Inventory Control with Time Delays in Deliveries Using Linear and Quadratic Criteria

Valery I. Smagin, Gennady M. Koshkin, Konstantin S. Kim

Institute of Applied Mathematics and Computer Science National Research Tomsk State University

Tomsk, Russia e-mail: vsm@mail.tsu.ru, kgm@mail.tsu.ru, kks93@rambler.ru

Abstract—Inventory control algorithm for the model of a warehouse with time delays is proposed. Inventory control algorithm based on optimization linear and quadratic criteria with using the Kalman filtering for systems with unknown input and smoothing procedures is constructed. Example to illustrate the proposed approach is given.

Keywords—inventory control; time delays; linear criterion; quadratic criterion; unknown input; Kalman filter; smoothing procedures.

I. INTRODUCTION Synthesis of controls based on optimization both linear and

quadratic criteria is applied in many approaches, for example, model predictive control [1], locally optimal control [2]. The main advantage of the method optimal control with such criteria is a significant simplification of the synthesis procedure. These methods have been applied to control in technical systems, chemical processes, inventory control, production-inventory system, and portfolio optimization [3–8].

In this paper, we consider a control algorithm for discrete model of warehouse with time delays. It is assumed that the demand model contains unknown additional terms. In contrast to [7], it is possible to have many delays in the model that means taking into account the deliveries of goods from different suppliers. Also, it is proposed to use the smoothing algorithms for calculating estimates of demand and forecast of demand.

II. SYNTHESIS OF CONTROLS BY QUADRATIC CRITERION WITH COMPLETE OBSERVATIONS

Consider the model of the warehouse, which is described by the discrete equation

0

( 1) ( ) ( ) ( )M

i ii

x k Ax k B u k h s k

,

00,1,2, ; (0) ; ( ) ψ( ), , ( 1), , 1;M Mk x x u j j j h h K K

1 0 0M Mh h h K (1)

where ( ) nx R is a vector, and ix is a product volume for the

i-th nomenclature, ( ) mu R is a vector of deliveries, and ( )iu is a volume of delivery of the i-th nomenclature, ih is a

time delay, ( ) ns k R is a vector of demand at the k -th step ( ( )is is a demand for the product of the i-th nomenclature),

0x and ( )j ( , ( 1),M Mj h h , 1)K are the known vectors. Matrices and iA B define characteristics and the structure of a warehouse.

Take the quadratic criterion in the following form:

( ) M{ ( 1) ( )) ( ( 1) ( ))I k x k z k C x k z k

0

( ) ( )}M

i i ii

u k h D u k h

(2)

where M{} is the mathematical expectation, 0, 0iC D are weight matrices, z is the tracked vector. In this section, we assume that all the components of the vectors x and s are measured exactly.

Find the optimal control from the equation

( ) 0; 0, .( )i

I k i Mu k h

Taking into account the representation of the model (1) as

1 1

1 0 1( ) ( ) ( ) ( )

M MM

h hMh l lM i i

l i lx k A x k h A B u k h l A s k l

and condition (3), we obtain the system of equations to find the optimal supplies

110 0 0 0 0( ) ( ) ( ( )Mh

Mu k B CB D B C A x k h

1 0 1 1( ) ( ) ( ) ( )),

M Mh hM Ml l

j j j jl j j l

A B u k h l B u k h A s k l z k

111 1 1 1 1( ) ( ) ( ( )Mh

Mu k B CB D B C A x k h

1 0 0, 11

( ) ( ) ( ) ( )),M Mh hM M

l lj j j j

l j j lj

A B u k h l B u k h A Fs k l z k

The reported study was funded by RFBR according to the research project № 17-08-00920 and the TSU Competitiveness Improvement Program.

This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Copyright © 2017, the Authors. Published by Atlantis Press.

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IV International Research Conference "Information Technologies in Science, Management, Social Sphere and Medicine"

11( ) ( ) ( ( )MhM M M M M Mu k h B CB D B C A x k h

M

1

1 0 0( ) ( )

Mh M Mlj j j j

l j jA B u k h l B u k h

1

( ) ( )).Mh

l

lA s k l z k

(4)

Since in the system (4) variable ( )s is not always known, there is the problem of its prediction for future time. This prediction may be implemented by various methods of time series forecasting [9-11].

III. SYNTHESIS OF CONTROLS BY QUADRATIC CRITERION WITH INCOMPLETE OBSERVATIONS

Let the model of demand have the form

( 1) ( ) ( ) , s k Rs k f q k 0(0) ,s s (5)

where R is the known matrix, f is the known vector, ( )q k is a random vector. Incomplete observations of the m1-vector of demand are defined by the formula

( ) ( ) ( ),w k Hs k k (6)

where H is a matrix, ( )k is a random vector of errors, ( )q k and ( )k are sequences of the Gaussian random vectors with the characteristics:

M{ ( )} 0, M{ ( )} 0,q k k M{ ( ) ( )} ,Tkjq k q j Q

M{ ( ) ( )} ,Tkjk j T M{ ( ) ( )} 0,Tq k j

where kj is the Kronecker symbol.

In this case, the control will be calculated according to (4) with making use of the filtering estimates ˆ ( )fs and the forecasts of demand ˆ ( )ps . For example, to determine (0)u , it is necessary to solve system (4) with Mk h , which, in this case, is represented in the form:

110 0 0 0 0( ) ( ) ( (0)Mh

Mu h h B CB D B C A x

1 0 1 00 0

( ) ψ( )M M

M j M M j

h hM Ml l

j M j j M jl j l j

h h l h h h l

A B u h h l A B h h l

1

1 1

ˆ ˆ( ) ( ) (0) ( )),M

M

hMhl

j M j p M f Mj l

B u h A Fs h l A s z h

111 1 1 1 1( ) ( ) ( (0)Mh

Mu h h B CB D B C A x

1 0 1 00 0

( ) ψ( )M M

M j M M j

h hM Ml l

j M j j M jl j l j

h h l h h h l

A B u h h l A B h h l

1

0, 11

ˆ ˆ( ) ( ) (0) ( )),M

M

hMhl

j M j p M f Mj lj

B u h h A s h l A s z h

11(0) ( ) ( (0)MhM M M Mu B CB D B C A x

M

1 0 1 00 0

( ) ψ( )M M

M j M M j

h hM Ml l

j M j j M jl j l j

h h l h h h l

A B u h h l A B h h l

11

0 1( ) ( ) (0) ( )).

MM

hMhl

j M j p M Mj l

B u h h A s h l A s z h

(7)

At the next step to find (1)u , it is necessary to solve the system (4) with 1Mk h , then for 2Mk h , 3Mk h , and so on.

The estimates ˆ ( )fs are the estimates of the optimal Kalman filtering:

ˆ ˆ( ) ( 1) ( )[ ( )f f fs k Rs k f K k w k ,

ˆ( ( 1) ]fH Rs k f , 0ˆ (0) ,fs s (8)

where

( ) ( / 1) TfK k P k k H 1( ( / 1) )THP k k H T

( / 1) ( 1) TP k k RP k R Q ,

( ) ( ( ) ) ( / 1),n fP k E K k H P k k 0(0)P P . (9)

In (9), the matrix En is the identity matrix of the ( n n )- dimension. So, here it is necessary to use extrapolator, which will calculate the estimate of the forecast demand for the 1-st step:

ˆ ˆ( 1) ( ) ( )( ( )p p ps k Rs k f K k w k 0ˆ ˆ( )), (0)p pHs k s s (10)

where

1( ) ( ) ( ( ) ) ,T Tp p pK k RP k H HP k H T

( 1) ( ( ) ) ( )( ( ) )Tp p p pP k R K k H P k R K k H

( ) ( ),Tp pQ K k TK k 0(0)pP P

The forecasts for the next steps 2,..., 1Мj h are determined by the formula

ˆ ˆ( ) ( 1) .p М p Мs k h j Rs k h j f (11)

IV. SYNTHESIS OF CONTROLS FOR DEMAND MODEL WITH

UNKNOWN PARAMETERS Here, we assume that the demand model contains unknown

additional parameters:

( 1) ( ) ( ) ( )s k R R s k f f q k , 0(0)s s , (12)

where R is the known matrix, f is the known vector, R and f are some addition unknown matrix and vector

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(parameters), which can be interpreted as errors in the model of parameters (12). Model (12) may be interpreted as a dynamic model with an unknown input

( 1) ( ) ( ) ( )s k Rs k f r k q k , 0(0)s s (13)

where ( ) ( )r k Rs k f is an unknown input. Obtain the filtering estimate on the base of the Kalman

filtering algorithm with unknown input [12, 13]: ˆ ˆ ˆ( ) ( 1) ( 1)f fs k Rs k f r k

+ ˆ ˆ( )[ ( ) ( ( 1) ( 1))]f fK k w k H Rs k f r k , 0ˆ (0)fs s , (14)

( ) ( / 1)fK k P k h k H 1( ( / 1) )HP k h k h H , (15)

( / 1) ( 1)P k k RP k R Q , (16)

1

( ) ( ( ) ) ( / 1)n fP k E K k H P k k P(0) = P0,

where ˆ( )r is given in (24).

The extrapolator, which will estimate the forecast for the 1-st step ˆ ( 1)ps k , we define as follows:

ˆ ˆ ˆ( 1) ( ) ( )p ps k Rs k f r k

ˆ( )( ( ) ( ))p pK k w k Hs k , 0ˆ (0)ps s (18)

1( ) ( ) ( ( ) )p pr prK k RP k H HP k T , (19)

( 1) ( ( ) ) ( )(pr p prP k R K k H P k R

( ) ) ( ) ( )p p pK k H Q K k TK k , 0(0)prP P . (20)

According to (8), the estimates of forecasts ˆ ( )ps k j for 2j take the form

ˆ ˆ ˆ( ) ( 1) ( 1)p ps k j Rs k j f r k j . (21)

Note that in (29) ˆ( 1)r k j for 2j can be calculated using the analysis methods of time series [9–11].

The estimate r is calculated by the least square method by criterion

2 2

1

( ) ( 1)k

V Wi

J i r i

, (22)

where ( ) ( ) ( )i w i Hs i % ( ˆ( ) ( 1)s i Rs i f % ); V > 0, W ≥ 0 are weight matrices of the appropriate dimensions,

2( ) ( ) ( )V

i i V i . Then, minimizing (22), we obtain

1ˆ( ) [ ] M{ ( )}r k H VH W H V k (23)

where ˆ( ) ( ) [ ( 1) ]k w k H Rs k f

Now, let us calculate value M[ ( )]k using nonparametric smoothing algorithm [13, 14]. Applying the analog of the Nadaraya-Watson kernel regression estimates [14], we have

1 ˆˆ( ) [ ] M{ ( )}r k H VH W H V k ,

and the j-th component of the vector ˆ ( )k is given as

1

1

( ) 1

ˆ ( )1 1

kj

i j jj k

i j j

i k iK

kk iK

In the ratio (25), ( )K is a kernel function, j is a bandwidth parameter. We use the Gaussian kernels

2

exp2

( )2

u

K u

and the bandwidths are calculated by the cross-validation method [15].

Note, the estimates (14), (18), and (24) are used by synthesis of controls (7) for system with unknown parameters defined by equations (1) and (13).

V. MINIMIZATION OF EXPENSES COST Define the cost of storage of products in the sliding time

interval [k, k + T] using the additional linear criterion

11

( , ) ( , )n k T

i ii t k

J k z c x t z

(26)

with the following restrictions:

( ) ,i ix k X [ , ],k k k T 1,..., ,i n (27)

where ic is a storage cost of product unit for the i-th nomenclature in the unit time interval, iX is a safety stock for the i-th nomenclature. In (26), the dependence ( , )ix t z on z is due control (11) for system (1), (13).

Minimization of the criterion (26) under the restrictions (27) is carried out with respect to the vector z by numerical method, and at every step the control ( )u k is recalculated. The obtained value of the optimal vector z provides the minimum cost in the interval [k, k + T]. Vector z determines the volume of deliveries, and then, by analogy, we solve the problem of minimizing the new criterion 1( 1, )J k z under the constraints (27) ( [ 1, 1]k k k T ). Such procedure is realized recursively.

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VI. SIMULATION RESULTS Define the data describing the warehouse model and

criteria:

0.997 0

0 0.8A

, 0 1

0.1 0.5R

, 0.8

,0.8

f

diag{0.05 0.02},Q 0 11, 2,h h diag{0.05 0.05},T

0,D W 0 1 0 2 ,B B H C V P E

Simulations were carried out with the matrix R and vector f :

0 0.02

0.05 0.06R

1

0.9, if 0 50,( ) 0.3, if 50 100,

1.7, if 100 150,

kf k k

k

2

1.2, if 0 50,( ) 0.2, if 50 100,

1.7, if 100 150.

kf k k

k

The deliveries are determined for one vehicle under the following restrictions:

( )( )

(0 0) if ( ( )) min,( ) ( ) if min ( ( )) max,

if ( ( )) max,i

u kk

G u k Gu k u k G G u k G

G u k G

where maxG is a vehicle capacity, 2

1( ( )) ( )i ii

G u k p u k

(pi is a weight of product unit of the i-th nomenclature),

( ( ))max( ) G u k

Gk is the compression ratio. The value of minG is usually defined by the condition

0.8 max min maxG G G

which provides the high efficient on using of the vehicle.

The simulation results are shown in Fig. 1–7.

Figure 1. The volume of the stored product demand forecast

,2ˆps for the second nomenclature x1 for the first nomenclature

Figure 2. The volume of the stored product x2 for the second nomenclature

Figure 3. The diagram of the load of the vehicle

Figure 4. The volume of the demand s1, estimate s f,1, and demand forecast ,1ˆps for the first nomenclature

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Figure 5. The volume of the demand s2, estimate s f,2, and

Figure 6. The results of evaluation of unknown input r1 in the model of demand

Figure 7. The results of evaluation of unknown input r2 in the model of demand

VII. CONCLUSION

Algorithm of the inventory control with allowance for the vehicle delays from the different supplies for demand models

with unknown parameters is developed. It is proposed to use the Kalman filtering algorithms and nonparametric smoothing for calculation of estimate of demand and forecast of demand.

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