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Reprint of “Ergo-lot-sizing: An approach to integrate ergonomic and economic objectives in manual materials handling”
International Journal of Production Economics 194 (2017) 32–42
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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe
Reprint of “Ergo-lot-sizing: An approach to integrate ergonomic and economic objectives in manual materials handling”☆ Daria Battini a, Christoph H. Glock b, Eric H. Grosse b, Alessandro Persona a, Fabio Sgarbossa a, * a b
Department of Management and Engineering, University of Padova, Stradella San Nicola 3, 36100 Padova, Vicenza, Italy Institute of Production and Supply Chain Management, Technische Universit€ at Darmstadt, Hochschulstr. 1, 64289 Darmstadt, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Lot-sizing Ergonomics Rest allowance In-house logistics Manual material handling
Over the last decades, academics and practitioners have paid much attention to lot-sizing, which determines economic order and production quantities by balancing inventory holding and setup cost. Recently, researchers have started to integrate sustainability issues into lot-sizing models. The focus of these works has been on environmental and economic dimensions of sustainability, however, while only few contributions studied the social aspect of this problem. Especially in in-house logistics, where a high amount of manual material handling is performed, lot-sizing decisions can have a significant impact on workload and human performance, which can have a strong influence on ergonomic parameters and thus worker welfare. The paper at hand extends previous research on ergonomic lot-sizing and introduces a new mathematical model that integrates ergonomic and economic aspects. A rest allowance function is used to take account of recovery periods that help to maintain low levels of fatigue and ergonomic risks. As recovery periods represent nonproductive time, the developed integrated model permits to estimate the economic impact of different workload levels. Finally, the model is applied in a numerical study, reflecting a typical manual material handling process. Based on the results of a parametrical analysis, we illustrate the applicability and validity of this approach to different industrial contexts.
1. Introduction The lot-sizing problem ranks among the most important decision problems in production and supply chain management. Over the last 100 years, a myriad of works has appeared on this topic, which made lotsizing one of the most extensively studied decision problems in operations management (see, for recent reviews, Glock et al. (2014) and Andriolo et al. (2014)). The dominant objective in lot-sizing research is the minimization of total cost, which is usually achieved by balancing inventory holding, setup and other costs affected by the lot-size. Recently, a new research stream that integrates sustainability issues into lot-sizing models has emerged. Examples in this area are works that consider: a) emissions resulting from transportation (Absi et al., 2013; Battini et al., 2014a),
b) carbon emissions as a function of the production lot-size or production rate (Jaber et al. 2013), c) how lot-sizing decisions affect warehousing operations and thus warehousing's environmental impact (Fichtinger et al. 2015), or d) that emissions and scrap can be controlled by varying production rates (Glock et al. 2012). The focus of prior research on sustainable lot-sizing has, however, mainly been on the economic and environmental dimensions of sustainability (such as emissions reduction). The social component (such as worker welfare) has, in contrast, not received much attention, even though the social component is frequently referred to as the third pillar of the ‘triple bottom line’-approach (Elkington, 1997). Within the social component, human factors and ergonomics are of major importance. According to Sanders and McCormick (1993), the objective of ergonomics research is “to enhance the effectiveness and efficiency with which
DOI of original article: https://doi.org/10.1016/j.ijpe.2017.01.010. ☆ This article is a reprint of a previously published article. For citation purposes, please use the original publication details; Journal of Production Economics, 185, pp. 230-239. * Corresponding author. E-mail addresses: [email protected] (D. Battini), [email protected] (C.H. Glock), [email protected] (E.H. Grosse), [email protected] (A. Persona), [email protected] (F. Sgarbossa). https://doi.org/10.1016/j.ijpe.2017.11.003 Available online 20 November 2017 0925-5273/© 2017 Published by Elsevier B.V.
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International Journal of Production Economics 194 (2017) 32–42
handling process. The remainder of the paper is structured as follows. The next section reviews literature on sustainable lot-sizing. Section 3 describes the problem under study in detail and explains necessary assumptions. Section 4 presents the mathematical formulation of the ergo-lot-size model, and Section 5 summarizes the results of a numerical experiment. Section 6 concludes the paper.
work and other activities are carried out. Included here would be such things as increased convenience of use, reduced errors, and increased productivity. The second objective is to enhance certain desirable human values, including improved safety, reduced fatigue and stress, increased comfort, greater user acceptance, increased job satisfaction, and improved quality of life.” Briefly, ergonomics aims at optimizing human well-being and overall system performance (Karwowski, 2005). Even though lot-sizing has mainly been studied in a manufacturing (economic production quantity) or purchasing (economic order quantity) context, lot-sizing decisions occur in other areas as well. In warehousing, for example, lot-sizes can be interpreted as the number of items that a warehouse worker has to handle in a single operation (e.g., when refilling the forward area in a warehouse or during line feeding processes). In such a typical in-house transportation scenario, it is clear that lot-sizes can have a significant impact on the amount of manual material handling required and the workloads assigned to workers (cf. Neumann and Medbo, 2010). Obviously, lot-sizing models that only consider the direct economic impact of lot-sizes may lead to wrong decision support in such a scenario by generating excessive workloads and increasing worker fatigue and, consequently, injury risks. For example, large lots that have to be handled (e.g., packed, labelled) or transported manually (e.g., by using a trolley or a cart) by the company's workers may lead to excessive pushing/pulling of a vehicle or lifting/carrying of products, which may result in worker fatigue and injuries (Jung et al., 2005; Knapik and Marras, 2009). Possible risk factors for injuries, in particular musculoskeletal disorders, include repetitive work with forceful exertions, awkward body postures, heavy lifting, insufficient recovery time, and rapid work pace during manual material handling (Punnett and Wegman, 2004). Clearly, all these risk factors may be influenced by lot-sizing decisions. A closer look at official statistics highlights the importance of these interdependencies: In the European Union, for example, musculoskeletal disorders account for over 38% of occupational diseases, with the lifting of loads being one of the most critical risk factors (EASHW, 2010). The economic burden of work-related musculoskeletal disorders accounts for up to 2% of the Gross National Product in the EU. For other regions, similar trends can be observed. For example, in the US, musculoskeletal disorders accounted for 32% of all worker injury cases in 2014, and stock and material movers incurred the highest number of these cases (BLS, 2014). This leads to an economic impact of musculoskeletal diseases in the US of about 5% of the Gross National Product (MEPS, 2014). Although authors called for the integration of ergonomics into managerial decision support models for in-house logistics due to the high amount of manual material handling and related risk factors (Grosse et al., 2015), interdependencies as the ones described above have thus far mainly been overlooked in the literature on lot-sizing. The work at hand contributes to closing this research gap by integrating the rest allowances of workers during task execution as an ergonomic variable into a lot-sizing model (for the rest allowance concept, the reader is referred to Price (1990) and Garg et al. (1978) and the literature overview in Section 2.2). The basic idea underlying the rest allowances concept is that manual work leads to fatigue of the workers involved, which makes it necessary to assign rest periods (rest allowance) to them from time to time to avoid high levels of fatigue which can lead to injury risks. As the lot-size directly influences workload and thus worker fatigue, the rest allowance (as an unproductive time) is integrated into the total cost function of the lot-sizing model, which links the ergonomic variables to the cost minimization objective. The developed integrated model permits to estimate the influence of the lot-size on the worker's workload level and the economic impact of the workload level on the total cost of the system. This paper is the first to propose a simple closed-form expression for the optimal ergonomic lot-size. In the optimal solution, the total cost of the process consisting of the direct material handling cost (picking, travelling and storing) and the cost of the unproductive rest periods necessary to maintain a low fatigue level are minimized. The model is applied in a numerical study, reflecting a typical manual material
2. Literature review Sustainable supply chain management has attracted an increased attention in recent years (Seuring and Müller 2008). Especially the manufacturing sector has been active in implementing sustainability initiatives, where both manufacturing strategies as well as the tactical and operational execution of production tasks have been influenced by sustainability objectives (Sarkis, 2001). In the area of production planning and control, this trend has given rise to the development of lot-sizing models that consider sustainability issues besides the minimization of total operational cost. Wahab et al. (2011), for example, developed an Economic Order Quantity (EOQ) model for a two-stage closed-loop supply chain that considers CO2 emissions. The authors assumed that two types of CO2 emissions occur in this system: fixed emissions, which are a function of the distance between the vendor and the buyer, the fuel efficiency of the vehicle used and other system-related factors, and variable emissions, which are a function of the lot-size. CO2 emissions were assumed to result in an environmental cost in this paper. Chen et al. (2013) developed a carbon-constrained EOQ model and assumed that a constraint on the maximum amount of CO2 emissions generated in production exists. Glock et al. (2012) developed a sustainable Economic Production Quantity (EPQ) model, but again concentrated on the environmental and economic dimensions of sustainability by incorporating an emission-based environmental quality index in their lot-sizing model. Other inventory models that consider greenhouse gas emissions are the works of Jaber et al. (2013), Battini et al. (2014a), Gurtu et al. (2015), Bouchery (2012), or He et al. (2015). Surprisingly, prior research on sustainable lot-sizing has almost unanimously interpreted “sustainable” as “environmentally friendly”. While sustainability clearly includes an ecological objective, there is more and more consent that sustainability also has an economic and a social dimension (e.g., Elkington, 1997). Especially the social component, which includes such aspects as worker welfare and ergonomics, for example, has largely been ignored in prior research on lot-sizing. Various different concepts and methods are generally suitable for integrating ergonomic aspects into managerial decision support models. Reviewing the extensive literature on ergonomic assessment methods is, however, not within the scope of this paper (the reader may refer to Chiasson et al. (2012) for an overview of ergonomic assessment methods). An analytical assessment model is the concept of energy expenditure, which allows the estimation of energy required to execute a manual task as a function of oxygen consumption (Garg et al., 1978). Garg et al. (1978) differentiated between base postures (such as sitting, standing and standing in bent position) of the worker during task execution and the movements he/she performs while completing the task (such as lifting, lowering, walking, carrying loads). The authors assumed that the energy expenditure in kilocalories per minute can be calculated by summing up the energy required to maintain different body posture and for performing the task, divided by the time required for performing the task in minutes. Other energy expenditure models for walking and carrying that consider worker characteristics and load weight were proposed by Aberg et al. (1967), Pimental and Pandolf (1979), Taboun and Dutta (1989), and Battini et al. (2015b). Examples of frequently used ergonomic assessment methods in practice are the NIOSH lifting equation (Dempsey, 2002), the OWAS method (De Bruijn et al., 1998), or the European assembly worksheet (Schaub et al., 2013). In addition, real-time integrated motion capture 33
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cost and ergonomic risks, where the latter was assessed using a lifting index. Andriolo et al. (2016) did not consider the rest allowance concept, but instead compared the cost objective and the ergonomic objective using Pareto frontiers. Also in the work of Andriolo et al. (2016), no closed-form to calculate the exact value of the optimal lot-size considering ergonomic aspects was proposed. Our brief overview of the literature showed that only a few works analysed manual material handling processes (such as lot-sizing decisions) from an integrated perspective that includes both economic and ergonomic aspects. With a few notable exceptions (Battini et al., 2015a; Andriolo et al., 2016), there appears to be a large gap in the literature regarding works that consider ergonomic aspects in managerial decision support models, in particular in determining optimal lot-sizes. The paper at hand contributes to closing this research gap by developing an economic lot-sizing model that considers both ergonomic and economic aspects. In particular, this work extends previous studies (Battini et al., 2015a; Andriolo et al., 2016) by:
Fig. 1. Illustration of a single job cycle.
systems can be used to measure muscle activity and workload (Battini et al., 2014b). To the best of the authors’ knowledge, only a few papers considered ergonomic issues in calculating optimal lot-sizes. One example is the work of Battini et al. (2015a), which included human energy expenditure in a lot-sizing model. The authors assumed that larger lot-sizes lead to higher energy expenditure at the worker, which makes breaks necessary from time to time to give the worker time to recover. They introduced a model to estimate the total time spent in the process, including the rest allowance, but they did not consider cost associated with the lot-size. Their approach consists in some simplified equations used to estimate the fatigue of the worker, based on the energy expenditure rate. The optimal solution is not calculated using a closed-form expression in their paper, but instead a set of suitable scenarios is evaluated to identify the best production strategy. Andriolo et al. (2016) proposed another lot-sizing model that considers ergonomic aspects. In contrast to Battini et al. (2015a), the authors proposed a multi-objective optimization approach that calculates Pareto-optimal solutions that minimize both
1) modelling the overall ergonomics level of a traditional manual material handling process through the use of energy expenditure rate equations (Garg et al., 1978); 2) integrating the rest allowances (including threshold levels for accumulating fatigue) of workers during task execution as an ergonomic variable into a lot-sizing model; 3) introducing a general economic model (total cost function) of the ergonomic lot-sizing problem and estimating the economic impact of different workload levels in manual material handling; 4) deriving a closed-form expression for calculating the optimal lot-size considering ergonomic aspects by minimizing the total cost function. We refer to this closed-form expression as the “ergo-lot-size” in the following. 3. Problem description This section describes the problem studied in this paper and summarizes the necessary assumptions and notations. In addition, the rest allowance formulation as well as the objective of the developed lot-sizing model are described.
Table 1 List of notations. Notations
Description
Q q q w Wmax qmax ¼ Wwmax d Nt ðqÞ
total amount of items to be handled in a given period [pcs] lot-size [pcs] optimal ergo-lot-size [pcs] item weight [kg] maximum load weight transported by the cart [kg] capacity of the cart [pcs]
s tps ¼ tp ¼ ts Tp ðqÞ ¼ q⋅tp Ts ðqÞ ¼ q⋅ts Tt ¼ 2d s Rt=ps cop RAðqÞ Ė tot ðqÞ Ė t Ė ps ðqÞ¼ E ̇ p ðqÞ¼ E ̇ s ðqÞ a, b α CðqÞ
3.1. Scenario under study The problem studied in this paper is a typical manual material handling and transportation process, where a certain amount of products (lot-size) has to be handled in a production or logistics facility. This scenario occurs when transporting products from the final stage of a production system to a storage facility, when refilling an order picking warehouse from the reserve area, or when feeding an assembly line, for example. Within this scenario, we study the following single job cycle (see Fig. 1):
distance between stock point “A” and “B” [m] number of trips necessary to ship the total amount of items Q from point “A” to point “B” constant travel speed [m/s] unitary picking/storing time [s] total time required to pick the lot q from point “A” and load it on the cart [s] total time required to unload the lot q form the cart and stock it at point “B” [s] total travel time required to transport the lot q from point “A” to point “B” and for returning to point “A” [s] ratio between the travel time and the unitary picking and storage 1 time, expressed by 2d s ⋅2⋅tps
1) We consider a stocking point “A” where items are produced on a machine and temporarily stored until they are shipped to the next stocking point “B”. 2) Human workers are responsible for transporting the items from point “A” to point “B”. They manually pick a certain quantity of products at point “A” and put them on an electric cart, which is used for transporting the items (lot-size) to point “B”. 3) At point “B”, all items are removed from the cart and stored in a rack (e.g., in an order picking warehouse or another storage facility). 4) After having stocked the full lot-size in the rack at point “B”, the worker returns to point “A” to pick up a new lot there. We refer to the whole process of picking the lot at point “A”, transporting it to point “B” and unloading and storing it there as well as returning to point “A” as a job cycle.
unitary worker cost [€/s] rest allowance necessary to maintain a low level of fatigue and to avoid ergonomic risks [%] average metabolic cost for completing each cycle [kcal/min] metabolic cost for transportation activity [kcal/min] metabolic cost for picking/storing activity [kcal/min] parameters of the linear function to estimate Ė ps ðqÞ cost impact of the rest allowance, as a multiplier of the unitary worker cost. Its value is greater than or equal to 1 total cost function [€]
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w⋅q, and it depends on the physical layout of the stocking locations “A” and “B”. As reported in the previous section, we assume that the worker is in standing position during transportation activities and that an electric cart is used to transport the lot-size q, which leads to fixed metabolic cost of this activity E_ t equal to 1.86 kcal/min. Eq. (3) estimates the average metabolic cost per cycle as the weighted average of the metabolic cost per task performed in the entire cycle as follows:
5) After finishing one job cycle, i.e. after returning to point “A”, the worker needs to rest (note the rest allowance model is explained in detail in Section 3.3). 3.2. Assumptions To develop the mathematical model based on the scenario under analysis, the following assumptions are made: after finishing a single job cycle, the worker needs to rest; an electric cart is used to transport the items from “A” to “B”; during transportation, the worker maintains a standing position; picking and storing activities consume the same amount of time, as the same movements and body postures have to be performed for both tasks, and as the weight of items to be stored or to be retrieved is the same; only the specific job cycle (see Section 3.1) is considered, and it is assumed that the worker in question is responsible only for the tasks associated with the job cycle at hand; only a single worker is considered; the hourly cost of the break could be higher than the worker's wage cost due to a possible opportunity cost. We assume that during the break, the worker rests exclusively without performing any valueadding tasks.
E_ tot ðqÞ ¼
q¼
3.4. Rest allowance
w¼
We model the rest time as a percentage of the total time spent on completing a cycle using the relaxation formulas introduced by Price (1990), where the relaxation allowance depends on the acceptable work load level of 300 W, which is equal to 4.3 kcal/min. Other approaches could be used, such as the one developed by Rohmert (1973), where the rest time after task execution is exponential in the time spent on performing the task and the metabolic cost. Based on the formulation introduced by Price (1990), the rest allowance can be defined as:
4:3 b a
(5)
8 > < E_ tot ðqÞ 4:3 if w > w and q > q RAðqÞ ¼ 4:3 E_ t > : 0 otherwise
(1)
otherwise
if E_ tot ðqÞ > 4:3 kcal=min
(4)
Using the equations developed by Garg et al. (1978), the parameters can be estimated as a ¼0.17403 and b ¼3.7206 (see Appendix A), so the threshold value w is equal to 3.3 kg. This is in line with the general international guidelines for occupational safety in manual material handling, where the weight of about 3 kg usually is the limit for low loads (ISO 11228-1, 2009; ISO 11228-2, 2009; ISO 11228-3, 2009). Finally, using the two developed threshold values, the rest allowance can be defined as:
(6)
In case w > w and q > q, the rest allowance can be rewritten as:
where MWRðqÞ is the Mean Work Rate, expressed in Watt, and RR is the Relaxation Rate, which equals 130 W (about 1.86 kcal/min) for the body posture “standing”, for example. One frequently used approach to estimate the metabolic cost of a manual task was developed by Garg et al. (1978). Garg et al. (1978) introduced simple equations to assess the metabolic cost of several manual material handling jobs based on observations made in laboratory experiments. Modifying Eq. (1) by expressing metabolic cost in kcal/min (1 W¼69.77 kcal/min, ISO 80000-5, 2007) leads to the following equation:
8 > < E_ tot ðqÞ 4:3 RAðqÞ ¼ 4:3 E_ t > : 0
2d 1 4:3 E_ t ⋅ ⋅ s 2⋅tps E_ ps 4:3
This is verified for E_ ps 4:3 > 0, so the metabolic cost for picking and storing activities has to be larger than 4.3 kcal/min. As was mentioned before, the equations developed by Garg et al. (1978) can be applied to estimate the value of E_ ps . Given the physical configuration of stocking points “A” and “B”, worker weight, walking speed and task time, E_ ps can be expressed by a linear function of w, such as E_ ps ¼ a⋅w þ b, where the parameters a and b depend on the layout of locations “A” and “B” (see the Appendix A for more details). Based on this assumption, the constraint E_ ps 4:3 > 0 changes to a⋅w þ b 4:3 > 0 and another threshold value expressed as a lower bound of item weight can be defined:
Table 1 summarizes the notations used in the mathematical model.
if MWRðqÞ > 300 Watt
(3)
Note that the relaxation time is necessary if E_ tot ðqÞ>4.3 kcal/min, so a threshold value expressed as a function of the lot-size and item weight can be defined as:
3.3. Notations
8 < MWRðqÞ 300 300 RR RAðqÞ ¼ : 0
2⋅E_ ps ⋅q⋅tps þ E_ t ⋅2ds 2⋅q⋅tps þ 2ds
E_ tot ðqÞ 4:3 RAðqÞ ¼ ¼ 4:3 E_ t
2⋅E_ps ⋅q⋅tps þE_t ⋅2d s 2⋅q⋅tps þ2d s
4:3
4:3 E_ t
(7)
The energy expenditure concept (Garg et al., 1978; Price, 1990) has frequently been used to estimate metabolic cost of repetitive manual material handling tasks in the literature (e.g., Battini et al., 2016; Calzavara et al., 2016). It is an easy-to-understand and easy-to-use approach for considering ergonomic conditions in a formal model as the energy expenditure is expressed in kilocalories (kcal) per unit of time (minute). This makes it possible to consider ergonomic aspects in a time/cost model. Other ergonomic risk measurement methods, such as the NIOSH lifting equation or the OWAS index, have limitations as they produce semi-quantitative risk indices for performing specific manual tasks in different body postures, which are difficult to consider in the objective function of an optimization model. The NIOSH lifting equation, for example, calculates recommended weight limits for lifting tasks only, which makes it unsuitable for the job cycle considered in this paper (see
(2)
otherwise
We assume that the time spent for picking and storing activities is a function of q, and that the worker needs the same unitary time for picking and storing, i.e. tps ¼ tp ¼ ts (see also Battini et al. (2015a)). The metabolic cost E_ ps ðqÞ increases with the handled load during task execution,
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lower than a few thousands Euro and can be used for many years, so the cost of the equipment per order is negligible as compared to the unitary worker cost. Eq. (12) considers the rest allowance RAðqÞ necessary for workers in each cycle to recover from fatigue. In addition, a cost factor α (greater than or equal to 1) that takes account of the opportunity cost of the resting phase is included in the objective function to take account of the fact that the worker does not work during rest times. As a result, our model does not only consider the direct cost of the worker, but also the potential cost impact of the production loss (non-productive time). In fact, the resting phase is not productive and it will impact the total cost if the worker has a high saturation level. In this case, the worker needs an extra-time to complete the process with an added cost or the process is not completed on time with potentially expensive effects on the following step of the logistic process. This additional cost impact of the resting phase is modelled using the factor α. The objective function can be rewritten as follows:
Section 3.1). OWAS, in turn, is very restrictive in formulating weight limits. More specifically, it considers just three categories for the handled load – namely less than 10 kg, between 10 kg and 20 kg and over 20 kg. Its use is therefore not helpful for coordinating a manual material handling process and for calculating the optimal ergo-lot-size. 3.5. Objective The objective of the model is to find the optimal ergo-lot-size that minimizes the total cost resulting from the material handling process under study. Thus, the quantity that the worker handles in each job cycle (i.e., the lot-size q) is the decision variable, and its value depends on: the standard time spent for each activity, i.e. picking, travelling and storing (see phases 1–4 of the job cycle in Section 3.1); the rest allowances necessary to guarantee an acceptable fatigue level, which is a function of the energy expenditure/metabolic cost (Garg et al., 1978; Price, 1990) of the worker during the execution of each task (see phase 5 of the job cycle in Section 3.1); the unitary worker wage cost.
2d Q ⋅cop ⋅ þ ts ⋅cop ⋅Qþ s q Q þRAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop q CðqÞ ¼ tp ⋅cop ⋅Q þ
We model this problem in the next section using an analytical formulation of the total cost function for the job cycle described above, which depends on the total time spent to perform this typical in-house logistics process. We use a rest allowance function to integrate ergonomic aspects into lot-sizing decisions. By minimizing the total cost function, the optimal ergo-lot-size can be derived.
(13)
Typically, the picking and storing activities are very similar in time, as the same movements and body postures have to be performed, and as the weight of items to be stored or to be retrieved is the same. We therefore suggest using the same unitary time for both activities, i.e. tps ¼ tp ¼ ts . The model can thus be simplified as follows:
CðqÞ ¼ 2tps ⋅cop ⋅Q þ
4. Model description and mathematical formulation of the ergolot-size
Q 2d Q ⋅cop ⋅ þ RAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop : s q q (14)
4.1. Model description 4.2. Optimal solution of the integrated model Based on the problem description introduced in Section 3, we model the total cost function as the sum of four cost terms related to the main activities performed during the process under study, i.e. picking Cp ðqÞ, travelling Ct ðqÞ, storing Cs ðqÞ, and resting CRA ðqÞ. We define the objective function (Eq. (8)) and the related individual cost terms (Eqs. (9)–(12)) as follows:
CðqÞ ¼ Cp ðqÞ þ Ct ðqÞ þ Cs ðqÞ þ CRA ðqÞ;
As we consider two different thresholds w and q, the integrated model has two different solutions, based on the presence of the rest allowance. Case 1: w w. If w w, the rest allowance is equal to 0, so the cost function simplifies to the traditional lot-sizing model that considers only picking, storing and transportation activities:
(8) CðqÞ ¼ 2tps ⋅cop ⋅Q þ
where
Cp ðqÞ ¼ Tp ðqÞ⋅cop ⋅Nt ðqÞ ¼ q⋅tp ⋅cop ⋅
Ct ðqÞ ¼ Tt ⋅cop ⋅Nt ðqÞ ¼
Q ¼ tp ⋅cop ⋅Q; q
2d Q ⋅cop ⋅ ; s q
(10)
(11)
Q CRA ðqÞ ¼ RAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop : q
(12)
(15)
Minimizing the cost function leads to an optimal ergo-lot-size of q ¼ ∞, so q ¼ qmax , where qmax is the maximum number of items that can be transported by the cart. In a practical application, the capacity of the cart would usually vary between 500 kg and 1000 kg, depending on the type of cart used. Thus, if the item weight is lower than 3.3 kg (see Section 3.4), the worker can concentrate on optimizing the number of trips without having to take account of others factors such as travelled distance, picking/storage time, or the cost impact of the rest allowance. Case 2: w > w Since the total cost function depends on the variable q, it is necessary to study the behaviour of Eq. (14) for q q and for q > q. In the first case, with q q, the total cost function is expressed by Eq. (15) because the E_ tot ðqÞ>4.3 kcal/min and then RAðqÞ ¼ 0 (see Eq. (6)). Under this condition, the total cost function does not include the rest allowance cost term, so it is a strictly decreasing function due to its
(9)
Q Cs ðqÞ ¼ Ts ðqÞ⋅cop ⋅Nt ðqÞ ¼ q⋅ts ⋅cop ⋅ ¼ ts ⋅cop ⋅Q; q
2d Q ⋅cop ⋅ s q
Eqs. (9) and (11) calculate the cost for picking and storing activities. They are expressed as the total time spent on picking and storing the total amount of items Q, multiplied with the unitary worker cost. As can be seen, these cost functions do not depend on the lot-size, but instead only on the total amount of items processed. Eq. (10), in turn, describes the cost of travelling. This cost term does not explicitly consider the cost of the used equipment because of its low value. The cost of a typical electric cart used in handling activities as the one considered here are usually
∂Ct ðqÞ Q 2d negative first partial derivative: ∂CðqÞ ∂q ¼ ∂q ¼ s ⋅cop ⋅q2 ;.
When q > q, the worker needs to rest after each job cycle in order to avoid high accumulated fatigue and thus ergonomic risks, as indicated by Eq. (6). In this case, the total cost function includes CRA ðqÞ (see Eq. (14)). The cost term related to the rest allowance can be written as:
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the end of a production system (point “A”) to a storage facility (point “B”) using an electric cart operated by a human worker. The weight w of each item is 6 kg. The distance between point “A” and point “B” is 100 m. The unitary picking time and the unitary storage time are both 8 s per item, while the travel speed is 1 m/s. The cost impact of the rest allowance α is equal to 4. The unitary worker cost cop is equal to 15 €/h (or 15/3600 €/s). The metabolic cost of the transportation activity, E_ t , is fixed and equal to 1.86 kcal/min, while the metabolic cost of the picking and storing activities is modelled as E_ ps ¼ 0:1743⋅w þ 3:7206(see Appendix A). In this case, w > w holds, and thus the optimal ergo-lot-size q can be calculated according to Eq. (4):
q ¼ q ¼
2⋅d 1 4:3 E_ t 2⋅100 1 4:3 1:86 ⋅ ⋅ ⋅ ⋅ ¼ s 2⋅tps E_ ps 4:3 1 2⋅8 0:17403⋅6 þ 3:7206 4:3
¼ 65 items Using this lot-size, we obtain the total weight for each trip as 390 kg, which needs to be compared to the capacity of the cart, which is typically between 500 kg and 1000 kg. The minimal total cost for transporting 1000 items from point “A” to point “B”, Cðq Þ, equals 78.49 € according to Eq. (14), and it is the sum of Cp ðq Þ ¼ Cs ðq Þ ¼33.33 € and Ct ðq Þ ¼12.83 €, while the rest allowance cost CRA ðq Þ ¼0 €. This is due to the fact that w > w and the optimal ergo-lot-size is equal to the threshold value q, where E_ tot ðqÞ ¼ 4:3 kcal=min, so RAðqÞ¼0. Fig. 2 shows the total cost curve CðqÞ for alternative lot-sizes q. As discussed before, under the traditional lot-sizing approach in materials handling, it is common to minimize the number of trips under the constraint of the capacity of the cart, which depends on the size and weight of the items (cf. Grosse et al., 2014). In this case, the maximum load weight transported by the cart is the upper bound of the lot-size. In the numerical example considered here, the cart has a capacity of 750 kg and can thus transport a maximum of 125 items each weighting 6 kg per trip, referred to as qmax. We calculate the total cost for qmax ¼125 and compare it to the minimum value obtained for q*. In the case where qmax ¼125, the total cost equals 97.64 € as the sum of Cp ðqmax Þ ¼ Cs ðqmax Þ ¼33.33 €, Ct ðqmax Þ ¼6.67 € and CRA ðqmax Þ ¼24.31 €. The percentage savings in total cost using the optimal ergo-lot-size q* equals 19.6%. We note that the picking and storing cost are not affected by the value of the lot-size, as is demonstrated by Eqs. (9) and (11), while the transportation and rest allowance cost are different in this case. It is clear that the transportation cost is lower in the second case where the traditional approach tends to saturate the cart to minimize the number of trips, but it implies a higher fatigue level of the worker and consequently an increase in cost linked to unproductive recovery time.
Fig. 2. Total cost curve for alternative lot-sizes.
Table 2 Input values used in the parametrical analysis. Variable
Value (s)
Q cop w d s tps ¼ tp ¼ ts α Wmax
1000 15 €/h ¼ 15/3600 €/s from 1 to 20 kg, each 0.5 kg from 25 to 250 m, each 25 m 1 m/s 8s 1; 2; 4 500, 750 and 1000 kg
Q CRA ðqÞ ¼ RAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop q
(16)
Substituting Eqs. (7) and (12) into Eq. (16), Eq. (16) can be reformulated as follows (the simplification steps can be obtained from the authors upon request):
E_ ps 4:3 2d Q α⋅cop ⋅ ⋅ CRA ðqÞ ¼ α⋅cop ⋅2⋅tps ⋅Q⋅ s q 4:3 E_ t
(17)
The total cost function can now be written as follows:
E_ ps 4:3 2d Q 2d Q α⋅cop ⋅ ⋅ CðqÞ ¼ 2tps ⋅cop ⋅Q þ ⋅cop ⋅ þ α⋅cop ⋅2⋅tps ⋅Q⋅ s q s q 4:3 E_ t (18)
5.2. Parametrical analysis
The first partial derivative of Eq. (18) with respect to q is positive as:
∂CðqÞ 2d Q 2d Q 2d Q ¼ ⋅cop ⋅ 2 þ α⋅cop ⋅ ⋅ 2 ¼ cop ⋅ ⋅ 2 ⋅ðα 1Þ ∂q s q s q s q
In this section, we apply the developed mathematical model to several scenarios using input parameter values that were inspired by observations made in an industrial application (see Table 2), and we analyse their impact on the optimal ergo-lot-size. In particular, we change the weight of the item from 1 to 20 kg in order to consider the handling of different types of materials, such as small and heavy boxes. The distance values considered in this analysis cover many different cases, e.g. scenarios where these locations are close to each other as well as scenarios where the picking and storing points are far away from each other, such as in different shop-floors or in different warehousing areas (reserve and forward area). The speed of the cart and the picking/storing time are assumed constant. The maximum load weight transported by the cart is fixed to three different values representative of the carts available in typical material handling systems catalogues. Concerning the cost impact of the rest allowance, we analyse the effect of three levels. When α is equal to 1, the rest period of the worker has
(19)
Finally, since the total cost function is strictly a decreasing function for q q and strictly increasing for q > q, the optimal ergo-lot-size q* can then be defined as follows:
q ¼ q ¼
2⋅d 1 4:3 E_ t ⋅ ⋅ s 2⋅tps E_ ps 4:3
(20)
5. Numerical example and parametrical analysis 5.1. Numerical example First, we describe the application of the new model to an illustrative example, where Q¼1000 items have to be moved from a buffer stock at 37
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be achieved, with the savings value depending on the total cost Cðqmax Þ. In this analysis, the savings are subject to the cost impact of the rest allowance and the maximum capacity of the cart. For an increasing cost impact of the rest allowance, we can observe a proportional increase in the percentage of savings. The savings thereby depend on the ratio of the manual activities’ execution time and the capacity of the cart. Following the explanation given in the analysis of the optimal ergo-lot-size value, the savings are larger in case of shorter transportation distances. In this case, the capacity of the cart has a relevant impact on the savings. In particular, there is no difference between the two approaches when the capacity of the cart is low and the distance between points “A” and “B” is long. In case the travelled distance is long, the ratio Rt=ps is high and the optimal ergo-lot-size is large (see Fig. 3), especially for low item weights. This results in an optimal total weight per trip larger than the capacity of the cart, so the latter will also be the optimal ergo-lot-size. For example, in case of a cart capacity of 500 kg and a travel distance of 200 m with an item weight of 6 kg, the ergo-lot-size using Eq. (4) should be 131 items per trip, corresponding to 786 kg on the cart; here, the lot-size clearly exceeds the capacity of the cart. In this particular case, the ergo-lot-size model obtains savings as compared to the traditional approach if the item weight exceeds 11 kg. This is due to the low capacity of the cart, which has a higher impact on savings if the item weight is high. A higher capacity of the cart allows higher savings also for lower item weights. A special case occurs when the cost impact of the rest allowance α equals 1, which leads to constant total cost for q q and no savings. In fact, Eq. (19), as the derivative of the total cost function, is equal to 0 for α¼1. In Fig. 5, the results of the numerical example are plotted with α ¼1.
Fig. 3. Impact of the time required to perform a job cycle on the ergo-lot-size for item weights ranging from 0 to 20 kg.
no significant impact on production and consequently the cost for the process owner is just related to the hourly worker cost. The case where α is larger than 1 was also considered to take account of potential opportunity cost defined in the previous section. First, we calculate the optimal ergo-lot-size q and then the savings obtained by using the ergo-lot-size. These savings are calculated by comparing the minimal total cost of the ergo-lot-size model, Cðq Þ, to the total cost resulting from the traditional approach that minimizes the number of trips, Cðqmax Þ. The following parameters are varied during the analysis:
6. Conclusion
the ratio between the travel time and the total picking and storage 1 time, Rt=ps , calculated as 2⋅d s ⋅2⋅tps ; the item weight w; the cost impact of the rest allowance α.
Inventory management and lot-sizing decisions are key aspects for business success in many companies. Although the calculation of optimal lot-sizes has received much attention in academic research over the last decades, the implication of lot-sizing decisions on the human worker with regard to workload and accumulated fatigue have mainly been overlooked so far. Although it is undisputed that manual material handling processes pose a risk for workers to develop injuries, and in particular musculoskeletal disorders, works that consider ergonomic aspects in managerial lot-sizing decision support models are rare. This is surprising given the fact that lot-sizes determine the amount of manual material handling required, e.g. by influencing the quantity of items that has to be picked and stored or that has to be transported in a single trip between stocking locations. Considering only cost and neglecting the health impact of the lot-size on the workers leads to wrong managerial decisions, as lot-sizing policies that are disadvantageous from an ergonomic point of view can lead to increased worker fatigue and thus increased risks for workers developing occupational diseases. In the long run, this will lead to reduced performance and higher cost. To contribute to closing this research gap, this paper presented an approach for considering ergonomic aspects in a managerial lot-sizing model. A lot-sizing model was developed that integrates the concept of rest allowance, which implies that workers have to rest after performing a manual job cycle that led to physical strain. In this case, recovery is needed to avoid the accumulation of fatigue, which in turn would increase the risk of injuries. During the time of recovery, the worker is unproductive however, which may lead to production losses. The developed model considers the percentage of rest time necessary for different quantities of handled items and item weights. After introducing the rest allowance function, the total cost function was developed and the closed formula for the optimal ergo-lot-size was derived. The behaviour of the model was analysed in a comprehensive numerical analysis based
Fig. 3 illustrates the optimal ergo-lot-size for different item weights w. Please note that the y-axis uses a logarithmic scale. We limit the plots to three cases to demonstrate the behaviour of the model (the value of the ratio Rt=ps is equal to 6.25, 12.50 and 25.00, resulting for the travel distances of 50, 100 and 200 m). As can be seen, the optimal solution is directly proportional to the parameter Rt=ps and inversely proportional to the item weight w. No values are depicted for item weights lower than the threshold value w. This is due to the high number of movements required to fill the cart to full capacity, which leads to a high level of fatigue. For this reason, the model suggests to reduce the number of items in the cart and thus items per trip such that no rest allowance is required. In this case, the ratio of the time spent for travelling to the time required for picking/storing, Rt=ps , is the relevant factor for determining the optimal ergo-lot-size. In addition, the optimal value of q is proportional to the travel time 2d s . In fact, the higher the time required for transporting the lot-size from point “A” to point “B”, the lower is the number of trips to be performed in a certain period in order to reduce the total time required for transporting the items and consequently the travelling cost Ct ðqÞ. Moreover, the worker spends more time at a lower metabolic rate, reducing the average metabolic cost for completing a cycle. Finally, the optimal value of q is inversely proportional to the picking and storage time 2⋅tps , as the fatigue level increases due to higher metabolic cost of these activities. Regarding the savings obtained using the developed ergo-lot-size model as compared to the traditional approach that minimizes the number of trips, Fig. 4 illustrates that a general reduction in total cost can
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Fig. 4. Savings that result from using the ergo-lot-size model as compared to the traditional approach.
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Fig. 5. Special case where the cost impact of the rest allowance is equal to 1.
fatigue, such as maximum endurance time, or modelling other manual handling activities. In addition, future research could employ a case study collecting field data and employ biomechanical modelling to assess risk values for the manual tasks related to handling and transporting lotsizes. These results could then be integrated into an extension of the proposed model. In addition, it might be worth integrating ergonomic aspects, such as the concept of energy expenditure/metabolic cost, into the classical economic production/order quantity model that also considers setup/ordering cost and inventory carrying cost in a production/ purchasing context. Finally, considering the metabolic cost of manually pushing and pulling a hand trolley during transportation activities could be a valuable extension of this paper.
on an illustrative example, and several scenarios were investigated in a parametrical analysis. It was shown that the developed method using the rest allowance concept is suitable to integrate ergonomic aspects into a formal lot-sizing model. The developed approach can help to improve current industrial practices as companies usually do not take ergonomic assessment methods into account when defining rest times. Our model can provide valuable decision support to managers to offer rest times to employees when needed and as a result of their actual work content, instead of offering general break times. Considering rest allowances in the predictive planning of lot-sizes can help to reduce ergonomic risks associated with musculoskeletal disorders resulting from strenuous manual material handling. As for further managerial insights it can be noted that the developed model is helpful for managers in defining lot-sizes from an integrated perspective, which also facilitates the ergonomic classification of items and the evaluation of the distance between two stocking points. The model proposed in this paper could be used as a starting point for further research on ergonomic lot-sizing. Possible areas for future investigations could be the integration of other functions representing
Acknowledgements The authors are grateful to the editor and two anonymous referees for their valuable comments on an earlier version of this paper. The second and third authors also wish to thank the Carlo and Karin Giersch Stiftung for funding their research.
Appendix A. : Illustration of the modelling of metabolic cost using the equations introduced by Garg et al. (1978) We estimate the metabolic cost of the picking and storing activities of items with different weights, while we make several assumptions for the physical layout of the stocking locations “A” and “B”, the human worker, the time required to transport items from location “A” to the cart (and from the cart to location “B”), and execution time. We also assume that storing tasks are equal to picking tasks. In this Appendix A, we explain in more detail how the metabolic cost can be calculated for the picking activity. We assume a male worker with a weight of 75 kg. The items to be picked are stocked at different height levels (0.15, 0.40, 0.65, 0.90, 1.15, 1.40 m from the floor), and they can be loaded on two different levels of the cart (0.40, 1.40 m from the floor). The walking time from cart to location “A” is set to 1.25 s. The same value is used for the time required to carry items from location “A” to the cart. The total pick time depends on the height levels where the items to be retrieved are stored, and it is assumed to be 9, 8, 7, 7, 8 and 9 s for the heights 0.15, 0.40, 0.65, 0.90, 1.15, 1.40 m from the floor, respectively. The worker performs the following elementary movements for picking a single item at point “A” in the sequence as listed below, where the metabolic cost can be estimated using the equation introduced by Garg et al. (1978) and the keywords written in italics: – – – –
walking from the cart to location “A” (walking); picking one item from location “A” (squat lift for height lower than 0.81 m and arm lower for height higher than 0.81 m); carrying the item to the cart (carrying); stocking the item into the cart (squat lower for putting the item at 0.40 m or arm lift for putting it at 1.40 m).
The reader can find the necessary equations in Garg et al. (1978) or can use the Predetermined Energy Motion Systems developed by Battini et al. (2015b). Varying the item weight, we estimate the metabolic cost/energy expenditure rate as reported in Table A1.
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Table A1 Metabolic cost of picking and storing items for different item weights. w [kg] ̇
Eps [kcal/min]
0.5 3.80
1 3.90
2 4.07
5 4.6
10 5.46
15 6.33
20 7.21
25 8.08
Since the equations proposed by Garg et al. (1978) are directly proportional to the handled load (item weight), we can fix the other input variables (such as workers’ execution time etc.) and model the metabolic cost using a simple linear function of item weight: E_ ps ¼ a⋅w þ b. We estimate the parameters a and b of the linear function using the least squares method with a correlation index equal to 1 (see Fig. A1).
Fig. A1. Fitting of the metabolic cost function to the values provided by Garg et al. (1978) for different item weights.
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Contents lists available at ScienceDirect
International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe
Reprint of “Ergo-lot-sizing: An approach to integrate ergonomic and economic objectives in manual materials handling”☆ Daria Battini a, Christoph H. Glock b, Eric H. Grosse b, Alessandro Persona a, Fabio Sgarbossa a, * a b
Department of Management and Engineering, University of Padova, Stradella San Nicola 3, 36100 Padova, Vicenza, Italy Institute of Production and Supply Chain Management, Technische Universit€ at Darmstadt, Hochschulstr. 1, 64289 Darmstadt, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Lot-sizing Ergonomics Rest allowance In-house logistics Manual material handling
Over the last decades, academics and practitioners have paid much attention to lot-sizing, which determines economic order and production quantities by balancing inventory holding and setup cost. Recently, researchers have started to integrate sustainability issues into lot-sizing models. The focus of these works has been on environmental and economic dimensions of sustainability, however, while only few contributions studied the social aspect of this problem. Especially in in-house logistics, where a high amount of manual material handling is performed, lot-sizing decisions can have a significant impact on workload and human performance, which can have a strong influence on ergonomic parameters and thus worker welfare. The paper at hand extends previous research on ergonomic lot-sizing and introduces a new mathematical model that integrates ergonomic and economic aspects. A rest allowance function is used to take account of recovery periods that help to maintain low levels of fatigue and ergonomic risks. As recovery periods represent nonproductive time, the developed integrated model permits to estimate the economic impact of different workload levels. Finally, the model is applied in a numerical study, reflecting a typical manual material handling process. Based on the results of a parametrical analysis, we illustrate the applicability and validity of this approach to different industrial contexts.
1. Introduction The lot-sizing problem ranks among the most important decision problems in production and supply chain management. Over the last 100 years, a myriad of works has appeared on this topic, which made lotsizing one of the most extensively studied decision problems in operations management (see, for recent reviews, Glock et al. (2014) and Andriolo et al. (2014)). The dominant objective in lot-sizing research is the minimization of total cost, which is usually achieved by balancing inventory holding, setup and other costs affected by the lot-size. Recently, a new research stream that integrates sustainability issues into lot-sizing models has emerged. Examples in this area are works that consider: a) emissions resulting from transportation (Absi et al., 2013; Battini et al., 2014a),
b) carbon emissions as a function of the production lot-size or production rate (Jaber et al. 2013), c) how lot-sizing decisions affect warehousing operations and thus warehousing's environmental impact (Fichtinger et al. 2015), or d) that emissions and scrap can be controlled by varying production rates (Glock et al. 2012). The focus of prior research on sustainable lot-sizing has, however, mainly been on the economic and environmental dimensions of sustainability (such as emissions reduction). The social component (such as worker welfare) has, in contrast, not received much attention, even though the social component is frequently referred to as the third pillar of the ‘triple bottom line’-approach (Elkington, 1997). Within the social component, human factors and ergonomics are of major importance. According to Sanders and McCormick (1993), the objective of ergonomics research is “to enhance the effectiveness and efficiency with which
DOI of original article: https://doi.org/10.1016/j.ijpe.2017.01.010. ☆ This article is a reprint of a previously published article. For citation purposes, please use the original publication details; Journal of Production Economics, 185, pp. 230-239. * Corresponding author. E-mail addresses: [email protected] (D. Battini), [email protected] (C.H. Glock), [email protected] (E.H. Grosse), [email protected] (A. Persona), [email protected] (F. Sgarbossa). https://doi.org/10.1016/j.ijpe.2017.11.003 Available online 20 November 2017 0925-5273/© 2017 Published by Elsevier B.V.
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handling process. The remainder of the paper is structured as follows. The next section reviews literature on sustainable lot-sizing. Section 3 describes the problem under study in detail and explains necessary assumptions. Section 4 presents the mathematical formulation of the ergo-lot-size model, and Section 5 summarizes the results of a numerical experiment. Section 6 concludes the paper.
work and other activities are carried out. Included here would be such things as increased convenience of use, reduced errors, and increased productivity. The second objective is to enhance certain desirable human values, including improved safety, reduced fatigue and stress, increased comfort, greater user acceptance, increased job satisfaction, and improved quality of life.” Briefly, ergonomics aims at optimizing human well-being and overall system performance (Karwowski, 2005). Even though lot-sizing has mainly been studied in a manufacturing (economic production quantity) or purchasing (economic order quantity) context, lot-sizing decisions occur in other areas as well. In warehousing, for example, lot-sizes can be interpreted as the number of items that a warehouse worker has to handle in a single operation (e.g., when refilling the forward area in a warehouse or during line feeding processes). In such a typical in-house transportation scenario, it is clear that lot-sizes can have a significant impact on the amount of manual material handling required and the workloads assigned to workers (cf. Neumann and Medbo, 2010). Obviously, lot-sizing models that only consider the direct economic impact of lot-sizes may lead to wrong decision support in such a scenario by generating excessive workloads and increasing worker fatigue and, consequently, injury risks. For example, large lots that have to be handled (e.g., packed, labelled) or transported manually (e.g., by using a trolley or a cart) by the company's workers may lead to excessive pushing/pulling of a vehicle or lifting/carrying of products, which may result in worker fatigue and injuries (Jung et al., 2005; Knapik and Marras, 2009). Possible risk factors for injuries, in particular musculoskeletal disorders, include repetitive work with forceful exertions, awkward body postures, heavy lifting, insufficient recovery time, and rapid work pace during manual material handling (Punnett and Wegman, 2004). Clearly, all these risk factors may be influenced by lot-sizing decisions. A closer look at official statistics highlights the importance of these interdependencies: In the European Union, for example, musculoskeletal disorders account for over 38% of occupational diseases, with the lifting of loads being one of the most critical risk factors (EASHW, 2010). The economic burden of work-related musculoskeletal disorders accounts for up to 2% of the Gross National Product in the EU. For other regions, similar trends can be observed. For example, in the US, musculoskeletal disorders accounted for 32% of all worker injury cases in 2014, and stock and material movers incurred the highest number of these cases (BLS, 2014). This leads to an economic impact of musculoskeletal diseases in the US of about 5% of the Gross National Product (MEPS, 2014). Although authors called for the integration of ergonomics into managerial decision support models for in-house logistics due to the high amount of manual material handling and related risk factors (Grosse et al., 2015), interdependencies as the ones described above have thus far mainly been overlooked in the literature on lot-sizing. The work at hand contributes to closing this research gap by integrating the rest allowances of workers during task execution as an ergonomic variable into a lot-sizing model (for the rest allowance concept, the reader is referred to Price (1990) and Garg et al. (1978) and the literature overview in Section 2.2). The basic idea underlying the rest allowances concept is that manual work leads to fatigue of the workers involved, which makes it necessary to assign rest periods (rest allowance) to them from time to time to avoid high levels of fatigue which can lead to injury risks. As the lot-size directly influences workload and thus worker fatigue, the rest allowance (as an unproductive time) is integrated into the total cost function of the lot-sizing model, which links the ergonomic variables to the cost minimization objective. The developed integrated model permits to estimate the influence of the lot-size on the worker's workload level and the economic impact of the workload level on the total cost of the system. This paper is the first to propose a simple closed-form expression for the optimal ergonomic lot-size. In the optimal solution, the total cost of the process consisting of the direct material handling cost (picking, travelling and storing) and the cost of the unproductive rest periods necessary to maintain a low fatigue level are minimized. The model is applied in a numerical study, reflecting a typical manual material
2. Literature review Sustainable supply chain management has attracted an increased attention in recent years (Seuring and Müller 2008). Especially the manufacturing sector has been active in implementing sustainability initiatives, where both manufacturing strategies as well as the tactical and operational execution of production tasks have been influenced by sustainability objectives (Sarkis, 2001). In the area of production planning and control, this trend has given rise to the development of lot-sizing models that consider sustainability issues besides the minimization of total operational cost. Wahab et al. (2011), for example, developed an Economic Order Quantity (EOQ) model for a two-stage closed-loop supply chain that considers CO2 emissions. The authors assumed that two types of CO2 emissions occur in this system: fixed emissions, which are a function of the distance between the vendor and the buyer, the fuel efficiency of the vehicle used and other system-related factors, and variable emissions, which are a function of the lot-size. CO2 emissions were assumed to result in an environmental cost in this paper. Chen et al. (2013) developed a carbon-constrained EOQ model and assumed that a constraint on the maximum amount of CO2 emissions generated in production exists. Glock et al. (2012) developed a sustainable Economic Production Quantity (EPQ) model, but again concentrated on the environmental and economic dimensions of sustainability by incorporating an emission-based environmental quality index in their lot-sizing model. Other inventory models that consider greenhouse gas emissions are the works of Jaber et al. (2013), Battini et al. (2014a), Gurtu et al. (2015), Bouchery (2012), or He et al. (2015). Surprisingly, prior research on sustainable lot-sizing has almost unanimously interpreted “sustainable” as “environmentally friendly”. While sustainability clearly includes an ecological objective, there is more and more consent that sustainability also has an economic and a social dimension (e.g., Elkington, 1997). Especially the social component, which includes such aspects as worker welfare and ergonomics, for example, has largely been ignored in prior research on lot-sizing. Various different concepts and methods are generally suitable for integrating ergonomic aspects into managerial decision support models. Reviewing the extensive literature on ergonomic assessment methods is, however, not within the scope of this paper (the reader may refer to Chiasson et al. (2012) for an overview of ergonomic assessment methods). An analytical assessment model is the concept of energy expenditure, which allows the estimation of energy required to execute a manual task as a function of oxygen consumption (Garg et al., 1978). Garg et al. (1978) differentiated between base postures (such as sitting, standing and standing in bent position) of the worker during task execution and the movements he/she performs while completing the task (such as lifting, lowering, walking, carrying loads). The authors assumed that the energy expenditure in kilocalories per minute can be calculated by summing up the energy required to maintain different body posture and for performing the task, divided by the time required for performing the task in minutes. Other energy expenditure models for walking and carrying that consider worker characteristics and load weight were proposed by Aberg et al. (1967), Pimental and Pandolf (1979), Taboun and Dutta (1989), and Battini et al. (2015b). Examples of frequently used ergonomic assessment methods in practice are the NIOSH lifting equation (Dempsey, 2002), the OWAS method (De Bruijn et al., 1998), or the European assembly worksheet (Schaub et al., 2013). In addition, real-time integrated motion capture 33
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cost and ergonomic risks, where the latter was assessed using a lifting index. Andriolo et al. (2016) did not consider the rest allowance concept, but instead compared the cost objective and the ergonomic objective using Pareto frontiers. Also in the work of Andriolo et al. (2016), no closed-form to calculate the exact value of the optimal lot-size considering ergonomic aspects was proposed. Our brief overview of the literature showed that only a few works analysed manual material handling processes (such as lot-sizing decisions) from an integrated perspective that includes both economic and ergonomic aspects. With a few notable exceptions (Battini et al., 2015a; Andriolo et al., 2016), there appears to be a large gap in the literature regarding works that consider ergonomic aspects in managerial decision support models, in particular in determining optimal lot-sizes. The paper at hand contributes to closing this research gap by developing an economic lot-sizing model that considers both ergonomic and economic aspects. In particular, this work extends previous studies (Battini et al., 2015a; Andriolo et al., 2016) by:
Fig. 1. Illustration of a single job cycle.
systems can be used to measure muscle activity and workload (Battini et al., 2014b). To the best of the authors’ knowledge, only a few papers considered ergonomic issues in calculating optimal lot-sizes. One example is the work of Battini et al. (2015a), which included human energy expenditure in a lot-sizing model. The authors assumed that larger lot-sizes lead to higher energy expenditure at the worker, which makes breaks necessary from time to time to give the worker time to recover. They introduced a model to estimate the total time spent in the process, including the rest allowance, but they did not consider cost associated with the lot-size. Their approach consists in some simplified equations used to estimate the fatigue of the worker, based on the energy expenditure rate. The optimal solution is not calculated using a closed-form expression in their paper, but instead a set of suitable scenarios is evaluated to identify the best production strategy. Andriolo et al. (2016) proposed another lot-sizing model that considers ergonomic aspects. In contrast to Battini et al. (2015a), the authors proposed a multi-objective optimization approach that calculates Pareto-optimal solutions that minimize both
1) modelling the overall ergonomics level of a traditional manual material handling process through the use of energy expenditure rate equations (Garg et al., 1978); 2) integrating the rest allowances (including threshold levels for accumulating fatigue) of workers during task execution as an ergonomic variable into a lot-sizing model; 3) introducing a general economic model (total cost function) of the ergonomic lot-sizing problem and estimating the economic impact of different workload levels in manual material handling; 4) deriving a closed-form expression for calculating the optimal lot-size considering ergonomic aspects by minimizing the total cost function. We refer to this closed-form expression as the “ergo-lot-size” in the following. 3. Problem description This section describes the problem studied in this paper and summarizes the necessary assumptions and notations. In addition, the rest allowance formulation as well as the objective of the developed lot-sizing model are described.
Table 1 List of notations. Notations
Description
Q q q w Wmax qmax ¼ Wwmax d Nt ðqÞ
total amount of items to be handled in a given period [pcs] lot-size [pcs] optimal ergo-lot-size [pcs] item weight [kg] maximum load weight transported by the cart [kg] capacity of the cart [pcs]
s tps ¼ tp ¼ ts Tp ðqÞ ¼ q⋅tp Ts ðqÞ ¼ q⋅ts Tt ¼ 2d s Rt=ps cop RAðqÞ Ė tot ðqÞ Ė t Ė ps ðqÞ¼ E ̇ p ðqÞ¼ E ̇ s ðqÞ a, b α CðqÞ
3.1. Scenario under study The problem studied in this paper is a typical manual material handling and transportation process, where a certain amount of products (lot-size) has to be handled in a production or logistics facility. This scenario occurs when transporting products from the final stage of a production system to a storage facility, when refilling an order picking warehouse from the reserve area, or when feeding an assembly line, for example. Within this scenario, we study the following single job cycle (see Fig. 1):
distance between stock point “A” and “B” [m] number of trips necessary to ship the total amount of items Q from point “A” to point “B” constant travel speed [m/s] unitary picking/storing time [s] total time required to pick the lot q from point “A” and load it on the cart [s] total time required to unload the lot q form the cart and stock it at point “B” [s] total travel time required to transport the lot q from point “A” to point “B” and for returning to point “A” [s] ratio between the travel time and the unitary picking and storage 1 time, expressed by 2d s ⋅2⋅tps
1) We consider a stocking point “A” where items are produced on a machine and temporarily stored until they are shipped to the next stocking point “B”. 2) Human workers are responsible for transporting the items from point “A” to point “B”. They manually pick a certain quantity of products at point “A” and put them on an electric cart, which is used for transporting the items (lot-size) to point “B”. 3) At point “B”, all items are removed from the cart and stored in a rack (e.g., in an order picking warehouse or another storage facility). 4) After having stocked the full lot-size in the rack at point “B”, the worker returns to point “A” to pick up a new lot there. We refer to the whole process of picking the lot at point “A”, transporting it to point “B” and unloading and storing it there as well as returning to point “A” as a job cycle.
unitary worker cost [€/s] rest allowance necessary to maintain a low level of fatigue and to avoid ergonomic risks [%] average metabolic cost for completing each cycle [kcal/min] metabolic cost for transportation activity [kcal/min] metabolic cost for picking/storing activity [kcal/min] parameters of the linear function to estimate Ė ps ðqÞ cost impact of the rest allowance, as a multiplier of the unitary worker cost. Its value is greater than or equal to 1 total cost function [€]
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w⋅q, and it depends on the physical layout of the stocking locations “A” and “B”. As reported in the previous section, we assume that the worker is in standing position during transportation activities and that an electric cart is used to transport the lot-size q, which leads to fixed metabolic cost of this activity E_ t equal to 1.86 kcal/min. Eq. (3) estimates the average metabolic cost per cycle as the weighted average of the metabolic cost per task performed in the entire cycle as follows:
5) After finishing one job cycle, i.e. after returning to point “A”, the worker needs to rest (note the rest allowance model is explained in detail in Section 3.3). 3.2. Assumptions To develop the mathematical model based on the scenario under analysis, the following assumptions are made: after finishing a single job cycle, the worker needs to rest; an electric cart is used to transport the items from “A” to “B”; during transportation, the worker maintains a standing position; picking and storing activities consume the same amount of time, as the same movements and body postures have to be performed for both tasks, and as the weight of items to be stored or to be retrieved is the same; only the specific job cycle (see Section 3.1) is considered, and it is assumed that the worker in question is responsible only for the tasks associated with the job cycle at hand; only a single worker is considered; the hourly cost of the break could be higher than the worker's wage cost due to a possible opportunity cost. We assume that during the break, the worker rests exclusively without performing any valueadding tasks.
E_ tot ðqÞ ¼
q¼
3.4. Rest allowance
w¼
We model the rest time as a percentage of the total time spent on completing a cycle using the relaxation formulas introduced by Price (1990), where the relaxation allowance depends on the acceptable work load level of 300 W, which is equal to 4.3 kcal/min. Other approaches could be used, such as the one developed by Rohmert (1973), where the rest time after task execution is exponential in the time spent on performing the task and the metabolic cost. Based on the formulation introduced by Price (1990), the rest allowance can be defined as:
4:3 b a
(5)
8 > < E_ tot ðqÞ 4:3 if w > w and q > q RAðqÞ ¼ 4:3 E_ t > : 0 otherwise
(1)
otherwise
if E_ tot ðqÞ > 4:3 kcal=min
(4)
Using the equations developed by Garg et al. (1978), the parameters can be estimated as a ¼0.17403 and b ¼3.7206 (see Appendix A), so the threshold value w is equal to 3.3 kg. This is in line with the general international guidelines for occupational safety in manual material handling, where the weight of about 3 kg usually is the limit for low loads (ISO 11228-1, 2009; ISO 11228-2, 2009; ISO 11228-3, 2009). Finally, using the two developed threshold values, the rest allowance can be defined as:
(6)
In case w > w and q > q, the rest allowance can be rewritten as:
where MWRðqÞ is the Mean Work Rate, expressed in Watt, and RR is the Relaxation Rate, which equals 130 W (about 1.86 kcal/min) for the body posture “standing”, for example. One frequently used approach to estimate the metabolic cost of a manual task was developed by Garg et al. (1978). Garg et al. (1978) introduced simple equations to assess the metabolic cost of several manual material handling jobs based on observations made in laboratory experiments. Modifying Eq. (1) by expressing metabolic cost in kcal/min (1 W¼69.77 kcal/min, ISO 80000-5, 2007) leads to the following equation:
8 > < E_ tot ðqÞ 4:3 RAðqÞ ¼ 4:3 E_ t > : 0
2d 1 4:3 E_ t ⋅ ⋅ s 2⋅tps E_ ps 4:3
This is verified for E_ ps 4:3 > 0, so the metabolic cost for picking and storing activities has to be larger than 4.3 kcal/min. As was mentioned before, the equations developed by Garg et al. (1978) can be applied to estimate the value of E_ ps . Given the physical configuration of stocking points “A” and “B”, worker weight, walking speed and task time, E_ ps can be expressed by a linear function of w, such as E_ ps ¼ a⋅w þ b, where the parameters a and b depend on the layout of locations “A” and “B” (see the Appendix A for more details). Based on this assumption, the constraint E_ ps 4:3 > 0 changes to a⋅w þ b 4:3 > 0 and another threshold value expressed as a lower bound of item weight can be defined:
Table 1 summarizes the notations used in the mathematical model.
if MWRðqÞ > 300 Watt
(3)
Note that the relaxation time is necessary if E_ tot ðqÞ>4.3 kcal/min, so a threshold value expressed as a function of the lot-size and item weight can be defined as:
3.3. Notations
8 < MWRðqÞ 300 300 RR RAðqÞ ¼ : 0
2⋅E_ ps ⋅q⋅tps þ E_ t ⋅2ds 2⋅q⋅tps þ 2ds
E_ tot ðqÞ 4:3 RAðqÞ ¼ ¼ 4:3 E_ t
2⋅E_ps ⋅q⋅tps þE_t ⋅2d s 2⋅q⋅tps þ2d s
4:3
4:3 E_ t
(7)
The energy expenditure concept (Garg et al., 1978; Price, 1990) has frequently been used to estimate metabolic cost of repetitive manual material handling tasks in the literature (e.g., Battini et al., 2016; Calzavara et al., 2016). It is an easy-to-understand and easy-to-use approach for considering ergonomic conditions in a formal model as the energy expenditure is expressed in kilocalories (kcal) per unit of time (minute). This makes it possible to consider ergonomic aspects in a time/cost model. Other ergonomic risk measurement methods, such as the NIOSH lifting equation or the OWAS index, have limitations as they produce semi-quantitative risk indices for performing specific manual tasks in different body postures, which are difficult to consider in the objective function of an optimization model. The NIOSH lifting equation, for example, calculates recommended weight limits for lifting tasks only, which makes it unsuitable for the job cycle considered in this paper (see
(2)
otherwise
We assume that the time spent for picking and storing activities is a function of q, and that the worker needs the same unitary time for picking and storing, i.e. tps ¼ tp ¼ ts (see also Battini et al. (2015a)). The metabolic cost E_ ps ðqÞ increases with the handled load during task execution,
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lower than a few thousands Euro and can be used for many years, so the cost of the equipment per order is negligible as compared to the unitary worker cost. Eq. (12) considers the rest allowance RAðqÞ necessary for workers in each cycle to recover from fatigue. In addition, a cost factor α (greater than or equal to 1) that takes account of the opportunity cost of the resting phase is included in the objective function to take account of the fact that the worker does not work during rest times. As a result, our model does not only consider the direct cost of the worker, but also the potential cost impact of the production loss (non-productive time). In fact, the resting phase is not productive and it will impact the total cost if the worker has a high saturation level. In this case, the worker needs an extra-time to complete the process with an added cost or the process is not completed on time with potentially expensive effects on the following step of the logistic process. This additional cost impact of the resting phase is modelled using the factor α. The objective function can be rewritten as follows:
Section 3.1). OWAS, in turn, is very restrictive in formulating weight limits. More specifically, it considers just three categories for the handled load – namely less than 10 kg, between 10 kg and 20 kg and over 20 kg. Its use is therefore not helpful for coordinating a manual material handling process and for calculating the optimal ergo-lot-size. 3.5. Objective The objective of the model is to find the optimal ergo-lot-size that minimizes the total cost resulting from the material handling process under study. Thus, the quantity that the worker handles in each job cycle (i.e., the lot-size q) is the decision variable, and its value depends on: the standard time spent for each activity, i.e. picking, travelling and storing (see phases 1–4 of the job cycle in Section 3.1); the rest allowances necessary to guarantee an acceptable fatigue level, which is a function of the energy expenditure/metabolic cost (Garg et al., 1978; Price, 1990) of the worker during the execution of each task (see phase 5 of the job cycle in Section 3.1); the unitary worker wage cost.
2d Q ⋅cop ⋅ þ ts ⋅cop ⋅Qþ s q Q þRAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop q CðqÞ ¼ tp ⋅cop ⋅Q þ
We model this problem in the next section using an analytical formulation of the total cost function for the job cycle described above, which depends on the total time spent to perform this typical in-house logistics process. We use a rest allowance function to integrate ergonomic aspects into lot-sizing decisions. By minimizing the total cost function, the optimal ergo-lot-size can be derived.
(13)
Typically, the picking and storing activities are very similar in time, as the same movements and body postures have to be performed, and as the weight of items to be stored or to be retrieved is the same. We therefore suggest using the same unitary time for both activities, i.e. tps ¼ tp ¼ ts . The model can thus be simplified as follows:
CðqÞ ¼ 2tps ⋅cop ⋅Q þ
4. Model description and mathematical formulation of the ergolot-size
Q 2d Q ⋅cop ⋅ þ RAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop : s q q (14)
4.1. Model description 4.2. Optimal solution of the integrated model Based on the problem description introduced in Section 3, we model the total cost function as the sum of four cost terms related to the main activities performed during the process under study, i.e. picking Cp ðqÞ, travelling Ct ðqÞ, storing Cs ðqÞ, and resting CRA ðqÞ. We define the objective function (Eq. (8)) and the related individual cost terms (Eqs. (9)–(12)) as follows:
CðqÞ ¼ Cp ðqÞ þ Ct ðqÞ þ Cs ðqÞ þ CRA ðqÞ;
As we consider two different thresholds w and q, the integrated model has two different solutions, based on the presence of the rest allowance. Case 1: w w. If w w, the rest allowance is equal to 0, so the cost function simplifies to the traditional lot-sizing model that considers only picking, storing and transportation activities:
(8) CðqÞ ¼ 2tps ⋅cop ⋅Q þ
where
Cp ðqÞ ¼ Tp ðqÞ⋅cop ⋅Nt ðqÞ ¼ q⋅tp ⋅cop ⋅
Ct ðqÞ ¼ Tt ⋅cop ⋅Nt ðqÞ ¼
Q ¼ tp ⋅cop ⋅Q; q
2d Q ⋅cop ⋅ ; s q
(10)
(11)
Q CRA ðqÞ ¼ RAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop : q
(12)
(15)
Minimizing the cost function leads to an optimal ergo-lot-size of q ¼ ∞, so q ¼ qmax , where qmax is the maximum number of items that can be transported by the cart. In a practical application, the capacity of the cart would usually vary between 500 kg and 1000 kg, depending on the type of cart used. Thus, if the item weight is lower than 3.3 kg (see Section 3.4), the worker can concentrate on optimizing the number of trips without having to take account of others factors such as travelled distance, picking/storage time, or the cost impact of the rest allowance. Case 2: w > w Since the total cost function depends on the variable q, it is necessary to study the behaviour of Eq. (14) for q q and for q > q. In the first case, with q q, the total cost function is expressed by Eq. (15) because the E_ tot ðqÞ>4.3 kcal/min and then RAðqÞ ¼ 0 (see Eq. (6)). Under this condition, the total cost function does not include the rest allowance cost term, so it is a strictly decreasing function due to its
(9)
Q Cs ðqÞ ¼ Ts ðqÞ⋅cop ⋅Nt ðqÞ ¼ q⋅ts ⋅cop ⋅ ¼ ts ⋅cop ⋅Q; q
2d Q ⋅cop ⋅ s q
Eqs. (9) and (11) calculate the cost for picking and storing activities. They are expressed as the total time spent on picking and storing the total amount of items Q, multiplied with the unitary worker cost. As can be seen, these cost functions do not depend on the lot-size, but instead only on the total amount of items processed. Eq. (10), in turn, describes the cost of travelling. This cost term does not explicitly consider the cost of the used equipment because of its low value. The cost of a typical electric cart used in handling activities as the one considered here are usually
∂Ct ðqÞ Q 2d negative first partial derivative: ∂CðqÞ ∂q ¼ ∂q ¼ s ⋅cop ⋅q2 ;.
When q > q, the worker needs to rest after each job cycle in order to avoid high accumulated fatigue and thus ergonomic risks, as indicated by Eq. (6). In this case, the total cost function includes CRA ðqÞ (see Eq. (14)). The cost term related to the rest allowance can be written as:
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the end of a production system (point “A”) to a storage facility (point “B”) using an electric cart operated by a human worker. The weight w of each item is 6 kg. The distance between point “A” and point “B” is 100 m. The unitary picking time and the unitary storage time are both 8 s per item, while the travel speed is 1 m/s. The cost impact of the rest allowance α is equal to 4. The unitary worker cost cop is equal to 15 €/h (or 15/3600 €/s). The metabolic cost of the transportation activity, E_ t , is fixed and equal to 1.86 kcal/min, while the metabolic cost of the picking and storing activities is modelled as E_ ps ¼ 0:1743⋅w þ 3:7206(see Appendix A). In this case, w > w holds, and thus the optimal ergo-lot-size q can be calculated according to Eq. (4):
q ¼ q ¼
2⋅d 1 4:3 E_ t 2⋅100 1 4:3 1:86 ⋅ ⋅ ⋅ ⋅ ¼ s 2⋅tps E_ ps 4:3 1 2⋅8 0:17403⋅6 þ 3:7206 4:3
¼ 65 items Using this lot-size, we obtain the total weight for each trip as 390 kg, which needs to be compared to the capacity of the cart, which is typically between 500 kg and 1000 kg. The minimal total cost for transporting 1000 items from point “A” to point “B”, Cðq Þ, equals 78.49 € according to Eq. (14), and it is the sum of Cp ðq Þ ¼ Cs ðq Þ ¼33.33 € and Ct ðq Þ ¼12.83 €, while the rest allowance cost CRA ðq Þ ¼0 €. This is due to the fact that w > w and the optimal ergo-lot-size is equal to the threshold value q, where E_ tot ðqÞ ¼ 4:3 kcal=min, so RAðqÞ¼0. Fig. 2 shows the total cost curve CðqÞ for alternative lot-sizes q. As discussed before, under the traditional lot-sizing approach in materials handling, it is common to minimize the number of trips under the constraint of the capacity of the cart, which depends on the size and weight of the items (cf. Grosse et al., 2014). In this case, the maximum load weight transported by the cart is the upper bound of the lot-size. In the numerical example considered here, the cart has a capacity of 750 kg and can thus transport a maximum of 125 items each weighting 6 kg per trip, referred to as qmax. We calculate the total cost for qmax ¼125 and compare it to the minimum value obtained for q*. In the case where qmax ¼125, the total cost equals 97.64 € as the sum of Cp ðqmax Þ ¼ Cs ðqmax Þ ¼33.33 €, Ct ðqmax Þ ¼6.67 € and CRA ðqmax Þ ¼24.31 €. The percentage savings in total cost using the optimal ergo-lot-size q* equals 19.6%. We note that the picking and storing cost are not affected by the value of the lot-size, as is demonstrated by Eqs. (9) and (11), while the transportation and rest allowance cost are different in this case. It is clear that the transportation cost is lower in the second case where the traditional approach tends to saturate the cart to minimize the number of trips, but it implies a higher fatigue level of the worker and consequently an increase in cost linked to unproductive recovery time.
Fig. 2. Total cost curve for alternative lot-sizes.
Table 2 Input values used in the parametrical analysis. Variable
Value (s)
Q cop w d s tps ¼ tp ¼ ts α Wmax
1000 15 €/h ¼ 15/3600 €/s from 1 to 20 kg, each 0.5 kg from 25 to 250 m, each 25 m 1 m/s 8s 1; 2; 4 500, 750 and 1000 kg
Q CRA ðqÞ ¼ RAðqÞ⋅ Tp ðqÞ þ Tt þ Ts ðqÞ ⋅ ⋅α⋅cop q
(16)
Substituting Eqs. (7) and (12) into Eq. (16), Eq. (16) can be reformulated as follows (the simplification steps can be obtained from the authors upon request):
E_ ps 4:3 2d Q α⋅cop ⋅ ⋅ CRA ðqÞ ¼ α⋅cop ⋅2⋅tps ⋅Q⋅ s q 4:3 E_ t
(17)
The total cost function can now be written as follows:
E_ ps 4:3 2d Q 2d Q α⋅cop ⋅ ⋅ CðqÞ ¼ 2tps ⋅cop ⋅Q þ ⋅cop ⋅ þ α⋅cop ⋅2⋅tps ⋅Q⋅ s q s q 4:3 E_ t (18)
5.2. Parametrical analysis
The first partial derivative of Eq. (18) with respect to q is positive as:
∂CðqÞ 2d Q 2d Q 2d Q ¼ ⋅cop ⋅ 2 þ α⋅cop ⋅ ⋅ 2 ¼ cop ⋅ ⋅ 2 ⋅ðα 1Þ ∂q s q s q s q
In this section, we apply the developed mathematical model to several scenarios using input parameter values that were inspired by observations made in an industrial application (see Table 2), and we analyse their impact on the optimal ergo-lot-size. In particular, we change the weight of the item from 1 to 20 kg in order to consider the handling of different types of materials, such as small and heavy boxes. The distance values considered in this analysis cover many different cases, e.g. scenarios where these locations are close to each other as well as scenarios where the picking and storing points are far away from each other, such as in different shop-floors or in different warehousing areas (reserve and forward area). The speed of the cart and the picking/storing time are assumed constant. The maximum load weight transported by the cart is fixed to three different values representative of the carts available in typical material handling systems catalogues. Concerning the cost impact of the rest allowance, we analyse the effect of three levels. When α is equal to 1, the rest period of the worker has
(19)
Finally, since the total cost function is strictly a decreasing function for q q and strictly increasing for q > q, the optimal ergo-lot-size q* can then be defined as follows:
q ¼ q ¼
2⋅d 1 4:3 E_ t ⋅ ⋅ s 2⋅tps E_ ps 4:3
(20)
5. Numerical example and parametrical analysis 5.1. Numerical example First, we describe the application of the new model to an illustrative example, where Q¼1000 items have to be moved from a buffer stock at 37
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be achieved, with the savings value depending on the total cost Cðqmax Þ. In this analysis, the savings are subject to the cost impact of the rest allowance and the maximum capacity of the cart. For an increasing cost impact of the rest allowance, we can observe a proportional increase in the percentage of savings. The savings thereby depend on the ratio of the manual activities’ execution time and the capacity of the cart. Following the explanation given in the analysis of the optimal ergo-lot-size value, the savings are larger in case of shorter transportation distances. In this case, the capacity of the cart has a relevant impact on the savings. In particular, there is no difference between the two approaches when the capacity of the cart is low and the distance between points “A” and “B” is long. In case the travelled distance is long, the ratio Rt=ps is high and the optimal ergo-lot-size is large (see Fig. 3), especially for low item weights. This results in an optimal total weight per trip larger than the capacity of the cart, so the latter will also be the optimal ergo-lot-size. For example, in case of a cart capacity of 500 kg and a travel distance of 200 m with an item weight of 6 kg, the ergo-lot-size using Eq. (4) should be 131 items per trip, corresponding to 786 kg on the cart; here, the lot-size clearly exceeds the capacity of the cart. In this particular case, the ergo-lot-size model obtains savings as compared to the traditional approach if the item weight exceeds 11 kg. This is due to the low capacity of the cart, which has a higher impact on savings if the item weight is high. A higher capacity of the cart allows higher savings also for lower item weights. A special case occurs when the cost impact of the rest allowance α equals 1, which leads to constant total cost for q q and no savings. In fact, Eq. (19), as the derivative of the total cost function, is equal to 0 for α¼1. In Fig. 5, the results of the numerical example are plotted with α ¼1.
Fig. 3. Impact of the time required to perform a job cycle on the ergo-lot-size for item weights ranging from 0 to 20 kg.
no significant impact on production and consequently the cost for the process owner is just related to the hourly worker cost. The case where α is larger than 1 was also considered to take account of potential opportunity cost defined in the previous section. First, we calculate the optimal ergo-lot-size q and then the savings obtained by using the ergo-lot-size. These savings are calculated by comparing the minimal total cost of the ergo-lot-size model, Cðq Þ, to the total cost resulting from the traditional approach that minimizes the number of trips, Cðqmax Þ. The following parameters are varied during the analysis:
6. Conclusion
the ratio between the travel time and the total picking and storage 1 time, Rt=ps , calculated as 2⋅d s ⋅2⋅tps ; the item weight w; the cost impact of the rest allowance α.
Inventory management and lot-sizing decisions are key aspects for business success in many companies. Although the calculation of optimal lot-sizes has received much attention in academic research over the last decades, the implication of lot-sizing decisions on the human worker with regard to workload and accumulated fatigue have mainly been overlooked so far. Although it is undisputed that manual material handling processes pose a risk for workers to develop injuries, and in particular musculoskeletal disorders, works that consider ergonomic aspects in managerial lot-sizing decision support models are rare. This is surprising given the fact that lot-sizes determine the amount of manual material handling required, e.g. by influencing the quantity of items that has to be picked and stored or that has to be transported in a single trip between stocking locations. Considering only cost and neglecting the health impact of the lot-size on the workers leads to wrong managerial decisions, as lot-sizing policies that are disadvantageous from an ergonomic point of view can lead to increased worker fatigue and thus increased risks for workers developing occupational diseases. In the long run, this will lead to reduced performance and higher cost. To contribute to closing this research gap, this paper presented an approach for considering ergonomic aspects in a managerial lot-sizing model. A lot-sizing model was developed that integrates the concept of rest allowance, which implies that workers have to rest after performing a manual job cycle that led to physical strain. In this case, recovery is needed to avoid the accumulation of fatigue, which in turn would increase the risk of injuries. During the time of recovery, the worker is unproductive however, which may lead to production losses. The developed model considers the percentage of rest time necessary for different quantities of handled items and item weights. After introducing the rest allowance function, the total cost function was developed and the closed formula for the optimal ergo-lot-size was derived. The behaviour of the model was analysed in a comprehensive numerical analysis based
Fig. 3 illustrates the optimal ergo-lot-size for different item weights w. Please note that the y-axis uses a logarithmic scale. We limit the plots to three cases to demonstrate the behaviour of the model (the value of the ratio Rt=ps is equal to 6.25, 12.50 and 25.00, resulting for the travel distances of 50, 100 and 200 m). As can be seen, the optimal solution is directly proportional to the parameter Rt=ps and inversely proportional to the item weight w. No values are depicted for item weights lower than the threshold value w. This is due to the high number of movements required to fill the cart to full capacity, which leads to a high level of fatigue. For this reason, the model suggests to reduce the number of items in the cart and thus items per trip such that no rest allowance is required. In this case, the ratio of the time spent for travelling to the time required for picking/storing, Rt=ps , is the relevant factor for determining the optimal ergo-lot-size. In addition, the optimal value of q is proportional to the travel time 2d s . In fact, the higher the time required for transporting the lot-size from point “A” to point “B”, the lower is the number of trips to be performed in a certain period in order to reduce the total time required for transporting the items and consequently the travelling cost Ct ðqÞ. Moreover, the worker spends more time at a lower metabolic rate, reducing the average metabolic cost for completing a cycle. Finally, the optimal value of q is inversely proportional to the picking and storage time 2⋅tps , as the fatigue level increases due to higher metabolic cost of these activities. Regarding the savings obtained using the developed ergo-lot-size model as compared to the traditional approach that minimizes the number of trips, Fig. 4 illustrates that a general reduction in total cost can
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Fig. 4. Savings that result from using the ergo-lot-size model as compared to the traditional approach.
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Fig. 5. Special case where the cost impact of the rest allowance is equal to 1.
fatigue, such as maximum endurance time, or modelling other manual handling activities. In addition, future research could employ a case study collecting field data and employ biomechanical modelling to assess risk values for the manual tasks related to handling and transporting lotsizes. These results could then be integrated into an extension of the proposed model. In addition, it might be worth integrating ergonomic aspects, such as the concept of energy expenditure/metabolic cost, into the classical economic production/order quantity model that also considers setup/ordering cost and inventory carrying cost in a production/ purchasing context. Finally, considering the metabolic cost of manually pushing and pulling a hand trolley during transportation activities could be a valuable extension of this paper.
on an illustrative example, and several scenarios were investigated in a parametrical analysis. It was shown that the developed method using the rest allowance concept is suitable to integrate ergonomic aspects into a formal lot-sizing model. The developed approach can help to improve current industrial practices as companies usually do not take ergonomic assessment methods into account when defining rest times. Our model can provide valuable decision support to managers to offer rest times to employees when needed and as a result of their actual work content, instead of offering general break times. Considering rest allowances in the predictive planning of lot-sizes can help to reduce ergonomic risks associated with musculoskeletal disorders resulting from strenuous manual material handling. As for further managerial insights it can be noted that the developed model is helpful for managers in defining lot-sizes from an integrated perspective, which also facilitates the ergonomic classification of items and the evaluation of the distance between two stocking points. The model proposed in this paper could be used as a starting point for further research on ergonomic lot-sizing. Possible areas for future investigations could be the integration of other functions representing
Acknowledgements The authors are grateful to the editor and two anonymous referees for their valuable comments on an earlier version of this paper. The second and third authors also wish to thank the Carlo and Karin Giersch Stiftung for funding their research.
Appendix A. : Illustration of the modelling of metabolic cost using the equations introduced by Garg et al. (1978) We estimate the metabolic cost of the picking and storing activities of items with different weights, while we make several assumptions for the physical layout of the stocking locations “A” and “B”, the human worker, the time required to transport items from location “A” to the cart (and from the cart to location “B”), and execution time. We also assume that storing tasks are equal to picking tasks. In this Appendix A, we explain in more detail how the metabolic cost can be calculated for the picking activity. We assume a male worker with a weight of 75 kg. The items to be picked are stocked at different height levels (0.15, 0.40, 0.65, 0.90, 1.15, 1.40 m from the floor), and they can be loaded on two different levels of the cart (0.40, 1.40 m from the floor). The walking time from cart to location “A” is set to 1.25 s. The same value is used for the time required to carry items from location “A” to the cart. The total pick time depends on the height levels where the items to be retrieved are stored, and it is assumed to be 9, 8, 7, 7, 8 and 9 s for the heights 0.15, 0.40, 0.65, 0.90, 1.15, 1.40 m from the floor, respectively. The worker performs the following elementary movements for picking a single item at point “A” in the sequence as listed below, where the metabolic cost can be estimated using the equation introduced by Garg et al. (1978) and the keywords written in italics: – – – –
walking from the cart to location “A” (walking); picking one item from location “A” (squat lift for height lower than 0.81 m and arm lower for height higher than 0.81 m); carrying the item to the cart (carrying); stocking the item into the cart (squat lower for putting the item at 0.40 m or arm lift for putting it at 1.40 m).
The reader can find the necessary equations in Garg et al. (1978) or can use the Predetermined Energy Motion Systems developed by Battini et al. (2015b). Varying the item weight, we estimate the metabolic cost/energy expenditure rate as reported in Table A1.
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Table A1 Metabolic cost of picking and storing items for different item weights. w [kg] ̇
Eps [kcal/min]
0.5 3.80
1 3.90
2 4.07
5 4.6
10 5.46
15 6.33
20 7.21
25 8.08
Since the equations proposed by Garg et al. (1978) are directly proportional to the handled load (item weight), we can fix the other input variables (such as workers’ execution time etc.) and model the metabolic cost using a simple linear function of item weight: E_ ps ¼ a⋅w þ b. We estimate the parameters a and b of the linear function using the least squares method with a correlation index equal to 1 (see Fig. A1).
Fig. A1. Fitting of the metabolic cost function to the values provided by Garg et al. (1978) for different item weights.
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