The purpose of this work is to contribute to the extended use of hybrid models to solve MRP issues dealing with stochastic demand over main stock-keeping units. The methodology development first apply SARIMA (Seasonal Autoregressive Integrated Moving Average Model), Long short term memory networks and Fb-prophet as forecasting methods to predict the demand for the master production schedule, next applies an integer programming model with JuMP (Julia Mathematical programming) for solving the MRP using the Lot for lot approach (L4L). The main contribution of this work is to show a way to solve dynamic demand problems over the Forecasting-MIP approach.
The heart of a Material Requirements planning system is the demand forecasting
for the main stock-keeping-unit. However, it is a complicated, time spending and
inaccurate task. The randomness in the forecast has led to improve them using
a variety of tools, such as the Box-Jenkins methodology [
The incremental use of computers in the 1960’s industry allows them to
improve complex scheduling and inventory control [
The development and adoption of Material requirements planning were slow
but in 1972 the American production and inventory control society APICS starts
to implement this model and initiates the mrp crusade that continues in our
time [
This section describes the basic concepts of Material requirement planning, Demand Driven Material requirement planning and Forecasting techniques needed to solve the dynamic demand problem presented in the case study. 2.1
This item first describes every part in a production system and later focuses on MRP implementation. A production plan describes in detail the quantity of principal sku (final product to sell) and produced in subsets of sku, the exact time of production and lot sizing. The production plan can be divided in master production schedule (MPS), Material requirement planning (MRP) and the detailed plan of jobs in the production floor. See Fig. 1.
The APICS dictionary defines MRP as: "A set of techniques that uses a bill
of material data, inventory data, and the master production schedule to calculate
requirements for materials" [
(b) The basic MRP record form: [
in the model proposed by [
P T minimize X X(T − t)Xi,t
x i=1 t=1 t−LT (i)
X τ=1
Xi,t δi,t − M δi,t ∈ 0, 1 Xi,t − δi,tLS(i) ≥ 0 ≥ 0
t Xi,τ + I(i, 0) ≥ X τ=1
P D(i, τ ) + X R(i, j)Xi,j j=1 ∀i ∈ 1, ..., P , ∀t ∈ 1, ..., T .
P Number of SKUs T Number of time buckets (i.e., planing horizon) LTi Lead time for each SKU i Ri,j Number i necesarios para hacer un j Di,t External demand for i over tperiod Ii, 0 Initial inventory for every SKU i LSi Minimum lot size for every SKU i M A large number
The planning horizon i calculated by [
The core of MPS is the forecasting job to predict future demands. This work uses the Box-Jenkins methodology to applying the seasonal ARIMA model in order to obtain the future demand of principal sku.
Time series analysis According to [
Box-Jenkins Methodology This method was given thanks to two known
statisticians [
The assumption of seasonality implies that the marginal distribution of two observations separated by the same time interval are the same. This means that Dt and Dt+1 they have the same distribution as Dt+m y Dt+m+1 for any m ≥ 1. This implies that the covariance of Dt and Dt+1 is exactly the same covariance of Dt+m and Dt+m+1. Therefore , the covariance of these two observations depends only on the number of periods that separate them. In this context, covariance is also known as self-covariance , we are comparing two values of the same series separated by a fixed delay .Leave Cov(Dt+m, Dt+m+k) be the covariance of Dt+m and Dt+m+k given by Cov(Dt+m, Dt+m+k) = E(Dt+mDt+m+k) − E(Dt+m)E(Dt+m+k)∀k, int ≥ 1.
There are four main steps needed to build BOX-JENKINS forecast models 1. Data transformations: The BOX-JENKINS method is based on the start of a series of stationary time. To be sure that the time series is stationary, several steps are needed preliminarily. We know that differentiation eliminates trend and seasonality. However, if the average of the series is relatively fixed, this may be the case in which the variance is not constant, so a transformation of the data is going to be required. 2. Identification of the model: This step refers exactly to which is the most appropriate ARIMA model. The identification of the type of model is both art and science, it is difficult if not impossible, to identify the model only the series is examined. It is much more effective to study the self-correlations of samples and partial self-correlations for the identification of patterns that coincide with those of the known processes. In some cases, the self-correlation structure will point to a simple AR or MA process, but it is more common to mix these two terms to have a better fit. 3. Parameter estimation: Once the appropriate model has been identified, the optimal values of the parameters of the model (i.e., a0, a1, . . . ., apyb0, b1, . . . ., bq) must be determined, usually this step is carried out by means of least squares adjustment methods or by the maximum likelihood method, this step is performed by a computer program. 4. Forecasts: Once the model has been identified and the values of the optimal parameters determined, the model provides forecasts of future values of the series. The BOX-JENKINS models are more effective in providing one-step forecasts, but can also provide multi- step forecasts. 5. Evaluation: The waste pattern (forecast errors) can give useful information to see the quality of the model. The residuals must form a white noise ( ie, random) process with zero means. When there are patterns in the waste, it is suggested that the model can improve. 2.3
The bright idea of introducing self-loops to produce paths where the gradient
can flow for long durations is a core contribution of the first long short-term
memory (LSTM) model [
The LSTM has been found hugely successful in many applications, such as
unconstrained handwriting recognition [
The LSTM block diagram is illustrated in Fig. 3. The corresponding forward propagation equations are given below, for a shallow recurrent network architecture.
Cells are connected recurrently to each other, replacing the usual hidden units of ordinary recurrent networks. An input feature is computed with a regular artificial neuron unit. Its value can be accumulated into the state if the sigmoidal input gate allows it. The state unit has a linear self-loop whose weight is controlled by the forget gate. The output of the cell can be shut off by the output gate. All the gating units have a sigmoid nonlinearity, while the input unit can have any squashing nonlinearity. The state unit can also be used as an extra input to the gating units. The black square indicates a delay of a single time step.
fi(t) = σ bif + X Uif,j x(jt) + X
(t−1) Wif,j hj (1) (2) (3) j , .
s(t) = fi(t) s(t−1) + gi(t−1)σ bi + X Ui,j x(jt) + X i i (t−1) Wi,j hj , j
j g(t) = σ big + X Uig,j x(jt) + X i
(t−1) Wig,j hj
h(t) = tanh sit qi(t),
i q(t) = σ bi0 + X Ui0,j x(jt) + X i
(t−1)
Wi0,j hj j j , 2.4
We use Prophet open-source software in Python [
y(t) = g(t) + s(t) + h(t) + t
Were g(t) is the trend function which models non periodic changes in the value of the time series, s(t) represents periodic changes, and h(t) represents the effects of holidays which occur on potentially irregular schedules over one or more days. The error term t represents any idiosyncratic changes that are not accommodated by the model; later we will make the parametric assumption that t is normally distributed. 3
Open source tools in optimization are increasing popularity over the last years
and can be observed in works like [
In this paper, we propose a methodology based on a combination of Forecasting and MIP techniques to help the managers of the fragrance industry to decide the best inventory and supply policies. The steps followed in the methodology are shown in the Fig. 4.
The methodology is applied to a particular manufacturing case study: to help the managers of the fragrance industry to achieve their goals related to minimizing inventory and minimizing significant deviations in their forecasts. First, the forecasting job is performed by using the box-Jenkins methodology although ASTSA package in R, the Fb-Prophet package and Lstmn with KERAS in Python, next, the Master Production Schedule, is created based on the previous forecasts, and the MRP optimization is performed to get the final MRP plan. The results of the MRP model are evaluated comparing their results with the real demand and improved by adjusting the historical data demand records. 4
We apply the solution method to a company dedicated to the creation development, manufacture, and marketing of flavors and fragrances for the Colombian industry. A step by step application is described as follows: 4.1
First, the historical demand data is plotted in order to isolate the data patterns. Later we apply the SARIMA model by using the R language Fig. 5, also the Fb-Prophet package and Lstmn using KERAS and Google Colab Fig. 6.
Table 1 shows the difference in forecasts generated and their associated errors in the last row. The techniques with the lowest associated error are SARIMA and LSTMN, so the predicted demand generated by them will be used, which can be seen in Table 1.
As it can be seen in Table 2 the future demand for 9 months is the principal input for the MRP formulation, and it will be represented on the subset Dj. (a) Historical demand data plot
(b) Forecasting for 2019 (monthly). In this item, we use the Julia programing language and the package JuMP for mathematical optimization, the code for the problem it can be seen as follows. using JuMP,Cbc,NamedArrays,DataFrames filas=size(D,1) col=size(D,2) mrp=Model(solver=CbcSolver()) @variables mrp begin x[1:filas,1:col]>=0 d[1:filas,1:col]>=0 end T=col @objective(mrp,Min,sum( x[i,j]*((T-j)) for i=1:filas, j=1:col)) for i=1:filas,t=1:col @constraint(mrp, sum(x[i,s] for s=1:t-LT[i,1])+I[i,1]>=sum(D[i,s] +sum(R[i,j]*x[j,s] for j=1:filas) for s=1:t)) end @constraint(mrp,x-d.*LS.>=0) @constraint(mrp,d-x/1000000000000000.>=0) status=solve(mrp) println(getobjectivevalue(mrp)) println(DataFrame(getvalue(x))) println(DataFrame(getvalue(d))) We use the Ri,j bill of manufacture that shows in the Table 3 as follows: t2 t3 t4 t5 t6 t7 t8
Future investigation must be related to DDMRP demand driven MRP and machine learning techniques related to forecasting methods. The first affirmations is related with methods for maintain a minimum and maximum intervals for inventory management over every sku. The second affirmation is related to method that helps us to obtain better and more accurates forecasts for future demand. 6
The use of Open source optimization and forecasting tools allows to solve optimization and predictive problems efficiently.
The forecasting problem is the principal problem in an MRP based company and can be solved at scale using open source tools.
The biggest contribution of this work is to show how to solve a real problem over forecasting and optimization using three programming languages in order to reduce error and improve the time scheduling for every stock keeping unit.
The company uses this implementation in order to plan his manufacturing process at a larger scale including every product of their portfolio.