Workforce planning in food packaging
Master’s Thesis SCMMSc Supply Chain Management Faculty of Economics and Business
University of Groningen Netherlands
First Supervisor: dr. O.A. Kilic Second Supervisor: prof.dr. D.P. van Donk
ACKNOWLEDGEMENTS:
First of all, I would like to appreciate dr. O.A. Kilic for his countless help during my thesis project. He always provided me with critical feedbacks on the content and extended my knowledge of mathematical modelling. Moreover, I want to thank prof.dr. D.P. van Donk for his significant insights on my topic. Furthermore, I want to thank the fellow students
ABSTRACT
We present a two-stage stochastic programming approach for the workforce planning problem in the food packaging industry. The problem lies in how to organize the workforce in food packaging line with demand uncertainty. Our model integrates the hiring/firing, overtime, and workforce allocation decisions, and it considers the cost tradeoffs between those decisions. The first stage corresponds to making hiring/firing decisions based on
forecasting demand. The second stage is the workforce allocation that plays a role in decision optimization. After the two-stage model is formulated, a case from a tea manufacturer is introduced. The computational results of the case illustrate the cost-saving performance of the model. Sensitivity analysis is conducted to generate several managerial insights on the
workforce planning problem. Keywords
Tables and Figures
Tables
Table 1 Differences between Processing and Packaging (van Dam et al., 1993) ... 10
Table 2 Workers' hourly salary rate (Bhatnagar et al., 2007) ... 12
Table 3 Parameters related to the food packaging lines ... 16
Table 4 Forecasting demand and unit selling price for three products ... 17
Table 5 Forecasting error distribution for Product 1 ... 17
Table 6 Forecasting error distribution for Product 2 ... 17
Table 7 Forecasting error distribution for Product 3 ... 18
Table 8 Administrative costs for each hiring/firing decision ... 18
Table 9 The maximum production capacity varies from the different number of temporary workers ... 19
Table 10 Total costs by different number of temporary workers ... 19
Table 11 Total costs comparison ... 21
Table 12 Number of permanent workers varies considering fixed salary ... 22
Table 13 Sensitivity analysis—Unit salary ratio varies ... 24
Table 14 Sensitivity analysis –Forecasting demand for Product 1 varies ... 25
Figures Figure 1 Workforce planning framework with demand uncertainty ... 9
Figure 2 Food processing process (Akkerman & van Donk, 2009) ... 10
Figure 3 Total costs by different number of temporary workers ... 20
Figure 4 Total costs by 4 ,6, 8, 10, and 12 temporary workers ... 21
Figure 5 Sensitivity analysis -- Number of permanent workers varies ... 23
Figure 6 Sensitivity analysis—Unit salary ratio varies (x=4,6,8,10,12) ... 25
Figure 7 Sensitivity analysis – Forecasting demand varies ... 26
1. INTRODUCTION
The food industry experiences uncertain demand, due to the short shelf life, fierce market competition, and sales promotions. Usually, manufacturers in most industries can use
inventory buffers to deal with the uncertain demand. However, inventory will not be suitable for food products. The perishable nature of the food products makes them difficult to stock. Instead, food manufacturers adjust the production capacity to meet the demand. However, the demand uncertainty leads to unpredictable production capacity requirement. The capacity shortage and the perishability might cause significant loss of revenue. Production capacity can be adjusted by workforce planning. Within the food industry, food packaging lines possess the characteristics of labour-intensive, low-skill requirements, and high product-variety. The packaging lines often suffer from understaffing for high demand, while overstaffing for low demand (Sureshkumar & Pillai, 2013). The workforce planning deals with the workforce needs associated with workforce availability, allocation, and transition (Zhu & Sherali, 2009). It’s difficult to know the actual demand precisely for each food product before making planning decisions. Thus, the workforce planning problem should be naturally treated as a stochastic problem. Failure to consider demand as a stochastic variable in the food industry may result in significant lost sales and a waste of labour costs. For the stochastic problem, we need to make long-term workforce planning when the demand is uncertain, while to do short-term workforce planning to recourse the long-term decisions when the demand is revealed.
Stochastic workforce planning problems with demand uncertainty have been well studied in the service industries, e.g., call centers, postal services, nurse management in hospitals, of which the primary focus is on reducing workforce costs and operational costs. Workforce allocation, as a corrective action, occurs at the second stage of the stochastic problem (Zhu & Sherali, 2009; Campbell, 2011; Parisio & Neil Jones, 2015). It helps to mitigate the risks of the unpredictable demand. However, the existing workforce planning literature has its limitations in the manufacturing industry, where lost sales should be the main consideration. Throughout the previous research related to workforce planning, little attention has been given to manage lost sales, which seriously influence the profits of the food manufacturers. Moreover, food managers also suffer from the huge profits loss caused by overstaffing or understaffing in the packaging lines. As a consequence, the contribution of this thesis is that we addressed a stochastic workforce planning problem in the food packaging industry by taking into consideration the lost sales and workforce costs. We implemented two-stage stochastic programming to achieve the minimum total costs and generate several managerial insights that will be discussed later on.
In the remainder of this thesis, we first reviewed the literature on stochastic workforce planning problems with uncertainty. In the third and fourth section, we designed the
2. THEORETICAL BACKGROUND
In this section, we start with the review of stochastic workforce planning frameworks, from which our model is derived. Then, we investigate the literature on stochastic programming associated with workforce planning. It has been proven to be a widely applied tool to solve the stochastic workforce planning problems. For the last part, we elaborate the necessities of hiring/firing and workforce allocation in dealing with uncertainty.
2.1 Stochastic workforce planning problems under demand uncertainty
2.1.1 Stochastic workforce planning framework
Incorporating demand uncertainty into the workforce planning for the food packaging industry is the main problem for this thesis. We combined hiring/firing and workforce
allocation into a stage workforce planning model. As can be seen from Figure 1, the two-stage workforce planning framework presented in this thesis is derived from the previous literature. The hiring/firing stage is quite in line with one of the sub-steps of the model proposed by Abernathy et al. (1973). They have provided a three-stage planning and scheduling solution for the staffing problem in the service organizations, where special service requirements with high demand variety exist. The staff-planning stage includes hiring/firing, training, and reallocation. Our model does not concern training since the tasks of the packaging lines are much more straightforward than service industries. Zhang et al. (2011) have proposed a two-stage production planning model for the international
manufacturers in the context of uncertain market growth and seasonal demand. The first stage incorporates purchase decisions, while the second stage deals with production, inventory and overtime problems. We did not consider inventory management in our planning problem because of the perishable nature of food products. However, we integrated overtime
Figure 1 Workforce planning framework with demand uncertainty
2.1.2 Stochastic programming on workforce planning
Stochastic programming with recourse is a general method employed in workforce planning problems with uncertainty. Compared with the deterministic models, the stochastic models have the advantages in two aspects: (1) reducing the expected costs (2) making informed risk management decisions (Robbins & Harrison, 2010). Tracing back to Dantzig (1955), the stochastic programming is suitable for problems where non-anticipative decisions have to be made before the actual data is observed, but only forecasting information is available. Stochastic programming has been used to deal with workforce planning problems with uncertainty in many contexts. For instance, in the retail industry, Parisio and Neil Jones (2015) approached two-stage stochastic programming to do personnel scheduling,
2.2 The necessity of workforce planning in food packaging
2.2.1 The necessity of hiring/firing in food packaging
The necessity of hiring/firing becomes evident by reviewing the literature associated with the characteristics of the food packaging lines. As shown in Figure 2, a typical food production line consists of a processing and a packaging stage. There is a high variety of end-products after being packaged, due to the different package sizes, labels, as well as packaging materials. Table 1 summarizes the main differences between the processing stage and the packaging stage. The packaging lines possess high labour-intensity but require low skills (van Dam et al., 1993). Workers in the packaging lines are responsible for performing product quality checks, logistics, changeovers, and operational condition checks. Most of these tasks are easy to handle, and only a few jobs require skilled workers. In the present packaging lines, combined high and low-skilled workers are widely used (Corominas et al., 2008). For instance, one high-skilled worker takes responsibility for one line, assisted by several low-skilled workers who are quite flexible to be assigned to different lines (Lagodimos & Mihiotis, 2006). Besides, the demand for food products is stochastic during the year.
Figure 2 Food processing process (Akkerman & van Donk, 2009)
Table 1 Differences between Processing and Packaging (van Dam et al., 1993)
Processing Packaging
Products variety Small Large
Order sizes Large Small
Labour intensity Low High
The high labour-intensity and stochastic demand in the food packaging industry make the hiring/firing relatively easy and necessary. Besides, the low skill-requirement of the food packaging line motivates the food manufacturers to hire temporary workers rather than permanent workers.
2.1.2 The necessity of workforce allocation with demand uncertainty
Sherali, 2009). Another example can be given in the surgery industry. To hedge against the demand uncertainties in the operating rooms for surgery, Denton et al. (2018) formulated a stochastic surgery allocation problem to minimize the overtime costs and operational costs for opening rooms. They implemented the daily surgery allocation as a recourse stage. Workforce allocation plays an important role when long-term workforce planning decisions have to be made ahead of the uncertain short-term demand. Several studies exist in the literature investigating workforce allocation to mitigate the risks of uncertainty. Jorne et al. (2012) have summarized three types of uncertainty in allocation problems: demand
uncertainty, arrival uncertainty, and capacity uncertainty. Demand uncertainty leads to the unpredictable working capacity requirements. In the face of the globalization, many international companies experience workforce demand uncertainty due to the market uncertainty in different areas. Zhu and Sherali (2009) proposed two-stage stochastic programming to cope with the workforce demand uncertainty, where the workforce
allocation occurs both at the first and second stage. The results have shown that the stochastic programming approach results in fewer changes to the workforce plan than the deterministic approaches. Arrival uncertainty usually happens in the service industry, as a form of demand uncertainty. For the postal service industry, Bard et al. (2007) have developed two-stage stochastic programming to improve personnel scheduling. The first stage is to decide the number of employees who work full-time or part-time. At the second stage, each full-time worker is given with a weekly schedule, while part-time workers are also assigned to shifts according to the real demand. Either full-time or part-time scheduling can ease the risks of workforce shortage. Capacity uncertainty refers to the difference between the planned and realized workforce. Easton (2014) has pointed out that the workforce capacity uncertainty might be caused by ignoring the worker attendance uncertainty. They proposed a two-stage stochastic program to make workforce scheduling and allocation decisions, considering the worker attendance uncertainty. They found that the approach is more profitable than those solutions ignoring the attendance uncertainty. The conclusions drawn from the previous literature are: (1) In the face of uncertainty, workforce allocation always acts as one step of the workforce planning problem to ease risks. (2) Stochastic programming approach is robust to the uncertainty that results in cost savings.
Overtime can mitigate the influence of understaffing (Parisio & Neil Jones, 2015). Despite hiring workers, overtime can also help to increase the workforce flexibility. Working overtime may improve the utilization of non-peak hours (Lagodimos & Mihiotis, 2006). However, as Buxey (2003) addressed, overtime is relatively expensive, although it can help to deal with the flexibility needs. Companies usually plan to have three shifts for one packaging line per workday (Lagodimos & Mihiotis, 2006). The hourly salary rate is different between overtime and regular shift. An example is shown in Table 2 from
Table 2 Workers' hourly salary rate (Bhatnagar et al., 2007)
Shift 1 (8:00-16:00) Shift 2 (16:00-0:00) Shift 3 (0:00-8:00)
Regular rate for temporary 125 131.25 137.5
Overtime rate for temporary 187.5 197.5 206.25
3. PROBLEM DEFINITION
The demand is stochastic in the food industry. Therefore, we adjust the total production capacity to meet the stochastic demand. The production capacity adjustment is achieved by using a two-stage stochastic programming approach. In this thesis, we consider a single-period workforce planning problem since the forecasting demand comes into the system only one-period in advance in our case. The hiring/firing decision is defined as the first-stage problem, while the workforce allocation is the second stage (as a recourse stage). The
workforce allocation in this thesis refers to the single period worker-task allocation which can help to make full use of the shifts in each packaging line. Either two permanent workers or one permanent and one temporary worker can perform one shift. The number of temporary workers will not exceed the number of permanent workers. Each worker is allowed to work eight regular hours and four overtime hours per day based on the labour law. Overtime will enter the workforce allocation problem when regular shifts cannot meet the revealed demand. Overtime can be arranged either on working days or weekends. The objective of the
workforce planning problem is to minimize the total costs, including hiring/firing administrative costs, salary costs, and lost sales. It is achieved by a two-stage stochastic programming approach which considers uncertain demand with a known demand
distribution. The total costs are restricted to the available number of permanent/temporary workers and overtime.
In general, the two-stage stochastic model developed in this thesis is originated from the following model, which is first given by Dantzig (1955), and later reviewed by (Birge & Louveaux, 1997):
min 𝑐'𝑥 + 𝐸
+[𝑄 𝑥, 𝜉 ]
𝑠. 𝑡. 𝐴𝑥 = 𝑏, 𝑥 ≥ 0.
Where 𝐸+[𝑄 𝑥, 𝜉 ] is the expectation of 𝑄 𝑥, 𝜉 . 𝑄 𝑥, 𝜉 is the optimal value of the second-stage problem. It can be defined as follows:
min 𝑞 𝜉 '𝑦
𝑠. 𝑡. 𝑇 𝜉 𝑥 + 𝑊 𝜉 𝑦 = ℎ 𝜉 , 𝑦 ≥ 0.
𝑥 is the decision variable vector for the first stage. In our problem, 𝑥 represents the number of temporary workers. 𝑐'𝑥 is the costs of hiring/firing. 𝜉 is a random vector whose distribution
recourse decision variable vector for the second stage. The number of each kind of shifts is the decision variable for the second stage in our problem. The recourse action can be deemed as a penalty function, of which the penalty costs consist of salary costs and lost sales.
Hiring/firing incurs administrative costs for temporary workers. Hence, at the first stage, hiring/firing decisions are made in consideration of the administrative costs. At the second stage, we aim to reduce salary costs and lost sales by assigning different number and type of shifts to each product. There are four types of shifts: regular shifts on working days, overtime shifts on working days, regular shifts on weekends, and overtime shifts on weekends. The unit salary cost is in ascending order, arising from regular shifts on working days to overtime shifts on weekends. The salary costs for one production period consists of all the hourly salaries paid to all the temporary workers. The lost sales of one product are determined by the lost units and unit selling price. The production priority will be given to the product with a higher selling price. The lost sales can be reduced either by adopting overtime or hiring more temporary workers. However, overtime will incur higher salary costs, while hiring leads to additional administrative costs. Overtime can substitute hiring and vice versa. So a tradeoff exists between hiring more temporary workers, which results in higher administrative costs and lower salary costs or hiring fewer temporary workers with the opposite effect. With the help of the two-stage stochastic programming, we investigate the optimal number of
temporary workers to get the minimum total costs. To better explain the problem, we formulated the objective function as follows:
𝑚𝑖𝑛 𝑤 𝑛𝑡, 𝑥 + 𝐸A (𝑔 𝑥, 𝑠 )
We have a fixed number of permanent workers. The current number of temporary workers (𝑛𝑡) is known. The first term represents the hiring/firing administrative costs when we decide to have 𝑥 number of temporary workers. 𝐸A (𝑔 𝑥, 𝑠 ) is the expectation of the minimum
salary costs and lost sales by 𝑥 temporary workers. 𝑠 is a random vector whose forecasting demand distribution is known. Scenarios are generated by taking different 𝑠.
Assumptions of the problem are given below:
(a) Each shift of each packaging line has the same capacity (units/shift). This assumption is justified by the fact that same machines are used for different packaging lines. (b) No worker will leave for a personal reason during the production period.
(c) Since the tasks are quite straightforward and similar to different lines, both temporary and permanent workers have the same processing efficiency.
4. MODEL DESIGN
In this section, related constants and variables listed below are for one specific period:
Constants
𝐶𝐴 capacity of the packaging line per shift (units/shift) 𝑁 number of products (n = 1, … , N)
𝐹 firing costs of each temporary worker
𝑀𝑆 fixed salary costs for each permanent worker per production period
𝑃A joint probability in scenario s (𝑠 = 1, … , 𝑆) 𝐷AQ actual demand for product n
𝐸AQ forecasting error for product n in scenario s
𝐹𝐷AQ forecasting demand for product n in scenario s
𝐶(A) unit salary per regular shift on working days
𝐶(R) unit salary per overtime shift on working days
𝐶(AS) unit salary per regular shift on weekends
𝐶(RS) unit salary per overtime shift on weekends
𝐵Q unit selling price for product n
𝐾SV number of working days within one production period
𝐾SW number of weekend days within one production period
Variables
𝑥 required number of temporary workers
𝑠AQ number of regular shifts on workdays for product n in
scenario s
𝑜AQ number of overtime shifts on workdays for product n
in scenario s
𝑠𝑤AQ number of regular shifts on weekends for product n in
scenario s
𝑜𝑤AQ number of overtime shifts on weekends for product n
in scenario s
We start the model by providing our objective function: Objective function
Stage 1:
𝑀𝑖𝑛 𝑤 𝑛𝑡, 𝑥 + 𝐸A (𝑔 𝑥, 𝑠 ) (1)
Stage 2:
𝑔 𝑥, 𝑠 = 𝑄 𝑥, 𝑠 + 𝐿(𝑥, 𝑠) (2)
The objective function aims to find the optimal number of temporary workers (𝑥) that can ensure the minimum value of hiring/firing administrative costs, salary costs, and lost sales for one specific period.
𝑤 𝑛𝑡, 𝑥 is the hiring/firing administrative costs determined by the decision variable 𝑥 and the current number of temporary workers (𝑛𝑡).
𝐸A (𝑔 𝑥, 𝑠 ) is the expectation of the minimum salary costs and lost sales in each scenario. In
order to achieve the value of 𝐸A (𝑔 𝑥, 𝑠 ), we first need to compute the vale of 𝑄 𝑥, 𝑠 and 𝐿(𝑥, 𝑠) separately. 𝑄 𝑥, 𝑠 is the minimized salary costs by 𝑥 number of temporary workers in scenario 𝑠. 𝐿(𝑥, 𝑠) is the minimized lost sales by x number of temporary workers in scenario s. 𝑃A is the probability in each scenario.
𝐸A (𝑔 𝑥, 𝑠 ) = 𝑃A×(𝑄 𝑥, 𝑠 + 𝐿 𝑥, 𝑠 ) A∈_ (4) 𝑄 𝑥, 𝑠 = 𝑀𝑖𝑛 𝐶(A) 𝑠 AQ+ 𝐶(R) 𝑜AQ+ 𝐶(AS) 𝑠𝑤AQ+ 𝐶(RS) 𝑜𝑤AQ Q∈` Q∈` Q∈` Q∈` (5) 𝐿 𝑥, 𝑠 = 𝑀𝑖𝑛 𝐵Q𝑀AQ Q∈` (6) 𝑃A = 1 A∈_ (7)
Once we know the actual demand (𝐷AQ) and total output (𝑌
AQ) for product n, we can compute
the missed demand (𝑀AQ) for product n. In scenario s, the actual demand for product n (𝐷 AQ)
equals to the sum of the forecasting demand (𝐹𝐷AQ) and the forecasting error (𝐸
AQ) for product
n. 𝑌AQ is the total output for product n by all types of shifts in scenario s.
𝑀AQ = 𝐷 AQ− 𝑌AQ (8) 𝐷AQ = 𝐹𝐷 AQ + 𝐸AQ (9) 𝑌AQ = 𝑠 AQ+ 𝑜AQ+ 𝑠𝑤AQ + 𝑜𝑤AQ ×𝐶𝐴 (10)
We compute the number of regular shifts and overtime shifts per day if we have 𝑥 temporary workers. Then, we give constraints to the decision variables: 𝑠AQ, 𝑜
𝑜𝑤AQ ≤ 𝐾SW𝑔(𝑥, 𝑛𝑝) Q∈` (18) 𝑠AQ 𝑜AQ, 𝑠𝑤AQ, 𝑜𝑤AQ ≥ 0 (19)
5. NUMERICAL EXAMPLE
5.1 Data collection from a case company
One tea manufacturer that motivated our study has a packaging shop floor of several packaging lines. One production period of the company contains 20 working days and four weekends. Based on the labour law, the legal working length per day per worker includes eight regular hours and four overtime hours. There are maximum three regular shifts per packaging line per day (8:00-16:00, 16:00-00:00, 00:00-08:00). The shift capacity (units/shift) is provided by the case company, considering the time for setups and
changeovers. Data is pre-processed by selecting three products, among which the demand and unit selling price varies. Hence, the results can be generalized to the whole packaging shop floor. The setting, in this case, is the production period before Christmas. During this period, demand multiplies but is still unpredictable.
The current workforce policy in our case-study company is that they always keep a fixed number of workers all year round. In the meanwhile, they do not use inventory buffers for their perishable products. As a result, they experience lost sales resulted from a lack of production capacity when the demand is high. On the contrary, they bear additional workforce costs when the demand is low.
For our stochastic workforce planning method, we make hiring/firing decisions
beforehand. When the demand is revealed, we allocate the available shifts to each product based on its unit selling price. At the end of this section, we compared the total costs from the current policy and our stochastic workforce planning method. The computational results indicate the cost-saving effect of our approach.
Table 3 Parameters related to the food packaging lines
First, we collected the data from three products with the descending forecasting demand and the ascending unit selling price (see Table 4). Product 1 has the highest forecasting demand but the lowest unit selling price. On the contrary, Product 3 has the highest unit selling price with the lowest forecasting demand. Those three products deserve to be investigated since they represent the boundary condition.
Table 4 Forecasting demand and unit selling price for three products
Product Empirical forecasting demand Unit selling price (𝐵Q)
Product 1 500,000 1 monetary units
Product 2 250,000 2.5 monetary units
Product 3 200,000 3 monetary units
Our approach generates the scenarios by adopting historical forecasting data, from which the validity is ensured. For each product, we assume the independence of the forecasting error distribution, which is taken from the empirical values. We use the discrete distribution with four chunks to ease the problem. They are listed in Table 5, 6, 7. We take the probabilities of each error distribution and combine them so that we have each scenario (64 scenarios in total). The company has stable market sales and a sophisticated forecasting system for those three products so that the actual demand, later on, will never come out of range.
Table 5 Forecasting error distribution for Product 1
Forecasting demand Forecasting errors Probabilities Actual demand 500,000 (−100,000, −50,000) 0.2 (400,000, 450,000)
500,000 (−50,000, 0) 0.3 (450,000, 500,000)
500,000 (0, 50,000) 0.3 (500,000, 550,000)
500,000 (50,000, 100,000) 0.2 (550,000, 600,000)
Table 6 Forecasting error distribution for Product 2
Forecasting demand Forecasting errors Probabilities Actual demand 250,000 (−20,000, −10,000) 0.25 (230,000, 240,000)
250,000 (−10,000, 0) 0.3 (240,000, 250,000)
250,000 (0, 10,000) 0.1 (250,000, 260,000)
Table 7 Forecasting error distribution for Product 3
Forecasting demand Forecasting errors Probabilities Actual demand 200,000 (−10,000, −5,000) 0.15 (190,000, 195,000)
200,000 (−5,000,0) 0.25 (195,000, 200,000)
200,000 (0,5,000) 0.3 (200,000, 205,000)
200,000 (5,000,10,000) 0.3 (205,000, 210,000)
5.2 Data analysis by two-stage stochastic programming
5.2.1 Hiring/firing administrative costs
The number of permanent workers in the case is 12, and the current number of temporary workers is 4 (𝑛𝑡 = 4, 𝑛𝑝 = 12). Based on the problem definition, the final number of temporary workers will not exceed the number of permanent workers. Thus, the number of temporary workers is available from 0 to 12 (see Table 8). The number grows by two each time since temporary workers cannot perform alone. The administrative costs will be one critical measurement when making the firing/hiring decisions.
Table 8 Administrative costs for each hiring/firing decision
𝑥 = 0 𝑥 = 2 𝑥 = 4 𝑥 = 6 𝑥 = 8 𝑥 = 10 𝑥 = 12 Hiring costs 0 0 0 2,000 4,000 6,000 8,000 Firing costs 2,000 1,000 0 0 0 0 0 Total costs 2,000 1,000 0 2,000 4,000 6,000 8,000 5.2.2 Production capacity
Table 9 The maximum production capacity varies from the different number of temporary workers 𝑥 = 0 𝑥 = 2 𝑥 = 4 𝑥 = 6 𝑥 = 8 𝑥 = 10 𝑥 = 12 𝑓(𝑥, 12) 6 7 8 9 10 11 12 𝑔(𝑥, 12) 3 3.5 4 4.5 5 5.5 6 Regular shifts on working days 120 140 160 180 200 220 𝟐𝟒𝟎 Overtime shifts on working days 60 70 80 90 100 110 120 Regular shifts on weekends 48 56 64 72 80 88 96 Overtime shifts on weekends 24 28 32 36 40 44 48
Total shifts (max.
capacity) 𝟐𝟓𝟐 294 336 378 420 462 504
We can see in Table 9 that the total production capacity consists of four types of shifts. The exact number of shifts to be used is based on the actual demand. If the demand exceeds the maximum capacity, there will be lost sales. On the contrary, in the low demand period, overtime shifts either on working days or weekends might not be entirely used.
5.2.3 Results analysis for the numerical study
In this section, we first discuss the workforce planning decisions made for the case company by the two-stage stochastic programming. Then, we compare the total costs from the tea manufacturer’s current policy and our model to see whether the costs have been saved. As described in the problem definition section, the minimum salary costs can be achieved by making the best of the regular hours. The lost sales can be reduced when the production is prioritized to the products with higher selling prices. The value of 𝐸A (𝑔 𝑥, 𝑠 ) is computed by summing up the minimum salary costs and lost sales in each scenario (64 scenarios in total) with respective probabilities.
Table 10 Total costs by different number of temporary workers
𝑥 = 0 𝑥 = 2 𝑥 = 4 𝑥 = 6 𝑥 = 8 𝑥 = 10 𝑥 = 12
𝑤(4, 𝑥) 2,000 1,000 0 2,000 4,000 6,000 8,000
𝐸A (𝑔 𝑥, 𝑠 ) 226,010 106,697 40,821 35,690 34,591 33,795 33,330 Total costs 228,010 107,697 40,821 𝟑𝟕, 𝟔𝟗𝟎 38,591 39,795 41,330 As shown in Table 10, the total costs reach the minimum level when the number of
temporary workers is six. That is to say, two more temporary workers are recommended to be hired. Lost sales account for a large proportion in the total costs when the number of
number of temporary workers exceeds the optimal value, the total costs are mainly influenced by the administrative costs. In this case, six number of temporary workers proves to be the best choice for the upcoming production period.
Figure 3 Total costs by different number of temporary workers
As can be seen in Figure 3, the hiring/firing administrative costs (the orange dots) keep increasing at a slower rate. The salary costs and lost sales (the blue dots) experience sharp decrease when 𝑥 increases from 0 to 4. Then, they continue to descend slowly after 𝑥 = 4. The first three grey dots for total costs almost overlap with the blue dots because of the high lost sales, which are caused by the short of production capacity even if overtime is entirely used. When the number of temporary workers increases, the lost sales can be avoided, and the overtime will be gradually reduced. Intuitively, the grey dots start to rise again when 𝑥 = 8 since the hiring/firing administrative costs take the leading position of the total costs.
0 50000 100000 150000 200000 250000 0 2 4 6 8 10 12 C os ts
Number of temporary workers
Total costs by different number of temporary workers
Figure 4 Total costs by 4 ,6, 8, 10, and 12 temporary workers
We take out the values of 𝑥 = 0 and 𝑥 = 2 from Figure 3 to make it more consistent with the reality (see Figure 4). The current number of temporary workers in the case-study company is 4. Firing is a worse decision in the face of the high demand. Moreover, the hiring/firing administrative costs are more readable in this figure when the boundary value in the y-axis is narrowed down. To show the trend of each cost, we use gradient lines. Similar to the previous analysis, the hiring/firing administrative costs increase at a constant rate. After 𝑥 = 6, the growth rate for the total costs is close to the administrative costs. It is reflected by those two parallel lines on the graph. When 𝑥 = 6, the total costs reach the bottom of the curve. With those in mind, an insight can be derived that we should stop hiring when 𝑥 = 6.
Table 11 compares the total costs from the company’s current policy and the two-stage stochastic programming. The current number of temporary workers is 4 (𝑛𝑡 = 4). The total cost-saving percentage of our approach is 7.7%. There are two reasons for the higher total costs from the current policy: (1) The production heavily relies on the overtime hours which increase the overall salary costs; (2) The limited production capacity, resulted from lack of temporary workers, leads to lost sales. The numerical study shows the applicability of the model to the real-world food packaging industry. When this model is generalized to the whole product families in the packaging shop floor, more costs will be saved.
Table 11 Total costs comparison
Current policy Two-stage stochastic programming 𝑤(4, 𝑥) 0 2,000 𝐸A (𝑔 𝑥, 𝑠 ) 40,821 35,690 Total costs 40,821 37,690 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 4 6 8 10 12 C os ts
Number of temporary workers
Total costs by different number of temporary workers
6. SENSITIVITY ANALYSIS AND DISCUSSION
In this section, we investigate the changes in the total costs by changing different parameter settings: number of permanent workers, unit salary ratios, and demand. The computational results show the cost-saving effect of our two-stage stochastic programming approach, as well as several managerial insights.
6.1 Change the number of permanent workers (
𝑛𝑝)
In the previous case, the number of permanent workers (𝑛𝑝) remains at 12. Theoretically, when 𝑛𝑝 increases, more regular shifts are formed. The overtime salary costs and lost sales will be reduced consequently. However, the total costs will not always be saved in practice. We need to realize that permanent workers are paid by fixed salaries, which will increase as the size of permanent workers expand. In this section, we consider to change the number of permanent workers, taking into account the fixed salary costs for permanent workers. We want to investigate whether the optimal number of temporary workers and the minimum value of total costs will change accordingly.
The unit salary cost still increases at a rate of 120% from the regular shift on working days to the overtime on weekends. The forecasting demand, forecasting errors, and probabilities in each scenario remain unchanged. The fixed salary for each permanent worker for one production period is 2,000 monetary units. We only consider the fixed salary for one period since we only do the workforce planning in one production period. Table 12 has provided the computational results related to the optimal number of temporary workers and the minimum total costs by different 𝑛𝑝. The number starts at eight. Otherwise, there will be unacceptable lost sales. Intuitively, 𝑥 does not decrease by two each time since firing administrative costs play a critical role in the final costs. The best case suggests the company should have ten permanent workers and eight temporary workers. Compared with the previous case (𝑛𝑝 = 12, 𝑥 = 6), 2,000 monetary units are saved. Temporary workers are cheaper than permanent workers in food packaging lines when providing the same production capacity.
Table 12 Number of permanent workers varies considering fixed salary
Figure 5 Sensitivity analysis -- Number of permanent workers varies
The dots in Figure 5 reflects that the number of permanent workers has a significant influence on the total costs. The best cost-saving performance is achieved by ten permanent workers. When 𝑛𝑝 grows above ten, the total costs increase infinitely. It can be explained by the fact that the lost sales and salary costs gradually become stable (see Table 12). More permanent workers only contribute to higher fixed salary costs.
Discussion of the changing of 𝑛𝑝:
(1) More permanent workers lead to fewer temporary workers, thereby save the hiring administrative costs. However, we observe that the number of temporary workers does not decrease at two each time as we expect. Consider the firing administrative costs, keeping those extra temporary workers appears to be more cost-saving than firing them. However, in reality, they might not be willing to stay in the company if they are not arranged for work. (2) More permanent workers can increase the production capacity and adopt less overtime at the same time. Thus, the salary costs for overtime can be saved. When the demand is met, in the meanwhile, overtime is no longer adopted, the increasing of permanent workers will only bring about additional fixed salary costs.
In all, permanent workers cannot be entirely substituted by temporary workers since they have to monitor the production. Increasing 𝑛𝑝 can improve the overall production capacity to avoid lost sales. Moreover, more regular shifts can be formed to reduce the salary costs of overtime production. However, the total costs continue to increase when 𝑛𝑝 is above ten (see Figure 4 &5). The reason is that the fixed salary costs for temporary workers account for a significant proportion of the total costs since then.
59000 60000 61000 62000 63000 64000 65000 66000 67000 8 10 12 14 16 T ot al cos ts
Number of permanent workers
Sensitivity analysis -- number of permanent workers varies
6.2 Change of unit salary ratios
In this experiment, we investigate the change of total costs regarding the change of unit salary ratios. In the previous case, the unit salary ratio is 120%. Now, we set various unit salary ratios (see Table 13). The number of permanent workers is 12. The forecasting
demand, forecasting errors, and probabilities in each scenario remain unchanged. In practice, we still take out the values of 𝑥 = 0 and 𝑥 = 2. The change of the unit salary ratios will alter the salary costs and ultimately influence the total costs. In general, the smaller the unit salary ratio, the lower the total costs.
Table 13 Sensitivity analysis—Unit salary ratio varies
Ratio 𝑥 = 4 𝑥 = 6 𝑥 = 8 𝑥 = 10 𝑥 = 12 105% 35,635 31,856 32,110 32,665 33,553 110% 37,278 33,704 34,163 34,924 36,017 115% 39,006 35,648 36,323 37,299 38,608 120% 40,821 37,690 38,591 39,795 41,330 125% 42,725 39,832 40,972 42,413 44,187 130% 44,721 42,078 43,467 45,158 47,182 Cost-saving 20.3% 24.3% 26.1% 27.7% 28.9%
The values listed in the last row of Table 13 is the cost-saving percentage for each number of temporary workers when the unit salary ratio decreases from 130% to 105%. Intuitively, the cost-saving percentage keeps ascending as 𝑥 increases. It can be interpreted by the fact that companies will benefit more from lower unit salary ratios when they have more temporary workers. Temporary workers are paid by hourly salaries. Lower unit salary ratio leads to lower salary costs for temporary workers. If the company has fewer temporary workers, the cost-saving effect by the changing of the ratio is small. For instance, the total costs are saved by 20.3% when there are four temporary workers, while the percentage for 12 temporary workers is almost 1.5 times higher. When 𝑥 = 6, the total costs are always the lowest
Figure 6 Sensitivity analysis—Unit salary ratio varies (x=4,6,8,10,12)
We eliminate the total costs of 𝑥 = 0 and 𝑥 = 2 in Figure 6. To show the change tendency of the total costs, we use the gradient lines. As can be seen from Figure 6, the dots get closer when the number of temporary workers decrease from 12 to 4. The implication is that
companies are more robust to the change of the unit salary ratio if they have fewer temporary workers. That is to say, companies with more workers will suffer more when facing new legislations of raising overtime salaries.
6.3 Change of demand
The third part is devoted to studying the optimal number of temporary workers and total costs concerning different forecasting demand for Product 1. The demand for Product 1 varies between 300,000 and 700,000 during the whole year due to the seasonality. Product 1 has the highest demand but the lowest unit selling price. It is the last to be produced among those three products. So it possesses the highest possibility of lost sales. The changing of the
forecasting demand for Product 1 will have the most considerable influence on the total costs.
Table 14 Sensitivity analysis –Forecasting demand for Product 1 varies
𝐹𝐷AQ 300,000 400,000 500,000 600,000 700,000 𝑥 𝑥 = 4 𝑥 = 4 𝑥 = 6 𝑥 = 8 𝑥 = 8 Total costs 27,268 32,028 37,690 43,358 48,439 30000 32000 34000 36000 38000 40000 42000 44000 46000 48000 50000 4 6 8 10 12 T ot al cos ts
Number of temporary workers
Sensitivity analysis -- unit salary ratio varies
As can be seen from Table 14, if a proper number of temporary workers is ensured, there will not be huge differences in total costs when the demand experiences sharp increase or
decrease. For instance, although the demand for Product 1 grows from 300,000 to 700,000, the total costs only increase by 21,171. Besides, we realize that the demand surge does not bring about sudden workforce shortage. Based on the computational results, frequent hiring/firing is not required even though the demand fluctuates. As an example, the number of temporary workers remains unchanged (𝑥 = 4) when the demand varies from 300,000 to 500,000. As shown in Figure 7, the total costs grow linearly at a slow rate.
Figure 7 Sensitivity analysis – Forecasting demand varies
The comparison of our experiments shows the following findings (managerial insights): (1) More permanent workers can reduce the hiring administrative costs since fewer temporary workers are required. Moreover, the increasing of permanent workers can save salary costs by adopting less overtime. However, the fixed salary costs for permanent workers will mainly restrict the number of permanent workers when other costs tend to be constant.
(2) By comparing different unit salary ratios, we find that companies with a large number of temporary workers will suffer more when facing the new legislation of raising overtime salaries.
(3) Workforce planning plays a critical role in the face of demand fluctuation. Instant workforce adjustment, e.g., hiring/firing temporary workers can avoid lost sales, in the meanwhile, control the total costs. In reality, a certain number of temporary workers remains even though the demand varies. It is proved by the computational results from the model. Based on that reasoning, frequent hiring/firing is not recommended in the food packaging lines. 0 10000 20000 30000 40000 50000 60000 300000 400000 500000 600000 700000 T ot al cos ts
Forecasting demand for Product 1
7. CONCLUSION AND FUTURE RESEARCH
In this thesis, we implemented a two-stage stochastic programming approach for the workforce planning problem in the food packaging industry with demand uncertainty. The computational results obtained from the case company have shown the cost-saving effect of the stochastic model. Extensive managerial insights have been achieved after the sensitivity analysis. It is undoubted that the proposed model applies to the whole product family on the packaging shop floor.
Current stochastic workforce planning literature focuses on minimizing the labour costs and operational costs. Research has seldom investigated the lost sales, which deserve to be considered in the food industry. We contribute to incorporating lost sales into the objective cost function of a two-stage stochastic program. Therefore, this model can be extended to the industries where lost sales are nontrivial.
In addition to the above aspects, the results can also be interesting for the food production managers. Demand uncertainty makes it difficult for them to make accurate workforce decisions based on the forecasting data. With the help of the two-stage stochastic
programming, food manufacturers can organize the workforce by the first step of making hiring/firing decisions. Then, they recourse the first-step decisions by allocating the workers to the production when the demand is revealed. They can achieve the minimum total costs consist of lost sales, hiring/firing administrative costs, and salary costs. Moreover, the extensive managerial insights on the relationship between the number of permanent workers, the demand, and the unit salary ratios are derived from the sensitivity analysis.
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