Lower-limit barrier in the problem of teh identification of a barrier in the functioning of a certain inventory storage and issue system

Nr 2

Jerzy ŚWIĄTEK* Tadeusz GALANC**

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 certain inventory system whose input is a non-aggregated dynamic-parameter process. The authors derive equations that define the distribution of conditional probabilities for the case of a lower-limit barier in subsystem L. They depend on the parameters of the functioning of transport subsystem and the parameters of the process of product supply to finite-volume storage. Keywords: system, barrier, inventory

1. Introduction

Paper [1] defined the notion of a barrier in the functioning of an inventory storage and issue system with non-aggregated dynamic-parameter input and presented the general operating principles of such a system and the conditional distributions for intermediate states of the subsystem L. These distributions were used in paper [2] to derive relations satisfied by the density functions for intermediate states.

Continuing the investigations reported in [1], [2], this paper contains derivation of the analytical forms of the conditional probabilities in the case of a lower-limit barrier in a process controlled by a dynamic-parameter input.

* Institute of Technical Computer Science, Wrocław University of Technology, ul. Janiszewskiego 11/17, 50-370 Wrocław, Poland, Jerzy.Swiatek@pwr.wroc.pl

** Institute of Industrial Engineering and Management, Technical University of Wrocław, ul. Smolu-chowskiego 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

The lower-limit barrier of the subsystem L at time t ∈ Tl is defined by the random event z(t) = 0. The state can be characterized by probabilities of the form:

. 0 , 1 , ), ) ( , ) ( , 0 ) ( ( ) }, 0 ({ t =P z t = x t =x v t =u t∈T u= Qul _{k l} k (1)

These probabilities satisfy the following relations: for u = 0,

∑

∫

+

∫

= = = + = + = + = + i V l k l kk V l i l ik k l k t dz Q t z q t dz Q t z q t v x t x t z P t Q ), , ( ) , }; 0 { , ( ) , ( ) , }; 0 { , ( ) 0 ) ( , ) ( , 0 ) ( ( ) }, 0 ({ 1 1 0 1 10 0 0 00 0 τ τ τ τ τ τ (2) where , , ], , 0 [ ) ( , ); ( ) 0 ) ( , ) ( 0 ], , [ for 0 ) ( ), , ( for jump one has ) ( ( ) , }; 0 { , ( 1 ) ( 1 00 l l l i l ik T t T t V t z z k i o t v x t x d z t t s s v t t t t x P t z q ∈ + ∈ ∈ = ≠ + = = ≤ − + ∈ = + ∈ + = τ τ τ τ τ θ τ (3) , , ], , 0 [ ) ( ); ( ) 0 ) ( , ) ( 0 ], , [ for 0 ) ( ], , [ , ) ( ( ) , }; 0 { , ( 1 ) ( 1 00 l l l k k l kk T t T t V t z z o t v x t x d z t t s s v t t s x s x P t z q ∈ + ∈ ∈ = + = = ≤ − + ∈ = + ∈ = = τ τ τ τ τ τ (4) , , ], , 0 [ ) ( ); ( ) 1 ) ( , ) ( 0 ) ( ) ( ), , ( for jump one has ) ( ], , [ for , ) ( ( ) , }; 0 { , ( 1 ) ( 1 10 l l l k k k l kk T t T t V t z z o t v x t x d x z h t t t t v t t s x s x P t z q ∈ + ∈ ∈ = + = = ≤ − − + + ∈ + + ∈ = = τ τ θ τ θ τ θ τ τ (5) for u = 1,

∑

∫

+

∫

= = = + = + = + = + i V l k l kk V l i l ik k l k t dz Q t z q t dz Q t z q t v x t x t z P t Q ), , ( ) , }; 0 { , ( ) , ( ) , }; 0 { , ( ) 1 ) ( , ) ( , 0 ) ( ( ) }, 0 ({ 1 1 0 0 01 0 1 11 1 τ τ τ τ τ τ (6) where , , ], , 0 [ ) ( , ); ( ) 1 ) ( , ) ( 0 ) ( ) ( ], , [ for 1 ) ( ), , ( for jump one has ) ( ( ) , }; 0 { , ( 1 ) ( 1 11 l l l i k i l ik T t T t V t z z k i o t v x t x x x z h t t s s v t t t t x P t z q ∈ + ∈ ∈ = ≠ + = = ≤ − − + + ∈ = + ∈ + = τ τ θ τ θ τ τ θ τ (7)

, , ], , 0 [ ) ( ); ( ) 1 ) ( , ) ( 0 ], , [ for 1 ) ( , ) ( ( ) , }; 0 { , ( 1 ) ( 1 11 l l l k k k l kk T t T t V t z z o t v x t x x z t t s s v x s x P t z q ∈ + ∈ ∈ = + = = ≤ + + ∈ = = = τ τ τ τ τ (8) . , ], , 0 [ ) ( ); ( ) 0 ) ( , ) ( 0 ) ( ) ( ), , ( for jump one has ) ( ], , [ for ) ( ( ) , }; 0 { , ( 1 ) ( 1 01 l l l k k k l kk T t T t V t z z o t v x t x x d z h t t t t v t t s x s x P t z q ∈ + ∈ ∈ = + = = ≤ − + − + ∈ + + ∈ = = τ τ θ τ θ τ θ τ τ (9)

In order to find the probabilities _Q1l({0};_t+τ)

k and Qk0l({0};t+τ), we have to find the conditional probabilities _q11l(_z,{0}; ,_t)

ik τ , q11kkl(z,{0};τ,t), qkk01l(z,{0};τ,t),

{ }

(

)

q

_ik00l

z

_,

₀

_{; ,}

_τ

t

_{, q}10l_{(z,{0}; ,t)}

kk τ , qkk00l(z,{0};τ,t). Taking into account relations (3) and (6) from [1], we obtain

), ( )) exp( 1 ( ) exp( ) , }; 0 { , ( () 1 ) ( ) ( ) ( * 1 11 _{π θ τ} π π τ π τ l l i B l i l ik l l ik z t d o q = −

∫

− − + (10) where . , ], , 0 [ ) ( , }, 0 ) ( ) ( ; 0 : { 1 l l k i T t T t V t z z k i x x z h B ∈ + ∈ ∈ = ≠ ≤ − + + < < = τ θ τ θ τ θ θ (11) If x_i >0, x_k <0,τ small,0<θ<τ,t∈T_l,t+τ∈T_l, then = < + ∩ ≤ − + + = ≤ − + + ) ( ) 0} { : ( ) 0} { : } ( : {θ h z θx_i τ θ x_k θ z θx_i τ θ x_k θ z θx_i V₁     ∅ − <       _< − ∩       − − − ≤ ≤ = . other , , , : 0 : 1 z x z x z V x x x z k i k i k _{θθ τ} τ θ θ Thus     ∅ − < ≤       − + ≤ < = . other , , 0 , 0 : z x z x x x z B k i k k _τ τ θ θ

Hence, from (10), by integration, we get

    _{+ ≤ <−} − + − = − . 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 (12)

Similarly it can be shown that the conditional probabilities are given by the fol-lowing formulas:

for x_k >0, x_i −any state,

, 0 ), ( ) , }; 0 { , ( () ₁ 1 11 _{z t o z V} q l l ik τ = τ ≤ ≤ (13) for any x_k,x_i,    − + ≤ < = − , other , 0 , 0 ), ( ) 1 ( ) ( ) , }; 0 { , ( () 0* () () 1 00 z d z o o t z q l l l ikl l ik τ τ τ π τ π τ τ (14)    − − + ≤ < = − , other , 0 , 0 ), ( ) 1 )( 1 ( ) ( ) , }; 0 { , ( () 0* () () 1 00 z d z o o t z q l l l kl l kk τ τ τ π τ π τ τ (15) for x_i<0, x_k =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 (16) for x_i<0,x_k<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 (17) for x_k ≤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 (18) for x_k >0, , 0 , 0 ) ( ) , }; 0 { , ( () ₁ 1 01 _{z t o z V} q l l kk τ − τ = ≤ ≤ (19)

for x_k =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 π τ π τ τ τ (20) for x_k <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 (21) for x_k <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 (22) for x_k >0,     _{+ ≤ <} + − − = − , 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 (23) for x_k =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 (24)

for x_k >0, . 0 , 0 ) ( ) , }; 0 { , ( () ₁ 1 11 _{z t o z V} q l l kk τ − τ = ≤ ≤ (25) The components ( ;...), ()( ), 1 ) (l _{τ o}l _τ

o which appear in equations (12)–(25) are in the vicinity of τ = 0 infinitely small quantities of an order higher than τ. The relations (12)–(25) will be used in the authors’ next paper to derive relations satisfied by the probabilities Qul({0};t+τ)

k . These will be used for quantitative identification of a barrier in the functioning of the system under consideration.

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, nr 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.

Bariera dolna procesu w zagadnieniu identyfikacji i bariery funkcjonowania pewnego systemu gromadzenia i wydawania zapasów

Badana jest bariera działania pewnego systemu gospodarki zapasami, którego wejście jest procesem niezagregowanym o dynamicznych parametrach. Wyprowadzono wzory wyrażające warunkowe rozkłady prawdopodobieństwa w przypadku bariery dolnej podsystemu L. Zależą one od parametrów funkcjono-wania podsystemu transportowego oraz parametrów procesu podaży produktu do magazynu o skończonej objętości.