Operative production planning utilising quantitative forecasting and Monte Carlo simulations

1 Introduction

The industrial production has undergone significant changes in recent decades. The changes relate to the scale and complexity of production and technologies used. Manufacturers to be competitive must produce high-quality products at low cost while being flexible in meeting rapidly changing customer needs [1]. The key role in solving this problem is the planning of production, where forecasting, optimization and simulation can be used.

The forecasting method is accurate when generating forecasts in line with actual requirements. Accurate forecasts in production planning reduce risk and improve business performance of a company [2]. The authors of an article [3] have expanded the use of Bootstrap (Bagging) combination aggregations and prognostic methods in the electric energy sector to obtain more accurate demand forecasts. Pedro and Coimbra [4] examined prognostic methods of the Persistent Model, Auto-Regressive Integrated Moving Average (ARIMA), k-Nearest Neighbours (kNN), Artificial Neural Networks (ANN) and ANN optimized by Genetic Algorithms (GA/ANN). Findings have shown that forecasting models based on the ANN principle have better results than other prognostic techniques. The article [5] compares the forecasting performance of state space models and ARIMA models. The prognosis was applied to a case study of five categories of women’s footwear. Authors [6] predict sales of plastic products using ARIMABox-Jenkins method. Measuring the accuracy of forecasting results was performed using the MAPE value (Mean Absolute Percentage Error). The accuracy of responses ranged from 68 to 74%. The application of the ARIMA method in practice is also addressed in [7, 8] and the seasonal ARIMA method in [9]. Pant and Attri [10] inform about -of movements in the index. It is simple to model, uses real-time data and provides accurate forecasts. The index is suitable for small businesses.

Production planning problems have been resolved as optimization problems since the early 1950s. Missbauer and Uzsoy [11] focused on optimization models that support decision-making on production volumes and the ordering process. Optimization of production processes using the Yamazumi method is solved by [12]. Authors [13] developed an analytical model for a multi-period production planning problem with dual supply sources of production capacity, where the supply price of one source is random, and the supply capacity of the other source is limited. The paper by [14] represented the methodology for the multi-criteria evaluation and selection of manufacturing processes at the stage of conceptual process planning.

Hierarchical production planning was dealt with by Gansterer et al. [15]. They surveyed three planning parameters, delivery times, insurance stocks and frequency, using optimization based on simulation. Six optimization methods were used to investigate parameters with systematic quantification of parameter combinations. The Variable Neighbourhood Search (VNS) lead to best results.

Jeon and Kim [1] processed an overview of simulation techniques used on production planning and control, from 2002 to 2014. The overview covers three types of simulation techniques (system dynamic, discrete event simulation and agent-based simulation) and eight production planning and control issues (facility resource planning, capacity planning, job planning, process planning, scheduling, inventory management, production and process design, purchase and supply management). Practical applications of computer simulation are dealt with in many research works, eg, for improvement of production logistics’ efficiency in [16], to identify the occurrence of failures in [17], to reach data for economic analysis of logistic systems in [18]. Li et al. [19] developed a metamodel Monte Carlo simulation method that captures the dynamic stochastic behaviour of a production system and enables real-time evaluation of performance of the release plan. The quality of this method and the results of optimization of the production plan were verified in semiconductor production. In the paper [20], the authors presented Data envelopment analysis (DEA) and Monte Carlo simulation approach that allows to find optimal assortment of production over a given period of time, due to inaccurate time series of data, and helps in prognosis and estimating of business efficiency. Monte Carlo simulation is also addressed by the authors of the articles [21, 22].

In this paper, the issue of the operational planning of mass production is addressed. We present and compare two approaches to creating an operational plan. The first approach uses quantitative forecasting, at which based on a time series the forecast of sales for the next planning period is determined. In the second approach, a method of production optimizing by using Monte Carlo simulations is applied.

2 Problem description

Creation of an operational plan is part of the operative - lowest level - management of the production process. However, from the point of view of ensuring market requirements and economic efficiency of production, it is a significant, yet key management level. The goal of each business is to reduce costs and increase the efficiency of capital, thus creating a better cash flow. Optimizing an operational production plan to minimize production and storage costs is one of the ways to achieve this goal. Obstacles are variable and difficult to predict demand, accidental impacts of the environment and other production factors.

2.3 Data used in the case study

Based on the production data of products A, B, C observed for 30 weeks (Table 1, Figure 1). The sale of these products is neither seasonal nor trendy character. In addition to sales, the input parameters are also capacity and time parameters of the production, price and scrap of the production (Table 2).

Figure 1

Real sales of the products A, B, C.

Table 1

Demand for products A, B, C observed for 30 weeks.

Week	Production [103 pcs]			Week	Production [103 pcs]
	A	B	C		A	B	C
1	6,370	6,350	3,295	16	10,965	6,415	2,899
2	7,908	5,387	1,814	17	9,807	6,801	3,764
3	7,659	5,122	2,091	18	10,768	6,938	2,403
4	12,541	8,624	3,963	19	6,964	10,969	4,616
5	11,484	10,411	6,205	20	9,339	13,011	3,484
6	7,610	12,749	5,382	21	7,952	9,552	2,929
7	11,508	9,965	5,108	22	6,907	9,402	2,328
8	9,433	4,367	2,960	23	11,133	10,021	1,849
9	6,623	7,660	4,968	24	9,197	8,947	4,105
10	6,495	6,842	2,953	25	10,372	14,364	3,522
11	9,089	13,404	3,971	26	8,996	9,772	1,508
12	8,437	5,911	3,850	27	8,281	7,534	2,297
13	11,488	10,395	3,450	28	8,822	9,919	4,062
14	12,186	6,092	2,732	29	8,234	12,275	2,503
15	11,201	6,546	3,144	30	10,898	8,482	1,712

Table 2

Entry variables related to the production of products A, B, C.

Parameter	Unit	Value
Line production capacity A	ths.pcs/h	168
Line production capacity B	ths.pcs/h	153
Line production capacity C	ths.pcs/h	173
Average time of rebuilding	h	4
Price A	EUR/ths.pcs	125
Price B	EUR/ths.pcs	118
Price C	EUR/ths.pcs	180
Scrap	coeff.	0
Minimal production batch A	ths.pcs	100
Minimal production batch B	ths.pcs	100
Minimal production batch C	ths.pcs	70

3 Methodology and methods

The purpose of this paper is to present two approaches to creation of a production plan for a production line. The first approach is based on forecasts and the second is based on simulations. The suitability of these approaches is compared on the basis of the selected criterion - total cost. The cost of rebuilding the line and the cost of storing finished production is considered to be the total cost.

When designing a production plan proposal, we use these two approaches:

The first approach utilizes quantitative forecasting. On the basis of the time series data analysis, the demand forecast for the production cycle time period and input raw material supply cycle, i.e. 2 and 6 weeks, is calculated. It is assumed to enter production in 1-week intervals in a volume based on a 2-week forecast. The level of safety inventory of finished products is on the level of 80% of the maximum difference of the 6-week forecast and actual sales. Total costs consist of the line breakdown costs (value of unprocessed production) and storage costs of finished production (1). An extra group of costs is the cost of safety stock. These are not dependent on production intervals and are maintained at a constant level of 80% of maximum forecast deviation from actual sales (2).

Where: C_T is total cost, C _hPi is capacity per hour for i-th product, P_i is price for i-th product, T_R is average time for line rebuild in hours, n_i is number of rebuilds for the monitored period, B_i is batch size for i-th product, m is number of products, C_SF is safety cost based on forecasts, ᐃQ_i is maximum deviation of 2-week forecast and sales for i-th product, and F _i is sales forecast for i-th product.

The second approach implements cost optimization method based on Harris-Wilson’s model using Monte Carlo simulations. The basis for optimization is the mathematical model of the cost function of total costs (4). The total costs consist of the cost of rebuilding the line (5) and the storage costs (6). The goal of optimization is to determine a production batch at which the total cost will be minimal. Subsequently, additional dependent parameters such as the production input interval (7) and the continuous production time (8) result from the optimal production batch. In modelling the cost function, it is considered that the consumption of the finished production is uniform and its stocks are supplemented at regular intervals by a constant volume of the production batch. The production batch is considered as the mean value for a given distribution. The production plan is at the mean value. Safety stock will be considered at the level of the difference of 1.5 times the standard deviation and the mean value. This is a level comparable to the level of safety stock when forecasting. Their value represents the cost of safety stocks (9).

Where: C_Ri is cost to rebuild the line for i-th product, C_Si is storage cost for i-th product, D_i is average demand for i-th product for the monitored period, t_i is production input interval, t_Pi is continuous production time, C_SS is safety cost based on simulation, σ_i is standard deviation, and mean_i is mean value.

The basis of Monte Carlo simulations is the definition of uncertainty for variable parameters in the form of appropriate distribution functions. In order to optimize production, decision variables and forecast variables have to be determined. Since demand is considered a key uncertain input variable, the distribution function is defined by using the FIT function of the Crystal Ball simulation tool. This feature allows, based on a specific time series, to describe the stochastic character of the variable by the most appropriate distribution function, including its statistical characteristics. The distribution functions of other variables are determined by estimation. Defined distributions for input and output variables are presented in Table 3. Optimization is done through the OptQuest software tool using Monte Carlo simulations.

4 Results

4.1 Analysis of data time series

On the basis of time series analysis, an optimal prognostic method was selected to provide a prognosis for 2 and 6 weeks (Table 4, Figure 2-4) with min. MAPE (Mean Absolute Percentage Error) value. The ARIMA (2,0,2) was proposed as the most accurate predictive method. Prognoses were worked out only for weeks 17-30 due to the use of a sufficiently long time series (min. 15 data) to prepare the forecast. Subsequently, according to the formula (3), deviations of the forecast and demand for the determination of safety stock were evaluated. Table 4 presents the calculated forecasts and max. deviations of ᐃQi for individual products.

Figure 2

Real sale of the product A and forecasts by ARIMA.

Figure 3

Real sale of the product B and forecasts by ARIMA.

Figure 4

Real sale of the product C and forecasts by ARIMA.

Table 4

Forecast for 2 and 6 weeks.

Week			Forecast [103 pcs]
	A		B		C
	2w	6w	2w	6w	2w	6w
17	7,282		7,165		3,698
18	9,651		9,019		3,650
19	9,353		6,983		3,644
20	9,721		7,416		3,674
21	9,786	10,991	7,690	8,527	4,076	3,742
22	10,639	9,478	8,519	7,982	3,276	3,716
23	11,792	9,419	7,343	7,773	3,324	3,691
24	8,906	9,552	6,667	7,715	3,682	3,606
25	9,729	10,038	8,758	7,893	2,523	3,648
26	10,144	9,867	8,803	8,521	3,662	3,633
27	9,697	10,575	10,022	7,957	3,391	3,649
28	7,631	9,175	8,266	8,605	3,068	3,563
29	9,067	9,635	7,904	8,851	3,917	2,522
30	8,583	9,677	6,055	8,803	3,584	3,298
MAPE	17.26%	16.75%	25.62%	17.04%	47.09%	50.82%
Max deviation ᐃQi	2,525		5,606		999

4.3 Production plan based on simulations

The distribution functions of the "sell" variable are designed through FIT function. The best functions describing sale were selected based on the chi-square test. Sales of product A optimally expressed the beta distribution, sales of product B lognormal distribution and sales of product C weibull division. Figure 5-7 present distribution functions for product sales including 1.5 standard deviation.

Figure 5

Distribution function for sale of product A.

Using OptQuest, production has been optimized with the aim to achieve a minimum mean value for forecast variable - total costs. Simulation for OptQuest was set to 10,000 trials. As shown in the graph, already at an experiment of about 1,620 the optimal solution was reached, which did not change during further simulations. Figure 8 presents the results of optimization of production, including optimal production volume. Production parameters are in Table 6 and production costs are presented in Table 7. Costs quantified in Table 7 are for a period of 10 weeks.

Figure 6

Distribution function for sale of product B.

Figure 7

Distribution function for sale of product C.

Figure 8

Graph of production optimization with resulting values.

Table 6

Production parameters after optimization.

Parameter	Unit	Product A	Product B	Product C
Production batch	pcs 103	4,950	4,957	3,032
Interval of production input	week	0.53	0.52	0.91
Safety stock value	pcs 103	2,712	4,099	1,741

Table 7

Cost of production after optimization.

Costs item	For individual product [EUR]	For the whole production [EUR]
Line rebuild costs A	1,580,218	4,362,626
Line rebuild costs B	1,386,038
Line rebuild costs C	1,396,370
Storage costs A	1,546,875	4,267,390
Storage costs B	1,356,115
Storage costs C	1,364,400
Total costs	-	8,534,782
Safety stock value A	339,000	1,136,062
Safety stock value B	483,682
Safety stock value C	313,380

Figure 9 presents a Monte Carlo simulation histogram for Total costs. In addition to the mean value, the confidence interval is 80% is distinguished, which means that with 80% reliability it can be expected that the costs will not be less than about EUR 7,500,000 and will not exceed the level of EUR 9,600,000.

Figure 9

Forecast of Total costs.

Other outputs from Monte Carlo simulations allow an analysis of the impact of input variables on the output reliability. In our case study, sensitivity analysis showed that the reliability of simulation results was influenced mainly by the variability of the sale C (Figure 10). By reducing the impact of inputs variability on the output, we reduce the risk that our expected output will be different from reality. The main benefit of Monte Carlo simulation is its possible application in risk management. The identification of risk factors makes it possible to apply a selective approach in decisions about controlled variables. In our case study, there is high variability in the sale of C product, which is the strongest risk factor threatening the target by variability (Figure 10). Another strong risk factor is line rebuild time and sale of the product B.

Figure 10

Sensitivity analysis of total costs.

5 Conclusions and future directions

Two approaches to creating an operational plan were presented in the paper. The first method used quantitative forecasting, at which based on a time series the forecast of sales for the next planning period was determined. The ARIMA (2,0,2) prognostic method was selected based on MAPE as the most reliable. Because the forecast inaccuracy required the maintenance of safety stock, in our case study, for the reason given in section 2, the safety stock was set at 80% of the maximum deviation of the forecast from reality. Production was performed at weekly intervals. Uncertainty of other input variables such as production time, line rebuild time, costs and others have not been taken into account. Costs and the amount of safety stock are presented in Table 8.

The second method applied a method of optimizing production using Monte Carlo simulations aiming to minimize costs. The stochastic character of demand was utilizing the Crystal Ball software tool captured with a suitable distribution function. Based on the estimate, the variability of the other input variables was also determined. By the means of the OptQuest, production parameters were optimized so that the total costs associated with production line rebuild and storage of finished production were minimal. Total costs and safety stock are shown in Table 8.

Production planning based on demand forecasting is always associated with the risk of low forecast accuracy. Time-series forecasting is used in planning most frequently. In production planning, where many input variables are of stochastic character, Monte Carlo simulations allow to include this stochasticity in the plan preparation, optimization and as well as to use the simulation outputs in the risk analysis. We consider this as the highest benefit of applying of Monte Carlo simulations in planning. However, there are many limitations. Also in our case study, we saw that high variability of several variables sharply reduced the plan reliability. Simulations can be used effectively in the case where input variability is low. It is important to precisely select the input variables where it is reasonable to define their stochastic character and which of them consider as constants.

Utilization of simulations and optimization by means of software tools represents an alternative to traditional approaches to production planning. The availability of software tools allows enterprises to use simulations even in such processes where quantitative prognosis is applied that requires the availability of historical data (which is not always available) and does not allow for inclusion into planning of uncertainties of other variables. Monte Carlo simulations allow to implement the uncertainty of the planned parameter (based on estimates or historical data) along with the uncertainty of other variables in the plan creation.

In the case study, we presented planning for a constant data model when data is moving within a narrow range of values. This model occurs in stabilized processes. Another character has data in unstable or cyclic processes. Therefore, further research in this area will focus on the use of simulations in production planning of products with seasonal and trend-driven demand model.