Nr 1
Jerzy ŚWIĄTEK* Tadeusz GALANC**
PROBABILITIES OF A LOWER-LIMIT BARRIER
IN THE PROBLEM OF THE IDENTIFICATION OF A BARRIER
IN THE FUNCTIONING OF A CERTAIN INVENTORY STORAGE
AND ISSUE SYSTEM
The paper investigates a barrier in the functioning of certain inventory system. Assuming that the storage input is a non-aggregated dynamic-parameter process, the paper derives a system of differen-tial equations satisfied by the probabilities of a lower-limit barrier in subsysyem L. The system of equations expresses relations between the distributions of the probability of the lower-limit barrier on the one hand and the parameters of the product supply process and the parameters of the functioning of the transport subsystem on the other hand.
Keywords: inventory system, barrier, transport, system of differential equations
1. Introduction
This paper is a continuation of the investigations presented in papers [1]–[3]. The general operating principles of such a system, the definition of the notion of a barrier in its functioning, and an analysis of the behaviour of the subsystem L in intermediate states are given in [1] and [2]. Using the analytical forms of conditional probabilities in the case of a lower-limit barrier in this subsystem, obtained in [3], we will derive a system of differential equations satisfied by the probabilities of a lower-limit barrier in a process controlled by a non-aggregated input. The terminology and notations are the same as used in [1]–[3].
* Institute of Control and Systems Engineering, Wrocław University of Technology, ul. Janiszew-skiego 11/17, 50-370 Wrocław, Poland. Jerzy.Swiatek@pwr.wroc.pl
** Institute of Industrial Engineering and Management, Technical University of Wrocław, ul. Smoluchow-skiego 25, 50-370 Wrocław, Poland. Tadeusz.Galanc@pwr.wroc.pl
2. Conditional probabilities in the case of a lower-limit barrier
in the subsystem L
In order to obtain relations satisfied by the probabilities Qul({0},t)
k (cf. equation (1) in [1]), we will use the formulas for the conditional probabilities qrwl(z,{0}; ,t),
ik τ l l t T T t∈ , +τ∈ derived in paper [3]: for xi >0,xk<0, + ≤ <− − + − = − , other , 0 , 0 ), ; ( ) 1 ( ) ( ) , }; 0 { , ( ) ( ) ( * 1 ) ( 1 11 z x z z o x x x z o t z q l k i k k l ik l l l ik τ τ τ π τ π τ τ (1)
for xk>0,xi – any state,
, 0 ), ( ) , }; 0 { , ( () 1 1 11 z t o z V q l l ik τ = τ ≤ ≤ (2) for any xk, xi, − + ≤ < = − , other , 0 , 0 ), ( ) 1 ( ) ( ) , }; 0 { , ( ) ( ) ( * 0 ) ( 1 00 z d z o o t z q l l ik l l l ik τ τ τ π τ π τ τ (3) − − + ≤ < = − , other , 0 , 0 ), ( ) 1 )( 1 ( ) ( ) , }; 0 { , ( ) ( ) ( * 0 ) ( 1 00 z d z o o t z q l l k l l l kk τ τ τ π τ π τ τ (4) for xk <0, = − ( ) ) , }; 0 { , ( () 1 10l τ l τ kk z t o q > + < < − > + + + − − − < ≤ > + + − ≤ + < ≤ ≤ + + − = , other , 0 , 0 , , 0 ), ; ( ) 1 ( , 0 , 0 ), ( ) 1 ( , other , 0 , 0 , 0 , 0 ), ( ) 1 ( ) ( * 1 ) ( ) ( * 1 ) ( ) ( * 1 ) ( z x d d z x x d z o x d z d x z x d o z x d d z x d o k k k l k l l k k k l l l k k k l l l k τ τ τ τ π τ π τ τ τ π τ π τ τ τ π τ π (5) for xk >0, other z, other z, other z, other z, other z,
+ ≤ < + − − = − , other , 0 , 0 ), ; ( ) 1 ( ) ( ) , }; 0 { , ( ) ( * 1 ) ( ) ( 1 10 z d z z o x d z d o t z q l k l l k l l kk τ τ τ π τ π τ τ (6) for xk =0, − − + ≤ < = − , other , 0 , 0 ), ; ( ) 1 ( ) ( ) , }; 0 { , ( () () 1* () 1 10 z d z z o d z d o t z q l l kl l l kk τ τ τ π τ π τ τ (7) for xk >0, . 0 , 0 ) ( ) , }; 0 { , ( () 1 1 11 z t o z V q l l kk τ − τ = ≤ ≤ (8) for xk ≤0, − − + ≤ < = − , other , 0 , 0 ), ( ) 1 )( 1 ( ) ( ) , }; 0 { , ( ) ( * 1 ) ( ) ( 1 11 z x z o o t z q l l kl l l k kk τ τ τ π τ π τ τ (9) for xk >0, , 0 , 0 ) ( ) , }; 0 { , ( () 1 1 01 z t o z V q l l kk τ − τ = ≤ ≤ (10) for xk =0, < ≤ + − − = − , other , 0 , 0 ), ; ( ) 1 ( ) ( ) , }; 0 { , ( () () 0* () 1 01 z d z z o d z o t z q l l kl l l kk τ τ τ π τ π τ τ (11) for xk <0, = − ( ) ) , }; 0 { , ( () 1 01l τ l τ kk z t o q > + < < − > + + + + − − − < ≤ > + + − ≤ + < ≤ ≤ + + − = , other , 0 , 0 , , 0 ), ; ( ) 1 ( , 0 , 0 ), ( ) 1 ( , other , 0 , 0 , 0 , 0 ), ( ) 1 ( ) ( * 0 ) ( ) ( * 0 ) ( ) ( * 0 ) ( z x d d z x x d z o x d x z x z x d o z x d d z x d o k k k l k k l l k k k l l l k k k l l l k τ τ τ τ τ π τ π τ τ τ π τ π τ τ τ π τ π (12) other z, other z, other z, other z, other z, other z,
for xi<0,xk =0, − < ≤ + + − = − , other , 0 , 0 ), ; ( ) 1 ( ) ( ) , }; 0 { , ( () 1* () () 1 11 z x z z o x z z o t z q l i i l ik l l l ik τ τ π τ π τ τ (13) for xi<0,xk <0, = − ( ) ) , }; 0 { , ( () 1 11l τ l τ ik z t o q < − < < − < + − + − − − < ≤ < + − > − < < − > + − + − − < ≤ > + − = , other , , 0 , , ), ; ( ) 1 ( , 0 , ), ( ) 1 ( , other , , 0 , , ), ; ( ) 1 ( , 0 , ), ( ) 1 ( ) ( ) ( * 1 ) ( ) ( * 1 ) ( ) ( * 1 ) ( ) ( * 1 z x x x z x x x z o x x x z x z x x o z x x x z x x x z o x x x z x z x x o k i i k k i l i k k l ik l k k i l l ik l k i k i k i l i k k l ik l i k i l l ik l τ τ τ τ τ π τ π τ τ τ π τ π τ τ τ τ π τ π τ τ τ π τ π (14)
Using equation (6) from [3] and the relations (1)–(14) for xk <0,t∈Tl,t+τ∈Tl, we get
}
, ) , ; } 0 { , 0 ( ) }, 0 ({ ) , ; } 0 { , ( ) }, ({ ) , ( ) , ; } 0 { , ( ) , ; } 0 { , 0 ( ) }, 0 ({ ) , ; } 0 { , ( ) }, ({ ) , ( ) , ; } 0 { , ( ) }, 0 ({ 1 , 1 0 , 1 , 01 0 1 01 1 0 0 0 01 11 1 1 11 1 1 0 1 11 1 1 1 − + + + = + + + + + = +∫
∑ ∫
l k l k l k l kk l k l kk l k V l k l kk l ik l i l ik l i i V l i l ik l k B B B t q t Q t V q t V Q dz t z f t z q t q t Q t V q t V Q dz t z f t z q t Q τ τ τ τ τ τ τ (15) where ) , }; 0 { , 0 ( ) }, 0 ({ ) , ( ) , }; 0 { , ( 1 1 11 0 11 0 1 , 1 t q t Q dz t z f t z q B il il ikl V l ik xk i l k i τ τ + =∑
∫
< ≠ −∑ ∫
≥ ≠ − + − = + k i i x xk i l i l l ik l x l ik l i V t q V t o f z t dz Q ({ }, ) ( ,{0}; , ) [(1 * ) () ()( )] 1( , ) 1 0 1 11 1 1 τ π τ π τ τ τ other z, other z, other z,∫
∫
+ + − + − + − − 1 ) , ( ) ( ) , ( ) ; ( ) 1 ( * () ( ) 1 () 1 1 V x l i l l i l i k k l ik l x x k k i dz t z f o dz t z f z o x x x z τ τ τ τ τ τ π τ π + + − + ({0}, )[(1 * ) () ()( )] ()( )] 1({ 1}, ) 1 1 t o o Q V t Q l i l l l ik l l i π τ π τ τ τ∑ ∫
< ≠ − + − k i k x x k i l i l l ik l x dz t z f o ( )] ( , ) ) 1 [( * () () 1 1 0 τ τ π τ π τ dz t z f z o x x x z l i l i k k l ik l x x i k ) , ( ) ; ( ) 1 ( * () () 1 1 + − + − − +∫
− − τ τ τ π τ π τ τ + + − + +∫
1 ) }, ({ ) ( )] ( ) 1 )[( }, 0 ({ ) , ( ) ( * () () () 1 1 1 1 1 ) ( V x l i l l l ik l l i l i l i t V Q o o t Q dz t z f o τ τ τ τ π τ π τ , ), ( ) }, ({ )] ( ) 1 [( ) }, 0 ({ ) , ( ) ( ) , ( ) ; ( ) 1 ( ) , ( )] ( ) 1 [( ) ( ) }, ({ )] ( ) 1 )( 1 [( ) }, 0 ({ ) , ( ) ( ) , ( )] ( ) 1 )( 1 [( ) , ; } 0 { , ( ) }, ({ ) , ; } 0 { , 0 ( ) }, 0 ({ ) , ( ) , ; } 0 { , ( ) , ; } 0 { , ( ) }, ({ ) , ; } 0 { , 0 ( ) }, 0 ({ ) , ( ) , ; } 0 { , ( ) ( 1 0 ) ( * 0 ) ( 0 0 ) ( 0 ) ( * 0 ) ( 0 0 ) ( * 0 ) ( ) ( 1 1 ) ( * 1 ) ( 1 1 ) ( 0 1 ) ( * 1 ) ( 1 01 1 0 01 0 0 0 01 1 11 1 1 11 1 0 1 11 1 0 , 1 1 1 1 τ τ τ π τ π τ τ τ τ π τ π τ τ π τ π τ τ τ π τ π τ τ τ π τ π τ τ τ τ τ τ τ τ τ τ τ τ l l k l l l k l k V d l k l d x l k l k k l l k x l k l l l k l l k l l l k l k V x l k l x l k l l l k l kk l k l kk l k V l k l kk l kk l k l kk l k V l k l kk l k o t V Q o t Q dz t z f o dz t z f z o d x x z dz t z f o o t V Q o t Q dz t z f o dz t z f o t V q t V Q t q t Q dz t z f t z q t V q t V Q t q t Q dz t z f t z q B k k k k + + − + + + + + − − + + − + + + − − + + + − − = + + + + + = =∫
∫
∫
∫
∫
∫
∫
− − − −. ) ( ) }, ({ ) ( ) 1 ( ) }, 0 ({ ) , ( ) ( ) , ( ) 1 ( ) , ; } 0 { , ( ) }, ({ ) , ; } 0 { , 0 ( ) }, 0 ({ ) , ( ) , ; } 0 { , ( ) ( 1 1 ) ( ) ( * 1 1 0 0 1 ) ( 0 1 ) ( * 1 0 1 11 1 1 11 1 0 1 11 1 , 1 1 + + − − + + − + − = + + = =
∑
∫
∫
∑ ∫
≥ ≠ − ≥ ≠ + τ τ τ π τ π τ τ π τ π τ τ τ τ l l i l i k k l ik l l i x k i V l i l x l i i k k l ik l x k i l ik l i l ik l i V l i l ik l k o t V Q o x x x t Q dz t z f o dz t z f x x x z t V q t V Q t q t Q dz t z f t z q B i k iLet us now move Q1l({0},t)
k to the left-hand side of equation (15). By dividing both sides of the resulting equation by τ and going to the limit for τ → 0, we obtain
. ) }, 0 ({ ) }, 0 ({ ) , 0 ( ) }, 0 ({ ) ( ) }), 0 ({ ) }, 0 ({ 0 ) ( 1 0 ) ( 1 1 1 * 1 ) ( 0 * 0 1
∑
∑
≥ ≠ < ≠ − + + − + − = i i x k i l ik l i i k k x k i l ik l i l k k l k l l k l k l l k t Q x x x t Q t f x t Q t Q t t Q π π π π π ∂ ∂ (16) As , 0 for 0 ) }, 0 ({ 1 = > i l i t x Qequation (16) can be written in a simpler form. For xk = 0 equation (16) is derived in a similar way. The counterpart of equation (16) for Q0l({0},t)
k is derived analogously. Thus, the probabilities Qul({0},t)
k satisfy the following system of differential equations:
. , ..., , 2 , 1 , ) }, 0 ({ ) , 0 ( ) }, 0 ({ ) ( ) }), 0 ({ ) }, 0 ({ , , 0 , 0 ) }), 0 ({ , , 0 , ) }, 0 ({ ) , 0 ( ) }, 0 ({ ) ( ) }), 0 ({ ) }, 0 ({ ) ( 0 0 0 * 0 ) ( 1 * 1 0 1 0 ) ( 1 1 1 * 1 ) ( 0 * 0 1 l k i l ik l i l k l k l l k l k l l k l k l k l k xk i l ik l i l k k l k l l k l k l l k T t n k t Q t af t Q t Q t t Q T t x t Q T t x t Q t f x t Q t Q t t Q i ∈ = + + + − = ∈ > = ∈ ≤ + − + − =
∑
∑
≠ ≤ ≠ π π π π ∂ ∂ π π π π ∂ ∂ (17)The obtained relations (17) will be used in the author’s subsequent work for the quantitative identification of a barrier in the functioning of the system under consid-eration.
References
[1] GALANC T., Intermediate states of a process in the problem of the identification of a barrier in the functioning of a certain inventory storage and issue system, Systems Science, 1998, No. 2.
[2] ŚWIĄTEK J., GALANC T., Process density functions in the problem of the identification of a barrier in
the functioning of a certain inventory storage and issue system, Badania Operacyjne i Decyzje, 2004,
nr. 3–4.
[3] ŚWIĄTEK J., GALANC T., Lower-limit barrier in the problem of the identification of a barrier in the func-tioning of a certain inventory storage and issue system, Badania Operacyjne i Decyzje, 2005, nr 2.
Prawdopodobieństwa bariery dolnej procesu
w zagadnieniu identyfikacji bariery funkcjonowania pewnego systemu gromadzenia i wydawania zapasów
Obiektem badania jest bariera działania pewnego systemu gospodarki zapasami. Przyjmując, że wej-ście magazynu–zbiornika jest procesem niezagregowanym o dynamicznych parametrach, wyprowadzono układ równań różniczkowych, który spełniają prawdopodobieństwa bariery dolnej podsystemu L. Układ ten wyraża powiązania między rozkładami prawdopodobieństwa bariery dolnej a parametrami procesu podaży produktu oraz parametrami funkcjonowania podsystemu transportowego.
