process since the subset of POAs is directly reduced by those evaluations.

The contribution of this work relies on integrating two consolidated techniques for generating and evaluating a representative subset of Pareto’s frontier solutions for facilitating the decisions regarding APP problems in the perspective of manufacturing strategy, in which the benefits were presented in a case study.

Some benefits from this approach when compared to others found in the literature include: this approach provides a way of modelling the APP problem in a way that the manufacturing strategy is considered, also giving suggestions on how to define the multiobjective model parameters based on the available information. Plus, it uses a very efficient GA which has been widely applied in the literatures and may be easily obtained.

In addition, different from other methods, by applying the FITradeoff, a large subset of solutions that are dominated, when applying the scaling constants ranking information, is already eliminated so then the DM may focus on the solutions that really matters to them. Finally, the features of the FITradeoff DSS, which includes different types of visualizations and analysis, makes it easier to elicit the DM’s preferences and aspirations reducing both modelling and elicitation errors.

As suggestions for future lines of research, it would be useful to conduct sensitivity analysis studies on demand variations. In addition, some comparisons can be made to evaluate different objectives. Moreover, a Decision Support System can be implemented to allow the operationalization of the proposed approach in a single system.

Acknowledgements

This work had partial support from the Academic Qualifications of Higher Education Personnel - Brazil (CAPES), the Brazilian Research Council (CNPq - Grant 312695/2020-9; 309237/2023-8), and the Foundation of Support in Science and Technology of the State of Pernambuco (FACEPE - Grant APQ-0484-3.08/17).

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• Financial Interest:
  Authors declare they have no financial interests.

Publication Dates

• Publication in this collection
  28 Oct 2024
• Date of issue
  2024

History

• Received
  25 Mar 2024
• Accepted
  21 June 2024