Adaptive Fuzzy Inventory Control Algorithm for Replenishment Process Optimization in an Uncertain...

Adaptive Fuzzy Inventory Control Algorithm forReplenishment Process Optimization in an

Uncertain Environment

Davorin Kofjac, Miroljub Kljajic, Andrej Skraba, and Blaz Rodic

University of Maribor, Faculty of Organizational Sciences,Kidriceva cesta 55a, 4000 Kranj, Slovenia

{davorin.kofjac,miroljub.kljajic,andrej.skraba,blaz.rodic}@fov.uni-mb.sihttp://www.uni-mb.si

Abstract. This paper presents a real case study of warehouse replen-ishment process optimization on a selected sample of representative ma-terials. Optimization is performed with simulation model supported byinventory control algorithms. The adaptive fuzzy inventory control al-gorithm based on fuzzy stock-outs, highest stock level and total costis introduced. The algorithm is tested and compared to the simulationresults of the actual warehouse process and classic inventory control al-gorithms such as Least-unit cost, Part period balancing and Silver-Mealalgorithm. The algorithms are tested on historic data and assessed usingthe Fuzzy Strategy Assessor (FSA). Simulation results are presented andadvantages of fuzzy inventory control algorithm are discussed.

Keywords: Replenishment, optimization, fuzzy sets, stochastic model,simulation.

1 Introduction

In an organization even with moderate size, there may be thousands of inventorystock keeping units and the main warehouse task is to enable the undisturbedproduction by assuring the right amount of materials. Several different principlesof warehouse optimization are described in [1], [2]. One way of optimizing thewarehouse process is to find the right replenishment strategy, while reducing thecost of the warehousing processes to a minimum without stock-outs occurringand warehouse capacity being exceeded. Therefore, the operator is dealing withthe decision problem - when to order and how many? This decision problem isvast, especially if we consider the fact, that he has to find the right replenishmentstrategy for more than 10.000 components stored in the warehouse.

This paper presents a real case study of replenishment process optimizationin an automotive company. Optimization is based on an anticipatory simulationmodel, inventory optimization algorithms and fuzzy sets. In comparison to theother methods, a dynamic analysis of the considered system behavior is the mainadvantage of testing the strategy with the aid of simulation scenarios [3].

W. Abramowicz (Ed.): BIS 2007, LNCS 4439, pp. 536–548, 2007.c© Springer-Verlag Berlin Heidelberg 2007

http://www.uni-mb.si

Adaptive Fuzzy Inventory Control Algorithm 537

Modern business environment is dynamic and uncertain. Inventory controlin such an environment is a complex task. The source of uncertainty can beeither the demand or the supply process. The demand and supply uncertaintycan be divided into time and quantity deviations from the original plan. Thetime deviation occurs when customers place their orders earlier or later thanexpected. The quantity deviation occurs when order quantity is different thanthe one anticipated or delivered. Both deviations can also occur simultaneously.Because of the mentioned uncertainties, a new term must be introduced - awarehouse, which task is to compensate those variations [4]. In other words -it should be a warehouse with adaptive replenishment strategy as deterministicapproaches lead to problems in an uncertain environment [5].

Adaptive fuzzy inventory control has been recently researched and is presentedin [6], [7], [8], [9]. Our adaptive approach is different from those mentioned, thatit uses known heuristic inventory algorithms, e.g. Silver-Meal, and assesses theirsimulation results and chooses the appropriate replenishment strategy. Assess-ment is performed using fuzzy number of stock-outs, highest stock level and totalcost for inventory control. Demand is stochastic and discontinuous; its quan-tity is changing and time varying gaps occur between two successive demands.Lead times are long (two to three months) and stochastic. Our approach alsooptimizes a complex cost function consisting of: ordering cost, take-over cost,transportation cost, physical storage cost, capital cost and manipulation cost,without violation of the following restrictions: warehouse capacity must not beexceeded and stock-outs must not occur.

The paper is structured in the following way: Section 2 describes the ware-house model, Section 3 elaborates replenishment algorithms with examples whilesimulation results are presented and discussed in Section 4. The conclusion ispresented in Section 5.

2 Warehouse Problem Formulation

Dealing with problems of warehousing, we encounter several contradictory crite-ria. An overly large warehouse means a greater amount of stock, greater capitalcost and more staff. The space itself is very valuable today. An overly smallwarehouse can represent possible stock-outs, it demands a reliable supplier etc.The warehouse in our case is used for storing components for further build-in.The components are delivered into the warehouse from different vendors andused in the production line, where they eventually become a part of differentfinished products. Components belong to different categories according to ABCand XYZ classification, which is described thoroughly in [1].

As the production process and the delivery are stochastic, future stock dy-namics cannot be determined with a certainty and have to be anticipated. Thatis, anticipatory models will act according to the past states of the system andaccording to the desirable and possible future states [10]. Formally,

xt+1 = f(x1, x2, . . . , xt, xts) (1)

538 D. Kofjac et al.

where xts, t + 1 ≤ s ≤ t + k(t), is a prediction xs done at time t, and k(t)

is the size of the forecasting interval at time t. If x is the stock variable, thenEqn. 1 is describing its past, present and future dynamics. Anticipation is doneby running simulation scenarios, where replenishment algorithms predict futurestock dynamics based on production plans. Different algorithms yield differentpredictions and the algorithm with the best results (regarding Eqn. 5) is usedto change the state of a system, which will also affect its future states.

2.1 Mathematical Formulation

From control point of view, warehouse optimization problem can be describedwith the following equation:

x(k + 1) = x(k) + d(k) − p(k), k = 0, 1, 2, . . . (2)x(0) = x0

where x(k) represents stock variable, d(k) stochastic material delivery, p(k)stochastic production process and x0 the initial stock value at k = 0. The deliv-ery function d(k) is delayed for an average time of an order o(k) as defined byEqn. 3:

d(k) = o(k − ϕ(τd)) (3)

where ϕ(τd) represents discrete uniform probability density function. Time de-lays τd are stochastic and tend to be very long for some materials (e.g. two tothree months) thus representing a problem, because they are usually much longerthan the time period of six weeks in which the production plan can be predictedwith a certainty. To compensate the stochastic delivery delay, the replenishmentorder policy o(k) has to be defined as:

o(k) = f(x(k), d(k − τd), p(k + τp)) (4)

where τp represents a production plan horizon. It is necessary to find such o(k)to minimize the following cost function:

J(o(k)) =n∑

k=0

g(ccx(k), cpx(k), ct(w)d(k − τd), coo(k), cvd(k − τd), cmp(k)) (5)

xmin ≤ x(k) ≤ xmax

where g represents cost function consisting of: cost of capital cc, cost of physicalstorage cp, transportation costs ct, fixed ordering costs co, costs of taking overthe products cv and cost of manipulation in production process cm. The followingrestrictions have to be considered:

– maximal warehouse capacity (xmax) for a specific product must not be ex-ceeded,

– no stock-outs may occur (xmin).


Stock

Production

Productionplan

Ordering

Delivery

Stochasticdelay

Transportationcost

Totalcosts

Orderingcost

Manipulationcost

Min

imiz

ation

Min

imiz

ation

Cost ofphysicalstorage

Cost ofcapital

Holdingcost

Take-overcost

Fig. 1. Causal loop diagram (CLD) of the warehouse model for replenishment processoptimization

xmin can also be represented as a safety stock, where stock level should not fallunder the safety stock level.

The transportation cost is explicitly considered in the model. Rather thanassuming it to be a part of the fixed ordering cost or to be insignificant [11], wewill take the transportation cost as a function of the shipment lot size. In thiscase transportation cost ct can be described by the following function:

ct(w) =

⎧⎪⎪⎨

⎪⎪⎩

0, w = 0Ct1, w ∈ (0, W1]Ct2, w ∈ (W1, W2]Ctn, w ∈ (Wn−1, Wn]

(6)

0 < Ct1 < Ct2 < . . . < Ctn

0 < W1 < W2 < . . . < Wn

where Ct1, Ct2,..., Ctn are transportation cost constants, dependent on the trans-port weight w, and W1, W2, ..., Wn are weight constants.

2.2 The Warehouse Model

Fig. 1 represents the causal loop diagram (CLD) from which the influences of thewarehouse model elements can be observed. The arrow represents the directionof the influence and the “+” or “-” sign its polarity. The “+” sign is used between


Delivery Warehouse Production

Ordering

Adaptive

Inventory Control

Simulation

System

Replenishment

Algorithms

(FRP, LUC, PPB,

S-M, FZY)

... Material flow

... Information flow

Fig. 2. Block model of the warehouse simulation model for replenishment process op-timization

two elements, if the causal element has the same influence on the consequentialelement. The “-” sign is used, if the causal element has opposite influence on theconsequential element.

There are two negative feedback loops in the CLD. The first interconnectsStock, Ordering and Delivery and it represents the fact that less is ordered,if the stock level is high. The second interconnects Delivery and Ordering andrepresents the concept that less is ordered, if more was ordered before. This loopconsiders orders, which have not been delivered yet and will have impact on thestock level later.

Fig. 2 presents the simplified warehouse model. It is a warehouse model withadded adaptive inventory control simulation system (AICSS) supported by re-plenishment algorithms. The AICSS gathers information on production plan,stock-on-hand and delivery and runs different simulation scenarios according tothe replenishment algorithms to anticipate future stock dynamics. A fuzzy strat-egy assessor (FSA) is used to assess the results of simulation runs. Replenishmentalgorithms and FSA are described in the following sections. Based on the FSAassessment, the AICSS offers the ordering process information on how much andwhen to order. For the detailed description of the warehouse model one shouldsee [12].

3 Replenishment Algorithms

Optimization is based on the simulation model described previously and thefollowing inventory control algorithms:


– Fixed review period (FRP),– Least unit cost (LUC),– Part period balancing (PPB),– Silver-Meal algorithm (S-M),– Fuzzy control algorithm (FZY).

The Fixed review period algorithm (FRP) is similar to the (R, S) system(described in [1]), where, every R units of time, an order is made to adjust thestock level to the order-up-to-level S. In contrast to the (R, S) system, S is not afixed value in the FRP algorithm. The Fixed review period is described in detailin [12], while Least unit cost, Part period balancing and Silver-Meal algorithmsare described in detail in [1]. Following is the detailed description of the FZYalgorithm.

3.1 Fuzzy Control Algorithm

Fuzzy control algorithm uses Fuzzy strategy assessor (FSA) to assess replen-ishment optimization simulation results. The FSA is based on fuzzy sets intro-duced by Zadeh [13]. There have been several studies that applied fuzzy sets ininventory control ([14], [15], [16], [17], [18], [9]). These papers presented studiesof various fuzzy inference systems where fuzzy demand, inventory level, orderquantity, lead times and various fuzzy costs, e.g. shortage cost, were studiedand analyzed. This paper is different from those previously mentioned in that itdoes not deal with shortage cost because it is unknown, but instead investigatesthe fuzzy number of stock-outs, highest stock level and total cost for inventorycontrol.

z

c

S

FSAE

Fig. 3. Fuzzy strategy assessor (FSA)

In this case, the value of stock-out or shortage cost is difficult to assess sincematerials are often a part of several finished products. The shortage cost is notmerely the value of material quantity lacking for the production process. Thenumber of stock-outs are the only information available here and difficult toassess regarding costs. Since this problem can be described as soft, fuzzy setswere chosen to assess the replenishment strategies results.

The inputs of the FSA, as shown in Fig. 3, are matrices for number of stock-outs z, total cost c and highest stock level S, each matrix containing simulationresults of various simulation strategies. The output of the FSA is replenishment


strategy estimation matrix E. Matrix elements values for z, c and S are normal-ized in the interval [0,1] using the following equation:

bi =bi

max(b), i = 1, 2, . . . , n (7)

where b is either the matrix z, c or S, bi is the matrix element, max(b) is themaximal value among matrix elements and n is the number of elements in thematrix. Matrices are then classified (fuzzified) using equally spaced Gaussianmembership functions (GMFs) shown in Fig. 4.

r

0

1

1 r0.33 0.66

verylow low high

veryhigh

�r

(a)

r

0

1

1 r0.50.25 0.75

none few some many a lot

�r

(c) (d)

�r

0

1

1 r0.50.25 0.75

verylow low middle high

veryhigh

�r

(b)

�r� r

0

1

1 r0.50.25 0.75

excellentverygood

good poorverypoor

�r

Fig. 4. Gaussian membership functions for stock-outs (a), total cost (b), highest stocklevel (c) and replenishment strategy estimation (d)

The Gaussian curve is given by the following function:

f(x) = e−0.5(x−m)2

σ2 (8)

where m is mean and σ is variance. Means and variances for each GMF are givenin Table 1.

Table 1. Means and variances of GMFs for each variable used in FSA

V ariable σ m

z 0.06 0,0.25,0.5,0.75,1c 0.15 0,0.33,0.66,1S 0.12 0,0.25,0.5,0.75,1E 0.05 0,0.25,0.5,0.75,1


The FSA inference system contains the expert’s rule base consisting of 100(5 · 4 · 5 = 100) rules. Estimation of the ith replenishment strategy is calculatedaccording to the following equation:

IF zi AND ci AND Si THEN Ei, i = 1, 2, . . . , n (9)

where n is a number of simulation strategies. A few examples of rules are givenbelow:

– If zi is none and ci is very low and Si is very low then Ei is excellent,– If zi is many and ci is very high and Si is low then Ei is poor.

The rules are used to obtain the final output E, which is then defuzzified usinga Som (smallest value of minimum) function in the interval [0,1]. The strategywith the lowest grade is selected as the one yielding the best simulation results.If several strategies achieve the same lowest grade, they are assessed once more.Matrices z, c and S now contain only the strategies with the lowest grades andare again normalized using the Eqn. 7 and fuzzified. A new estimation matrix Eis calculated. This process is repeated until only one strategy has achieved thelowest grade.

The fuzzy control algorithm (FZY) is based on all algorithms mentioned be-fore. All algorithms are run in a simulation and the results are assessed usinga fuzzy strategy assessment (FSA), providing the estimation matrix E, uponwhich the order time and quantity are selected.

Table 2. Numerical example of FZY algorithm estimation

Algorithm Ej qj

FRP-14 0.22 137FRP-21 0.24 137FRP-28 0.45 577FRP-35 0.4 777FRP-42 0.17 777FRP-49 0.41 977FRP-56 0.42 977FRP-63 0.46 977FRP-70 0.46 977PPB 0.2 777LUC 0.46 1938S-M 0.46 577

Steps, performed during the simulation of the FZY algorithm, are:

1. Simulate the whole range of algorithms in m iterations2. Calculate average results of each algorithm3. Assess average results using FSA to obtain the estimation matrix4. Select algorithm with the lowest grade


5. Select the first order that occurred in the simulation of the selected algorithm(according to Eqn. 10) and apply it in simulation. The output of the fuzzycontrol algorithm is an order o(k) as defined by the following equation:

o(k) = f(qj), j = 1, 2, . . . , n (10)

where j represents the chosen strategy, n a number of simulation strategiesand q a vector of order quantities of all replenishment algorithms. The nu-merical example of estimation is provided in Table 2. The algorithm withthe lowest assessment is FRP −42 and the order quantity o(k) is 777 pieces.

4 Results

The simulation model has been built using Matlab with its Simulink and Fuzzylogic toolboxes. The experiment was performed with the historic data providedby the observed company. Altogether, four materials were examined in this studyand their details are described in Table 3.

Table 3. Material lead times in weeks, demand mean and demand interval

Material 1 2 3 4

Lead time 2-3 2-3 6-8 6-9Demand mean 2629.23 143.07 3500.32 1567.27Demand interval [0,62800] [0,744] [0,22000] [0,13168]

The simulation of the actual warehouse process (RP - Real process) was usingreal data of delivery and demand while the simulation with replenishment algo-rithms (VP - Virtual process) was using only real demand data. The orderingand delivery process in VP were controlled by replenishment algorithms. TheRP simulation was run only once, whereas ten simulation runs were executedfor every VP replenishment algorithm. Based on these simulation runs, averagecosts and average stock-outs were calculated. With several simulation runs anda calculation of average values, we have tried to minimize the influences of therandom generator, which represent the stochastic environment. Out of all simu-lation runs the maximum stock level was considered as the warehouse capacitylimit and the strategy with minimal highest stock level was favored.

Table 4. Summary of simulation results for all materials

Material RP Cost V P Cost Savings (%) V P Strategy

1 1.63 · 106 1.52 · 106 6.7 FZY2 3.48 · 106 3.11 · 106 10.6 FZY3 1.91 · 106 1.88 · 106 1.6 FZY4 2.62 · 106 1.89 · 106 27.8 FZY

Total Σ = 9.64 · 106 Σ = 8.4 · 106 12.7


Fig. 5. Frequency distributions of algorithms used during FZY algorithm simulationfor each material


The simulation was run for a period of seven years in RP and VP, but alsowith subperiods of 24 weeks in the VP; depending on the algorithm, at eachtime-step (continuous review) or at each-review period (periodic review), thesimulation was run for 24 weeks, since this is the production plan predictionperiod, thus providing a replenishment schedule until the next review.

A Monte Carlo simulation was used for variation of production plan unre-liability. The production plan variability was simulated by perturbations of itsquantity every two weeks for a certain amount, e.g. 5%. Variable lead times weresimulated by uniform random generator. If stock-outs occurred during the sim-ulation, the missing quantity was transferred into the next period. The safetystock was also considered; it was equal to the average weekly demand.

Results in Table 4 show the comparison of total cost of the RP and the strategychosen by the FSA among the VP strategies. The results presented in Table 4indicate that the FZY algorithm was chosen as the one yielding the best resultsfor all cases. FRP, LUC, PPB and S-M algorithms were never chosen. The totalcost saving is 12.7%, if all four materials are considered.

Frequency distributions of algorithms used during the simulation of FZY al-gorithm are presented in Fig. 5. One can notice that predominant algorithm isFRP - 14 days for Materials 1, 3 and 4 while algorithm FRP - 21 days is predom-inant for Material 2. It is also worth to point out that classic algorithms PPB,LUC and S-M were never chosen during the FZY simulation. Obviously, theirheuristics does not work well for the selected materials. The reason for that maybe a more complex cost function (Eqn. 5) than the cost function used by classicalgorithms in their optimization calculations, which uses only cost of capital andfixed ordering cost.

Table 5. Comparison of total cost of FZY (c1) and of predominant algorithm used inFZY simulation (c2)

Material Algorithm 1 c1 Algorithm 2 c2c2c1

1 FZY 1.52 · 106 FRP - 14 1.77 · 106 1.162 FZY 3.11 · 106 FRP - 21 3.93 · 106 1.263 FZY 1.88 · 106 FRP - 14 2.16 · 106 1.154 FZY 1.89 · 106 FRP - 14 2.06 · 106 1.09

Table 5 presents comparison of total cost of FZY (c1) and of predominantalgorithm used in FZY simulation (c2). If only predominant algorithm wouldbe used in simulation instead of FZY algorithm, it would produce higher totalcost for each material. The total cost would be higher for 16, 26, 15 and 9percent for Material 1, 2, 3 and 4 respectively. One can deduce that variabilityof replenishment algorithms produces lower total cost than using strictly onereplenishment policy.


5 Conclusion

The research presented in this paper is the real case study of replenishment pro-cess optimization on historic data of selected materials. Lead times are stochasticand so is demand. The demand has also a high dispersion from the mean.

Simulation results of the FZY algorithm yielded significant cost savings. Theintroduced adaptive fuzzy inventory control algorithm surpassed results of otheralgorithms, including classic ones like Least-unit cost, Part period balancing andSilver-Meal. It was also proven, that there is a need for an adaptive algorithms,which is able to follow the changing patterns in demand, because the days of astable market are long gone.

This paper also indicates that classic inventory control algorithms use inad-equate cost function, which is not appropriate for replenishment process op-timization for materials selected for this study. The cost function presented inthis paper is more complex than the one used by classic algorithms and contains:cost of capital, cost of physical storage, transportation costs, fixed ordering costs,costs of taking over the products and cost of manipulation in production process,while the cost of classic algorithms contains only the cost of capital and fixedordering cost.

Results of the study are encouraging and the direction of our future research isa thorough validation of the presented warehouse model and its implementationin the company’s replenishment process.

Acknowledgment. This research was supported by the Ministry of HigherEducation, Science and Technology of the Republic of Slovenia (Programme No.UNI-MB-0586-P5-0018). Our sincere thanks goes also to Mr. Valter Rejec andhis team.

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