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Estimating the Compound Poisson Demand’s Parameters for Single Period Problem for Large Lot SizeÕ
Proceedigs of the 15th IFAC Symposium on Proceedigs of the 15th IFAC Symposium on Information of Control Problems in Manufacturing Proceedigs the 15th IFAC Symposium on Available online at www.sciencedirect.com Proceedigs of the 15th IFAC Symposium on Information Control Problems in Manufacturing May 11-13, 2015. Canada Information Control Problems in Manufacturing Proceedigs of theOttawa, 15th IFAC Symposium on Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada
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IFAC-PapersOnLine 48-3 (2015) 1357–1361
Estimating the Compound Poisson Demand’s Parameters for Single Period Estimating the Compound Poisson Demand’s Parameters for Single Period Estimating the Compound Poisson Demand’s Parameters for Single Period Estimating the Compound Poisson Demand’s Parameters for Single Period Problem for Large Lot Size* Estimating the Compound Poisson Demand’s Parameters for Single Period Problem for Large Lot Size* Problem for Large Lot Size* Problem for Large Lot Size* Problem for**, Valentina Large Lot Size***, I. Subbotina Anna V. Kitaeva
I. Subbotina Anna V. Kitaeva Kitaeva**,, Valentina *, ** *, Valentina I. Anna Natalia V. Stepanova ** Valentina I. Subbotina Subbotina Anna V. V. Kitaeva *,V. *, Natalia Stepanova ** ,V. Valentina I. Subbotina , Anna V. Kitaeva ** Natalia Stepanova Natalia V. Stepanova** Natalia V. Stepanova * *National Research Tomsk State University, Tomsk, Russia, (e-mail: [email protected]) State University, Tomsk, Russia, (e-mail: [email protected]) *National Research ** Tomsk *National Research Tomsk State Tomsk, Russia, (e-mail: [email protected]) Economics and Law Institute, Barnaul, Russia **Altai Tomsk State University, University, Tomsk, Russia, (e-mail: [email protected]) *National Research Economics and Law Institute, Barnaul, Russia **Altai National Research Tomsk State University, Tomsk, Russia, (e-mail: [email protected]) **Altai Economics and Law Institute, Barnaul, Russia **Altai Economics and Law Institute, Barnaul, Russia Altai Economics and Law Institute, Barnaul, Russia Abstract. We consider aa single period problem when the demand is aa compound Poisson process with Abstract. We consider single period problem when the demand is compound Poisson process with Abstract. We consider a single period problem when the demand is a compound Poisson process with unknown intensity and continuous batch size distribution; and the lot size and the number of customers Abstract. intensity We consider a single period problem when the and demand is asize compound Poisson of process with unknown and continuous batch size distribution; the lot and the number customers Abstract. We consider a single period problem when the deviation demand isofasize compound Poisson process with unknown intensity and continuous batch size distribution; and the lot and the number of customers are large enough. The estimators of the mean and standard demand for unobserved lost sales unknown intensityThe andestimators continuous batch size and distribution; and the lotofsize and the number of customers are large enough. of the mean standard deviation demand for unobserved lost sales unknown intensity and continuous batch size distribution; and the lot size and the number of customers are large enough. The estimators of the mean and standard deviation of demand for unobserved lost sales proposed. TheThe procedures areof based on the observed selling period’s durations and the demands, demands, and are large enough. estimators the mean and standard deviation of demand for and unobserved lost sales are proposed. The procedures are based on the observed selling period’s durations the and large enough. The estimators of the mean and standard deviation of demand for and unobserved lost sales are proposed. The procedures are based on the observed selling period’s durations the demands, and use their marginal asymptotic distributions. Numerical results are given. are proposed. The procedures are based on the observed selling period’s durations and the demands, and use their marginal results are given. durations and the demands, and are proposed. The asymptotic procedures distributions. are based on Numerical the observed selling period’s use their marginal asymptotic distributions. Numerical results are use theirIFAC marginal asymptotic distributions. Numerical results are given. given. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, © 2015, (International Federation of Automatic Control) Hosting by Elsevier Ltd. All Selling rights reserved. use their marginal asymptotic distributions. Numerical results are given. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Duration, Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, Parameters’ Estimation. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, Parameters’ Estimation. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, Parameters’ Estimation. Parameters’ Estimation. Parameters’ Estimation. distribution to aa standard normal random variable N (0, 1) as 1. ASYMPTOTIC DISTRIBUTIONS distribution to standard normal random variable N (0, 1) as 1. ASYMPTOTIC DISTRIBUTIONS distribution to a standard normal random variable N (0, 1) λ T → ∞ . 1. ASYMPTOTIC DISTRIBUTIONS distribution to a standard normal random variable N (0, 1) as as λ T → ∞ . 1. ASYMPTOTIC DISTRIBUTIONS distribution to a standard normal random variable N (0, 1) as Consider aa 1.single-product chain consisting of aa λT →∞ ∞ .. ASYMPTOTICsupply DISTRIBUTIONS λ T → Consider single-product supply chain consisting of Let us find asymptotic distribution of the length u ( ⋅ ) Consider a single-product supply chain consisting of a λ T → ∞ . supplier, and customers. The duration Consider aaa buyer, single-product supply chain consistingof ofthe a Let us find asymptotic distribution u (⋅) of the length of of supplier, buyer, and customers. The duration of the Let us us find asymptotic distribution of the length of u ((⋅⋅)) the Consider a single-product supply chain consisting of a time supplier, a buyer, and customers. The duration of the product’s lifetime is T. At the beginning of a time period T τ it takes to sell the lot size for model under Let find asymptotic distribution of the length of u supplier, a buyer, and customers. The duration of the product’s lifetime T. At the beginning of aa time period T time ττ it takes to sell the lot size for model under Let us find asymptotic distribution of the length of u (⋅) the supplier, apurchases buyer,is and customers. The duration of the product’s lifetime is T. At the beginning of time period T time it takes to sell the lot size for the model under the buyer a quantity Q, and replenishment during consideration. For exponential distribution of batch size the product’s lifetime is T. At the beginning of a time period T time τ it takes to sell the lot size for the model under the buyer purchases a quantity Q, and replenishment during consideration. For exponential distribution of batch size the product’s lifetime is T. At the beginning of a time period T time τsolution it takes toexponential sell the lot size for the model under the buyer purchases aa quantity Q, and during consideration. For period is impossible. This is known as the exact is possible the purchases quantity Q, model and replenishment replenishment during consideration. For exponential distribution distribution of of batch batch size size the the the buyer period is impossible. This is known as the exact solution is possible the buyer purchases a see, quantity Q, model and replenishment during consideration. For exponential distribution of batch size the period is impossible. This model is known as the exact solution is possible newsvendor problem, for example, Arrow, Harris, and the period is impossible. This model Arrow, is known as and the exact solution is possible newsvendor problem, see, for example, Harris, the period is impossible. This model Arrow, is(1998). known as the exact solution is possible −λt −Q a λtQ newsvendor problem, see, for example, Harris, Marshak (1951); Silver, Pyke, Peterson newsvendor problem, see, forand example, Arrow, Harris, and and , tQ Marshak (1951); Silver, Pyke, Peterson (1998). uu ((tt )) = ee−λt −Q a11 II 0 22 λ λ newsvendor problem, see, forand example, Arrow, Harris, and tQ λ Marshak (1951); Silver, Pyke, and Peterson (1998). = λ −λt −Q a1 0 ,, tQ λ Marshak (1951);beSilver, Pyke,by and Petersonarriving (1998). according a −λt −Q a1 I 0 u t e = λ ( ) 2 1 Let the demand generated customers Marshak (1951);beSilver, Pyke,by and Petersonarriving (1998). according u (t ) = λe−λt −Q a1 I 0 2 λaatQ 1 Let the demand generated customers u (t ) = λe I 0 2 a1 ,, Let the demand be generated by customers arriving according and requiring aa Poisson process with unknown intensity λ 1 Let the demand be generated by customers arriving according requiring Poisson process with unknown intensity λ and a1 of the first kind Let the demand be generated by customers arriving according where the modified Bessel function I 0 ( ⋅) is and requiring aaamounts Poisson process with unknown intensity λ of varying size independent of the arrival process. where is the modified Bessel function of the first kind and requiring I ⋅ Poisson process with unknown intensity λ ( ) 0 of varyingwith size independent of the process. where II 0 ( ⋅⋅) is modified Bessel function and requiring aamounts Poisson process unknown intensity λarrival amounts of size independent of the process. where is the the modified function of of the the first first kind kind The amounts required at each arrival (batch sizes) are i.i.d. and zeroth Here Q is aa Bessel lot size. ) 0 ( order. amounts of varying varying size independent of the arrival arrival process. where is the modified Bessel function of the first kind The amounts required at each arrival (batch sizes) are i.i.d. and zeroth order. Here Q is lot size. I ⋅ ( ) 0 amounts of varying size independent of the arrival process. The amounts required at each arrival (batch sizes) are i.i.d. and zeroth order. Here Q is a lot size. continuous random variables with unknown distribution. The amountsrandom requiredvariables at each arrival (batch sizes) are i.i.d. and zeroththe order. Here for Q isany a lotbatch size. size’s distribution let us continuous with unknown distribution. To solve problem The amounts atcustomer each arrival (batch sizes) are and zeroththe order. Here for Q isany a lotbatch size. size’s distribution let us continuous random variables with unknown distribution. To solve problem the Denote X(t) aarequired random demand at [0,t], p (⋅) i.i.d. continuous random variables with unknown distribution. To solve the problem for any batch size’s distribution distribution let us us the Denote X(t) random customer demand at [0,t], p ( ⋅ ) consider following diffusion approximation of demand continuous random variables with unknown distribution. To solve the problem for any batch size’s let the Denote X(t) aa random customer demand at [0,t], pthe ((⋅⋅)) first consider following approximation of demand To solveX(t) the problem diffusion for any batch size’s distribution let us a probability density function of X(T) = X, a the Denote X(t) random customer demand at [0,t], consider following diffusion approximation of demand 1 and 2p process probability function of X(T) = X, aaat1 and following diffusion approximation of demand the consider Denote X(t)density a random customer demand [0,t],aa2 pthe (⋅) first process X(t) probability density of X(T) = consider following diffusion approximation of demand and second moments of batch respectively. process X(t) X(t) and a22 the the first first process probability density function function ofsizes X(T)distribution = X, X, a11 and and second moments of batch sizes distribution respectively. a2 the first process X(t) probability density function ofsizes X(T)distribution = X, a1 and and second moments of batch respectively. dX ((tt )) = aa1λ dt + aa2 λ dw ((tt ), andgeneral second moments of batchof sizes distribution respectively. In the distribution X is complicated and it is dX dt + dw andgeneral second moments of batchof sizes distribution respectively. In the distribution X is complicated and it is dX ((tt )) = = aa11λ λ dt + aa22 λ λ dw ((tt ), ), dX = λ dt + λ dw ), In general the distribution of X is complicated and it is difficult to solve the newsvendor problem in closed form. 1 2 In general the distribution of X is complicated and it is dX ( t ) = a λ dt + a λ dw ( t ), difficult to solve the newsvendor problem in closed form. is the Wiener process. where w ( ⋅ ) 1 2 In general the distribution of Xproblem is complicated and it is where w(⋅) is the Wiener process. difficult to solve the newsvendor in closed form. difficult to solve the newsvendor problem in closed form. is the Wiener process. where w ( ⋅ ) For example, for exponential batch sizes’ distribution where w(⋅) is the Wiener process. difficult to solve newsvendor problem in closed form. For example, for the exponential batch sizes’ distribution process. where Diffusion have been applied to inventory models in w(⋅methods ) is the Wiener For Diffusion methods have been applied to inventory models in For example, example, for for exponential exponential batch batch sizes’ sizes’ distribution distribution Diffusion methods have been applied to inventory models in a variety of domains to begin with the papers by Bather For example, for exponential batch sizes’ distribution Diffusion methods have been applied to inventory models in T Tx λ a variety of domains to begin with the papers by Bather − x a1 −λT λ −λ ( c )T 2 λTx , T Diffusion methods have been applied to inventory models in pp (( xx)) = x e I δ ( ) a variety of domains to begin with the papers by Bather −λ ( c )T + e − x a1 −λT λ (1966) and Puterman (1975). In Kitaeva (2014) the diffusion 1 T Tx λ λ a variety of domains to begin with the papers by Bather x e I = δ ( ) 2 , −λ ( c )T + e − x a1 −λT a (1966) and Puterman (1975). In Kitaeva (2014) the diffusion T Tx λ λ 1 x a −λ ( c )T + e − x a1 −λT a1 x I1 pp (( xx)) = x e δ ( ) 2 , a variety of domains to begin with the papers by Bather 1 (1966) and Puterman (1975). In Kitaeva (2014) the diffusion approximation of stock level process has been used to find λT I 2 λaTx , = δ( x ) e +e (1966) and Puterman (1975). In Kitaeva (2014) the diffusion approximation of stock level process has been used to find p ( x) = δ( x)e −λ (c )T + e − x a1 −λT aa111 xx I11 2 aa111 , (1966) and Puterman (1975). In Kitaeva (2014) the diffusion approximation of stock level process has been used to find the steady-state distribution and to solve the problem of approximation of stock level process has been used to find a x a the steady-state distribution and to solve the problem of 1 1 where is the modified Bessel function of the first kind I ( ⋅ ) approximation of stock leveltheand process hasofbeen used to find 1 (⋅) is the modified Bessel function of the first kind the steady-state distribution to solve the problem of where I on/off control minimizing variance the process. In 1 the steady-state distribution and to solve the problem of where is the modified Bessel function of the first kind I ( ⋅ ) on/off control minimizing the variance of the process. In 1 the steady-state distribution and to solve the problem of where modified Bessel function of the first kind on/off I1order, (⋅) is the is the Dirac delta function. and first δ ( ⋅ ) control minimizing the variance of the process. In Kitaeva, Subbotina, and Zmeev (2014) theoretical on/off control minimizing the variance of the process. In is the Dirac delta function. and first order, where is the modified Bessel function of the first kind δ ( ⋅ ) I ( ⋅ ) Kitaeva, Subbotina, and (2014) theoretical 1 and on/off control minimizing the Zmeev variance of the with process. In Kitaeva, Subbotina, and Zmeev (2014) theoretical justification of a diffusion approximation some is the the Dirac Dirac delta delta function. function. and first first order, order, δδ((⋅⋅)) is Kitaeva, Subbotina, and Zmeev (2014) with theoretical of aa diffusion approximation some the Dirac function. order quantity justification and first (⋅) isthis Let case delta the optimal λT order, >> 1 . δIn Kitaeva, Subbotina, and Zmeev (2014) with theoretical justification of diffusion approximation some numerical results is given. Let λT >> 1 . In this case the optimal order quantity justification of a diffusion approximation with some numerical results is given. Let . In this case the optimal order quantity λ T >> 1 minimizing also large. of isa given. diffusion approximation with some Let λT >>the In this profit case is the optimal order quantity justification 1 . expected numerical results minimizing the expected profit is also large. results is given. Let λT >>the In this profit case isthe optimal order quantity numerical 1 . expected minimizing also large. Denote τ(х) the first occurrence time of the crossing of level numerical results is given. minimizing the expected profit is X also large. Denote τ(х) the first occurrence time of the crossing of level − a T minimizing the expected profit is X also large. −of s τ ( xlevel ) Denote τ(х) the first occurrence time of the crossing 1λ − a λ T X ⋅ given X(0) = х; Q by Denote τ(х) the first occurrence time of the crossing of level g ( s , x ) = E e According central limit theorem converges in ( ) − s τ ( x )} = 1λT { X − a X ⋅ given X(0) = х; Q by g ( s , x ) = E e According central limit theorem converges in ( ) −of s τ ( xlevel )} = Denote τ(х) the first occurrence time of the crossing 1λ { X − a T a λ T X ⋅ − sτ ( x ) } = given X(0) = х; Q by g ( s , x ) = E e According central limit theorem converges in 1 ( ) { 2 aλ1λ T converges in ∞ by X ⋅ T given X(0) = х; Q g ( s , x ) = E e According central limit theorem X −a ( ) − sτ ( x ) } = { 2 a λT X ⋅ given X(0) = х; g ( s, x) = E {e Q ∞∞ by According central limit theorem }= a22 λ T converges in ________________________________ = ee −− sysy uu (( yy //(xx)))dy is the Laplace transform of conditional ∞ a2 λT ________________________________ ∫ = dy − sy is the Laplace transform of conditional ∞ ________________________________ ∫ = ________________________________ * This work is performed under the state order No. 1.511.2014/K of the = ∫∫00 ee −− sysy uu (( yy // xx))dy dy is is the the Laplace Laplace transform transform of of conditional conditional ________________________________ * This work is performed under the state order No. 1.511.2014/K of the ( / ) = e u y x dy is the Laplace transform of conditional 0 ∫ *Ministry This work is performed theofstate No.Federation 1.511.2014/K of the of Education andunder Science the order Russian
* This work is performed theofstate No.Federation 1.511.2014/K of the of Education andunder Science the order Russian *Ministry This work is performed theofstate No.Federation 1.511.2014/K of the Ministry of Education andunder Science the order Russian Ministry of Education and Science of the Russian Federation
0 0
Ministry of Education and Science of the Russian Federation
2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2015 IFAC 1423 Peer review©under of International Federation of Automatic Copyright 2015 responsibility IFAC 1423Control. Copyright © 2015 IFAC 1423 10.1016/j.ifacol.2015.06.275 Copyright © 2015 IFAC 1423 Copyright © 2015 IFAC 1423
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density u (⋅ / x) . Denote ∆x = X ( t + ∆t ) − X ( t ) , and E∆x {} ⋅ the expectation with respect to random value ∆x . Consider
2.1 Estimation by leftovers periods
g ( s, x) = E {e − s ( ∆t +τ ( x +∆x )) } = e − s∆t E∆x {g ( s, x + ∆x)} = g ( s, x) +
Let x1, x2, …, xm, ∀i xi ≤ Q be the observed demand in m periods.
a λ ∂ g ( s, x ) ∂g ( s, x) a1λ + 2 ∆t + o ( ∆t ) . 2 ∂x ∂x 2 2
+ − sg ( s, x) +
Suppose we have observed n sale periods, and at the end of m of them we have had leftovers and in n − m cases we have had lost sales.
We consider the demand as a normal random variable with E { x} = µ x = a1λT and the variance the mean
It follows as ∆t → 0 a ∂g ( s, x) 2 s ∂ g ( s, x ) +2 1 − g ( s, x) = 0; g ( s, Q) = 1. (1) a2 ∂x a2 λ ∂x 2 2
Denote the value of interest τ(0) = τ , and g ( s, 0) = g τ ( s ) . From (1) we get a a 2 2s g τ ( s ) = exp 1 Q − 12 + Q . a2 a2 a2 λ
Var { x} = σ2x = a2 λT . It follows Q − µx P{ X > Q} = 1 − Φ , σx where Φ ( x) =
1
−t 2
x
∫ exp 2 dt . 2p −∞
To estimate the probability P{ X > Q} we use the ratio
The inverse Laplace transform of g τ ( s ) u (t ) =
Consider the case
Q 2pa2 λt 3 2
n−m = 1− h , n
2 a2λ Q . exp − 1 t − 2a2 t a1λ
where h = m n . So we receive the first equation for the estimates µˆ x and σˆ x
E{τ}
a2 = q 1 >> 1 , Q a1 = q . a2 Var{τ}
µˆ x + σˆ x Ψ (h) = Q ,
2 a 3λ 2 a13λ 2 Q 1 exp − Then sup u (t ) − t − → 0 as 2a2 Q a1λ 2pa2Q t λa q → ∞ and we can consider ( τ − q / λ ) 1 as N (0, 1). a2 q
where Ψ ( ⋅) = Φ −1 (⋅) . By equating the theoretical and empirical conditional means, we obtain the second equation 2 1 exp − ( Q − µˆ x ) / σˆ x 2 =x, µˆ x − σˆ x 2pΦ ( Q − µˆ x ) / σˆ x
2. PARAMETERS’ ESTIMATION There are different approaches to handling of demand uncertainty. In Rossia et al. (2014) overview of parametric approaches to the newsvendor problem is provided. Note that most of the papers have focused on the case when the demand is fully observed. The censored demand case induced by lost sales is considered by Hill (1992), Nahmias (1994), Lau and Lau (1996); and Agrawal and Smith (1996). We estimate parameters of interest a1λT and a2 λT given two samples: the sale period’s duration, i.e. the time at which the vendor runs short, t1, t2, …, ∀i ti ≤ T , if there are lost sales; and the sizes of sales during time period T x1, x2, …, ∀i xi ≤ Q , if there are leftovers at the end of the period. We assume that the customer satisfy the demand as far as possible when the amount required is more than the remainder; and the lot sizes are the same for each period. We base the estimating procedure on the distributions obtained above.
where x =
1 m ∑ xi . m i =1
So we get following system
µˆ x − σˆ x F (h) = x , µˆ x + σˆ x Ψ (h) = Q,
1 exp − Ψ 2 (h) 2 . where F (h) = 2ph The estimates σˆ x =
µˆ x = 1424
Q−x , Ψ ( h) + F ( h)
x Ψ (h) + QF (h) . Ψ ( h) + F ( h)
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2.2 Estimation by lost sales periods
νˆ =
To our knowledge, no research exists using the observed sale period’s durations to the parameters estimation in the newsvendor problem framework.
τΨ (h) + F (h) 1− τ , sˆ = , Ψ ( h) + F ( h) Ψ ( h) + F ( h)
and
µˆ (xt ) = Q
Let t1, t2, …, tn-m, ∀i ti ≤ T be the observed sale period’s durations in n − m periods.
Let us introduce the dimensionless quantities ν = µt T and s = σt T .
Ψ ( h) + F ( h)
2
( τΨ (h) + F (h) )3
,
where µˆ (xt ) and σˆ (xt ) are the estimates of a1λT and a2 λT derived from the sample t1, t2, …, tn-m. Obtaining joint distribution (Ti , X i ) is one of the directions of future research.
We have
a1λT =
3. SIMULATION
Q Q2 s2 , a2 λT = 3 , ν ν
In this section, we simulate the system for three setups:
and
T − µt P{t < T } = Φ σt
1− ν = Φ . s
1. Batch size has uniform distribution at [2, 8]. It follows that a1 = 5 and a2 = 28. The intensity λ equals 1. 2. Batch size has exponential distribution with a1 = 4. It follows that a2 = 32. The intensity λ equals 1.
m = h. n
3. Batch size has beta distribution with shape parameters α = 0.429 and β = 0.286. It follows that a1 = 0.6 and a2 = 0.5. The intensity λ equals 4.
We estimate the probability P{t < T } by
So we get the first equation for estimates νˆ and sˆ
1 − νˆ Φ =h, sˆ or νˆ + sˆΨ (h) = 1 . Analogously the previous part we receive the second equation 2 1 exp − (T − µˆ t ) / σˆ t 2 =t , µˆ t − σˆ t 2pΦ (T − µˆ t ) / σˆ t
where t =
Ψ ( h) + F ( h) , τΨ (h) + F (h)
σˆ (xt ) = Q 2 (1 − τ )
We consider the sale period’s durations as a normal random variable with the mean E {t} = µt = Q a1λ and the variance
Var {t} = σt2 = a2 λQ (a1λ)3 .
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1 n−m ∑ ti . n − m i =1
We can rewrite (2) as 2 1 exp − (1 − νˆ ) / sˆ 2 = τ, νˆ − sˆ ˆ ˆ 2pΦ (1 − ν ) / s
(2)
We replicate the procedures eleven times for n = 300 and n = 50. Tables 1−3 report the mean absolute percent errors (MAPE) of corresponding estimates in percentages; m n − m (t ) m 2 n − m (t )2 µˆ = µˆ x + µˆ x and σˆ = are σˆ x + σˆ x n n n n weighted estimates. In Figures 1−3 and 4−6 the numerical results for exponential batch size’s distribution are given for n = 300 and n = 50 m is equal respectively. For Q = 540 the ratio n−m approximately to 3, i.e. the probability of leftovers is m ≈ 1 , i.e. the approximately 0.75, for Q = 500 the ratio n−m m 1 ≈ , lot size is close to optimal, for Q = 455 the ratio n−m 3 i.e. the probability of stock-out is approximately 0.75. Table 1. Uniform distribution, a1 λT = 625 ,
a2 λT = 59.16
where τ = t T .
MAPE(∙)% Q
We get the system of equations
µˆ x
µˆ (xt )
µˆ
σˆ x
σˆ (xt )
σˆ
n = 300
νˆ + sˆΨ (h) = 1, ˆ (h) = τ. νˆ − sF The estimates 1425
585
0.70
0.45
0.48
8.70
2.76
2.93
625
0.68
0.70
0.69
7.72
4.85
4.93
665
0.61
1.40
0.80
4.84
9.27
3.37
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n = 50 585
1,93
0,98
1,08
14,93
12,74
10,68
625
1,12
1,27
1,18
10,93
14,89
6,27
665
1,26
1,65
1,30
7,43
13,74
8,36
Table 2. Exponential distribution, a1 λT = 500 ,
a2 λT = 63.25 MAPE(∙)% Q
µˆ x
µˆ (xt )
µˆ
σˆ x
σˆ (xt )
σˆ
n = 300
Fig. 2. Exponential distribution, Q = 500, n = 300.
455
0.92
1.02
0.90
9.72
2.86
3.47
500
0.69
0.64
0.66
6.65
4.92
3.64
540
1.01
1.65
1.05
3.25
13.13
4.31
n = 50 455
2,60
2,10
2,12
16,94
10,88
9,14
500
1,37
1,39
1,36
10,78
11,21
8,05
540
1,50
2,35
1,63
6,57
15,66
5,40
Table 3. Beta distribution, a1 λT = 75, MAPE(∙)% Q
µˆ x
µˆ (xt )
70
1.27
0.68
75
0.89
80
a2 λT = 7.91
σˆ x
σˆ (xt )
σˆ
n = 300 0.79
9.79
4.81
3.78
0.85
0.87
7.65
7.31
6.07
0.45
0.86
0.54
4.75
8.70
4.14
70
1,69
0,73
n = 50 0,89
15,56
8,70
6,30
75
1,74
1,53
1,56
17,85
15,45
12,15
80
1,04
1,81
1,23
11,25
14,63
8,95
µˆ
Fig. 3. Exponential distribution, Q = 455, n = 300.
Fig. 4. Exponential distribution, Q = 540, n = 50.
Fig. 1. Exponential distribution, Q = 540, n = 300.
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of Application. In Dudin A. et al. (eds.), Lecture Notes in Computer Science, 189–196, Springer, Switzerland. Lau, H.-S., and Lau, A.H.-L. (1996). Estimating the demand distributions of single period items having frequent stockouts. European Journal of Operational Research, 92, 254–265. Nahmias, S. (1994). Demand estimation in lost sales inventory systems. Naval Research Logistics, 41, 739– 757. Rossia, R., Prestwichb, S., Tarimc, S.A., and Hnich, B. (2014). Confidence-based optimisation for the newsvendor problem under binomial, Poisson and exponential demand. European Journal of Operational Research, 239, 674–684. Silver, E.A., Pyke, D.F., and Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. Wiley, New York.
Fig. 5. Exponential distribution, Q = 500, n = 50.
Fig. 6. Exponential distribution, Q = 455, n = 50. So we recommend to use weighted estimates µˆ and σˆ . It allows us to increase significantly the accuracy of demand’s standard deviation’s estimation which is comparatively low, and the accuracy of µˆ does not differ significantly from the best one proposed. REFERENCES Agrawal, N., and Smith, S.A. (1996). Estimating negative binomial demand for retail inventory management with unobservable lost sales. Naval Research Logistics, 43, 839–861. Arrow, K.J., Harris, Th.E., and Marschak, J. (1951). Optimal Inventory Policy. Econometrica, 19 (3), 205–272. Hill, R.M. (1992). Parameter estimation and performance measurement in lost sales inventory systems. International Journal of Production Economics, 28, 211–215. Kitaeva, A.V. (2014). Stabilization of Inventory System Performance: On/Off Control. Proceedings of the 19th IFAC World Congress, 10748–10753, Cape Town, South Africa, 24-29 August 2014. Kitaeva, A., Subbotina, V., and Zmeev, O. (2014). Diffusion Appoximation in Inventory Management with Examples 1427
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IFAC-PapersOnLine 48-3 (2015) 1357–1361
Estimating the Compound Poisson Demand’s Parameters for Single Period Estimating the Compound Poisson Demand’s Parameters for Single Period Estimating the Compound Poisson Demand’s Parameters for Single Period Estimating the Compound Poisson Demand’s Parameters for Single Period Problem for Large Lot Size* Estimating the Compound Poisson Demand’s Parameters for Single Period Problem for Large Lot Size* Problem for Large Lot Size* Problem for Large Lot Size* Problem for**, Valentina Large Lot Size***, I. Subbotina Anna V. Kitaeva
I. Subbotina Anna V. Kitaeva Kitaeva**,, Valentina *, ** *, Valentina I. Anna Natalia V. Stepanova ** Valentina I. Subbotina Subbotina Anna V. V. Kitaeva *,V. *, Natalia Stepanova ** ,V. Valentina I. Subbotina , Anna V. Kitaeva ** Natalia Stepanova Natalia V. Stepanova** Natalia V. Stepanova * *National Research Tomsk State University, Tomsk, Russia, (e-mail: [email protected]) State University, Tomsk, Russia, (e-mail: [email protected]) *National Research ** Tomsk *National Research Tomsk State Tomsk, Russia, (e-mail: [email protected]) Economics and Law Institute, Barnaul, Russia **Altai Tomsk State University, University, Tomsk, Russia, (e-mail: [email protected]) *National Research Economics and Law Institute, Barnaul, Russia **Altai National Research Tomsk State University, Tomsk, Russia, (e-mail: [email protected]) **Altai Economics and Law Institute, Barnaul, Russia **Altai Economics and Law Institute, Barnaul, Russia Altai Economics and Law Institute, Barnaul, Russia Abstract. We consider aa single period problem when the demand is aa compound Poisson process with Abstract. We consider single period problem when the demand is compound Poisson process with Abstract. We consider a single period problem when the demand is a compound Poisson process with unknown intensity and continuous batch size distribution; and the lot size and the number of customers Abstract. intensity We consider a single period problem when the and demand is asize compound Poisson of process with unknown and continuous batch size distribution; the lot and the number customers Abstract. We consider a single period problem when the deviation demand isofasize compound Poisson process with unknown intensity and continuous batch size distribution; and the lot and the number of customers are large enough. The estimators of the mean and standard demand for unobserved lost sales unknown intensityThe andestimators continuous batch size and distribution; and the lotofsize and the number of customers are large enough. of the mean standard deviation demand for unobserved lost sales unknown intensity and continuous batch size distribution; and the lot size and the number of customers are large enough. The estimators of the mean and standard deviation of demand for unobserved lost sales proposed. TheThe procedures areof based on the observed selling period’s durations and the demands, demands, and are large enough. estimators the mean and standard deviation of demand for and unobserved lost sales are proposed. The procedures are based on the observed selling period’s durations the and large enough. The estimators of the mean and standard deviation of demand for and unobserved lost sales are proposed. The procedures are based on the observed selling period’s durations the demands, and use their marginal asymptotic distributions. Numerical results are given. are proposed. The procedures are based on the observed selling period’s durations and the demands, and use their marginal results are given. durations and the demands, and are proposed. The asymptotic procedures distributions. are based on Numerical the observed selling period’s use their marginal asymptotic distributions. Numerical results are use theirIFAC marginal asymptotic distributions. Numerical results are given. given. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, © 2015, (International Federation of Automatic Control) Hosting by Elsevier Ltd. All Selling rights reserved. use their marginal asymptotic distributions. Numerical results are given. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Duration, Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, Parameters’ Estimation. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, Parameters’ Estimation. Keywords: Compound Poisson Demand, Newsvendor, Asymptotic Distribution, Selling Duration, Parameters’ Estimation. Parameters’ Estimation. Parameters’ Estimation. distribution to aa standard normal random variable N (0, 1) as 1. ASYMPTOTIC DISTRIBUTIONS distribution to standard normal random variable N (0, 1) as 1. ASYMPTOTIC DISTRIBUTIONS distribution to a standard normal random variable N (0, 1) λ T → ∞ . 1. ASYMPTOTIC DISTRIBUTIONS distribution to a standard normal random variable N (0, 1) as as λ T → ∞ . 1. ASYMPTOTIC DISTRIBUTIONS distribution to a standard normal random variable N (0, 1) as Consider aa 1.single-product chain consisting of aa λT →∞ ∞ .. ASYMPTOTICsupply DISTRIBUTIONS λ T → Consider single-product supply chain consisting of Let us find asymptotic distribution of the length u ( ⋅ ) Consider a single-product supply chain consisting of a λ T → ∞ . supplier, and customers. The duration Consider aaa buyer, single-product supply chain consistingof ofthe a Let us find asymptotic distribution u (⋅) of the length of of supplier, buyer, and customers. The duration of the Let us us find asymptotic distribution of the length of u ((⋅⋅)) the Consider a single-product supply chain consisting of a time supplier, a buyer, and customers. The duration of the product’s lifetime is T. At the beginning of a time period T τ it takes to sell the lot size for model under Let find asymptotic distribution of the length of u supplier, a buyer, and customers. The duration of the product’s lifetime T. At the beginning of aa time period T time ττ it takes to sell the lot size for model under Let us find asymptotic distribution of the length of u (⋅) the supplier, apurchases buyer,is and customers. The duration of the product’s lifetime is T. At the beginning of time period T time it takes to sell the lot size for the model under the buyer a quantity Q, and replenishment during consideration. For exponential distribution of batch size the product’s lifetime is T. At the beginning of a time period T time τ it takes to sell the lot size for the model under the buyer purchases a quantity Q, and replenishment during consideration. For exponential distribution of batch size the product’s lifetime is T. At the beginning of a time period T time τsolution it takes toexponential sell the lot size for the model under the buyer purchases aa quantity Q, and during consideration. For period is impossible. This is known as the exact is possible the purchases quantity Q, model and replenishment replenishment during consideration. For exponential distribution distribution of of batch batch size size the the the buyer period is impossible. This is known as the exact solution is possible the buyer purchases a see, quantity Q, model and replenishment during consideration. For exponential distribution of batch size the period is impossible. This model is known as the exact solution is possible newsvendor problem, for example, Arrow, Harris, and the period is impossible. This model Arrow, is known as and the exact solution is possible newsvendor problem, see, for example, Harris, the period is impossible. This model Arrow, is(1998). known as the exact solution is possible −λt −Q a λtQ newsvendor problem, see, for example, Harris, Marshak (1951); Silver, Pyke, Peterson newsvendor problem, see, forand example, Arrow, Harris, and and , tQ Marshak (1951); Silver, Pyke, Peterson (1998). uu ((tt )) = ee−λt −Q a11 II 0 22 λ λ newsvendor problem, see, forand example, Arrow, Harris, and tQ λ Marshak (1951); Silver, Pyke, and Peterson (1998). = λ −λt −Q a1 0 ,, tQ λ Marshak (1951);beSilver, Pyke,by and Petersonarriving (1998). according a −λt −Q a1 I 0 u t e = λ ( ) 2 1 Let the demand generated customers Marshak (1951);beSilver, Pyke,by and Petersonarriving (1998). according u (t ) = λe−λt −Q a1 I 0 2 λaatQ 1 Let the demand generated customers u (t ) = λe I 0 2 a1 ,, Let the demand be generated by customers arriving according and requiring aa Poisson process with unknown intensity λ 1 Let the demand be generated by customers arriving according requiring Poisson process with unknown intensity λ and a1 of the first kind Let the demand be generated by customers arriving according where the modified Bessel function I 0 ( ⋅) is and requiring aaamounts Poisson process with unknown intensity λ of varying size independent of the arrival process. where is the modified Bessel function of the first kind and requiring I ⋅ Poisson process with unknown intensity λ ( ) 0 of varyingwith size independent of the process. where II 0 ( ⋅⋅) is modified Bessel function and requiring aamounts Poisson process unknown intensity λarrival amounts of size independent of the process. where is the the modified function of of the the first first kind kind The amounts required at each arrival (batch sizes) are i.i.d. and zeroth Here Q is aa Bessel lot size. ) 0 ( order. amounts of varying varying size independent of the arrival arrival process. where is the modified Bessel function of the first kind The amounts required at each arrival (batch sizes) are i.i.d. and zeroth order. Here Q is lot size. I ⋅ ( ) 0 amounts of varying size independent of the arrival process. The amounts required at each arrival (batch sizes) are i.i.d. and zeroth order. Here Q is a lot size. continuous random variables with unknown distribution. The amountsrandom requiredvariables at each arrival (batch sizes) are i.i.d. and zeroththe order. Here for Q isany a lotbatch size. size’s distribution let us continuous with unknown distribution. To solve problem The amounts atcustomer each arrival (batch sizes) are and zeroththe order. Here for Q isany a lotbatch size. size’s distribution let us continuous random variables with unknown distribution. To solve problem the Denote X(t) aarequired random demand at [0,t], p (⋅) i.i.d. continuous random variables with unknown distribution. To solve the problem for any batch size’s distribution distribution let us us the Denote X(t) random customer demand at [0,t], p ( ⋅ ) consider following diffusion approximation of demand continuous random variables with unknown distribution. To solve the problem for any batch size’s let the Denote X(t) aa random customer demand at [0,t], pthe ((⋅⋅)) first consider following approximation of demand To solveX(t) the problem diffusion for any batch size’s distribution let us a probability density function of X(T) = X, a the Denote X(t) random customer demand at [0,t], consider following diffusion approximation of demand 1 and 2p process probability function of X(T) = X, aaat1 and following diffusion approximation of demand the consider Denote X(t)density a random customer demand [0,t],aa2 pthe (⋅) first process X(t) probability density of X(T) = consider following diffusion approximation of demand and second moments of batch respectively. process X(t) X(t) and a22 the the first first process probability density function function ofsizes X(T)distribution = X, X, a11 and and second moments of batch sizes distribution respectively. a2 the first process X(t) probability density function ofsizes X(T)distribution = X, a1 and and second moments of batch respectively. dX ((tt )) = aa1λ dt + aa2 λ dw ((tt ), andgeneral second moments of batchof sizes distribution respectively. In the distribution X is complicated and it is dX dt + dw andgeneral second moments of batchof sizes distribution respectively. In the distribution X is complicated and it is dX ((tt )) = = aa11λ λ dt + aa22 λ λ dw ((tt ), ), dX = λ dt + λ dw ), In general the distribution of X is complicated and it is difficult to solve the newsvendor problem in closed form. 1 2 In general the distribution of X is complicated and it is dX ( t ) = a λ dt + a λ dw ( t ), difficult to solve the newsvendor problem in closed form. is the Wiener process. where w ( ⋅ ) 1 2 In general the distribution of Xproblem is complicated and it is where w(⋅) is the Wiener process. difficult to solve the newsvendor in closed form. difficult to solve the newsvendor problem in closed form. is the Wiener process. where w ( ⋅ ) For example, for exponential batch sizes’ distribution where w(⋅) is the Wiener process. difficult to solve newsvendor problem in closed form. For example, for the exponential batch sizes’ distribution process. where Diffusion have been applied to inventory models in w(⋅methods ) is the Wiener For Diffusion methods have been applied to inventory models in For example, example, for for exponential exponential batch batch sizes’ sizes’ distribution distribution Diffusion methods have been applied to inventory models in a variety of domains to begin with the papers by Bather For example, for exponential batch sizes’ distribution Diffusion methods have been applied to inventory models in T Tx λ a variety of domains to begin with the papers by Bather − x a1 −λT λ −λ ( c )T 2 λTx , T Diffusion methods have been applied to inventory models in pp (( xx)) = x e I δ ( ) a variety of domains to begin with the papers by Bather −λ ( c )T + e − x a1 −λT λ (1966) and Puterman (1975). In Kitaeva (2014) the diffusion 1 T Tx λ λ a variety of domains to begin with the papers by Bather x e I = δ ( ) 2 , −λ ( c )T + e − x a1 −λT a (1966) and Puterman (1975). In Kitaeva (2014) the diffusion T Tx λ λ 1 x a −λ ( c )T + e − x a1 −λT a1 x I1 pp (( xx)) = x e δ ( ) 2 , a variety of domains to begin with the papers by Bather 1 (1966) and Puterman (1975). In Kitaeva (2014) the diffusion approximation of stock level process has been used to find λT I 2 λaTx , = δ( x ) e +e (1966) and Puterman (1975). In Kitaeva (2014) the diffusion approximation of stock level process has been used to find p ( x) = δ( x)e −λ (c )T + e − x a1 −λT aa111 xx I11 2 aa111 , (1966) and Puterman (1975). In Kitaeva (2014) the diffusion approximation of stock level process has been used to find the steady-state distribution and to solve the problem of approximation of stock level process has been used to find a x a the steady-state distribution and to solve the problem of 1 1 where is the modified Bessel function of the first kind I ( ⋅ ) approximation of stock leveltheand process hasofbeen used to find 1 (⋅) is the modified Bessel function of the first kind the steady-state distribution to solve the problem of where I on/off control minimizing variance the process. In 1 the steady-state distribution and to solve the problem of where is the modified Bessel function of the first kind I ( ⋅ ) on/off control minimizing the variance of the process. In 1 the steady-state distribution and to solve the problem of where modified Bessel function of the first kind on/off I1order, (⋅) is the is the Dirac delta function. and first δ ( ⋅ ) control minimizing the variance of the process. In Kitaeva, Subbotina, and Zmeev (2014) theoretical on/off control minimizing the variance of the process. In is the Dirac delta function. and first order, where is the modified Bessel function of the first kind δ ( ⋅ ) I ( ⋅ ) Kitaeva, Subbotina, and (2014) theoretical 1 and on/off control minimizing the Zmeev variance of the with process. In Kitaeva, Subbotina, and Zmeev (2014) theoretical justification of a diffusion approximation some is the the Dirac Dirac delta delta function. function. and first first order, order, δδ((⋅⋅)) is Kitaeva, Subbotina, and Zmeev (2014) with theoretical of aa diffusion approximation some the Dirac function. order quantity justification and first (⋅) isthis Let case delta the optimal λT order, >> 1 . δIn Kitaeva, Subbotina, and Zmeev (2014) with theoretical justification of diffusion approximation some numerical results is given. Let λT >> 1 . In this case the optimal order quantity justification of a diffusion approximation with some numerical results is given. Let . In this case the optimal order quantity λ T >> 1 minimizing also large. of isa given. diffusion approximation with some Let λT >>the In this profit case is the optimal order quantity justification 1 . expected numerical results minimizing the expected profit is also large. results is given. Let λT >>the In this profit case isthe optimal order quantity numerical 1 . expected minimizing also large. Denote τ(х) the first occurrence time of the crossing of level numerical results is given. minimizing the expected profit is X also large. Denote τ(х) the first occurrence time of the crossing of level − a T minimizing the expected profit is X also large. −of s τ ( xlevel ) Denote τ(х) the first occurrence time of the crossing 1λ − a λ T X ⋅ given X(0) = х; Q by Denote τ(х) the first occurrence time of the crossing of level g ( s , x ) = E e According central limit theorem converges in ( ) − s τ ( x )} = 1λT { X − a X ⋅ given X(0) = х; Q by g ( s , x ) = E e According central limit theorem converges in ( ) −of s τ ( xlevel )} = Denote τ(х) the first occurrence time of the crossing 1λ { X − a T a λ T X ⋅ − sτ ( x ) } = given X(0) = х; Q by g ( s , x ) = E e According central limit theorem converges in 1 ( ) { 2 aλ1λ T converges in ∞ by X ⋅ T given X(0) = х; Q g ( s , x ) = E e According central limit theorem X −a ( ) − sτ ( x ) } = { 2 a λT X ⋅ given X(0) = х; g ( s, x) = E {e Q ∞∞ by According central limit theorem }= a22 λ T converges in ________________________________ = ee −− sysy uu (( yy //(xx)))dy is the Laplace transform of conditional ∞ a2 λT ________________________________ ∫ = dy − sy is the Laplace transform of conditional ∞ ________________________________ ∫ = ________________________________ * This work is performed under the state order No. 1.511.2014/K of the = ∫∫00 ee −− sysy uu (( yy // xx))dy dy is is the the Laplace Laplace transform transform of of conditional conditional ________________________________ * This work is performed under the state order No. 1.511.2014/K of the ( / ) = e u y x dy is the Laplace transform of conditional 0 ∫ *Ministry This work is performed theofstate No.Federation 1.511.2014/K of the of Education andunder Science the order Russian
* This work is performed theofstate No.Federation 1.511.2014/K of the of Education andunder Science the order Russian *Ministry This work is performed theofstate No.Federation 1.511.2014/K of the Ministry of Education andunder Science the order Russian Ministry of Education and Science of the Russian Federation
0 0
Ministry of Education and Science of the Russian Federation
2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2015 IFAC 1423 Peer review©under of International Federation of Automatic Copyright 2015 responsibility IFAC 1423Control. Copyright © 2015 IFAC 1423 10.1016/j.ifacol.2015.06.275 Copyright © 2015 IFAC 1423 Copyright © 2015 IFAC 1423
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density u (⋅ / x) . Denote ∆x = X ( t + ∆t ) − X ( t ) , and E∆x {} ⋅ the expectation with respect to random value ∆x . Consider
2.1 Estimation by leftovers periods
g ( s, x) = E {e − s ( ∆t +τ ( x +∆x )) } = e − s∆t E∆x {g ( s, x + ∆x)} = g ( s, x) +
Let x1, x2, …, xm, ∀i xi ≤ Q be the observed demand in m periods.
a λ ∂ g ( s, x ) ∂g ( s, x) a1λ + 2 ∆t + o ( ∆t ) . 2 ∂x ∂x 2 2
+ − sg ( s, x) +
Suppose we have observed n sale periods, and at the end of m of them we have had leftovers and in n − m cases we have had lost sales.
We consider the demand as a normal random variable with E { x} = µ x = a1λT and the variance the mean
It follows as ∆t → 0 a ∂g ( s, x) 2 s ∂ g ( s, x ) +2 1 − g ( s, x) = 0; g ( s, Q) = 1. (1) a2 ∂x a2 λ ∂x 2 2
Denote the value of interest τ(0) = τ , and g ( s, 0) = g τ ( s ) . From (1) we get a a 2 2s g τ ( s ) = exp 1 Q − 12 + Q . a2 a2 a2 λ
Var { x} = σ2x = a2 λT . It follows Q − µx P{ X > Q} = 1 − Φ , σx where Φ ( x) =
1
−t 2
x
∫ exp 2 dt . 2p −∞
To estimate the probability P{ X > Q} we use the ratio
The inverse Laplace transform of g τ ( s ) u (t ) =
Consider the case
Q 2pa2 λt 3 2
n−m = 1− h , n
2 a2λ Q . exp − 1 t − 2a2 t a1λ
where h = m n . So we receive the first equation for the estimates µˆ x and σˆ x
E{τ}
a2 = q 1 >> 1 , Q a1 = q . a2 Var{τ}
µˆ x + σˆ x Ψ (h) = Q ,
2 a 3λ 2 a13λ 2 Q 1 exp − Then sup u (t ) − t − → 0 as 2a2 Q a1λ 2pa2Q t λa q → ∞ and we can consider ( τ − q / λ ) 1 as N (0, 1). a2 q
where Ψ ( ⋅) = Φ −1 (⋅) . By equating the theoretical and empirical conditional means, we obtain the second equation 2 1 exp − ( Q − µˆ x ) / σˆ x 2 =x, µˆ x − σˆ x 2pΦ ( Q − µˆ x ) / σˆ x
2. PARAMETERS’ ESTIMATION There are different approaches to handling of demand uncertainty. In Rossia et al. (2014) overview of parametric approaches to the newsvendor problem is provided. Note that most of the papers have focused on the case when the demand is fully observed. The censored demand case induced by lost sales is considered by Hill (1992), Nahmias (1994), Lau and Lau (1996); and Agrawal and Smith (1996). We estimate parameters of interest a1λT and a2 λT given two samples: the sale period’s duration, i.e. the time at which the vendor runs short, t1, t2, …, ∀i ti ≤ T , if there are lost sales; and the sizes of sales during time period T x1, x2, …, ∀i xi ≤ Q , if there are leftovers at the end of the period. We assume that the customer satisfy the demand as far as possible when the amount required is more than the remainder; and the lot sizes are the same for each period. We base the estimating procedure on the distributions obtained above.
where x =
1 m ∑ xi . m i =1
So we get following system
µˆ x − σˆ x F (h) = x , µˆ x + σˆ x Ψ (h) = Q,
1 exp − Ψ 2 (h) 2 . where F (h) = 2ph The estimates σˆ x =
µˆ x = 1424
Q−x , Ψ ( h) + F ( h)
x Ψ (h) + QF (h) . Ψ ( h) + F ( h)
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2.2 Estimation by lost sales periods
νˆ =
To our knowledge, no research exists using the observed sale period’s durations to the parameters estimation in the newsvendor problem framework.
τΨ (h) + F (h) 1− τ , sˆ = , Ψ ( h) + F ( h) Ψ ( h) + F ( h)
and
µˆ (xt ) = Q
Let t1, t2, …, tn-m, ∀i ti ≤ T be the observed sale period’s durations in n − m periods.
Let us introduce the dimensionless quantities ν = µt T and s = σt T .
Ψ ( h) + F ( h)
2
( τΨ (h) + F (h) )3
,
where µˆ (xt ) and σˆ (xt ) are the estimates of a1λT and a2 λT derived from the sample t1, t2, …, tn-m. Obtaining joint distribution (Ti , X i ) is one of the directions of future research.
We have
a1λT =
3. SIMULATION
Q Q2 s2 , a2 λT = 3 , ν ν
In this section, we simulate the system for three setups:
and
T − µt P{t < T } = Φ σt
1− ν = Φ . s
1. Batch size has uniform distribution at [2, 8]. It follows that a1 = 5 and a2 = 28. The intensity λ equals 1. 2. Batch size has exponential distribution with a1 = 4. It follows that a2 = 32. The intensity λ equals 1.
m = h. n
3. Batch size has beta distribution with shape parameters α = 0.429 and β = 0.286. It follows that a1 = 0.6 and a2 = 0.5. The intensity λ equals 4.
We estimate the probability P{t < T } by
So we get the first equation for estimates νˆ and sˆ
1 − νˆ Φ =h, sˆ or νˆ + sˆΨ (h) = 1 . Analogously the previous part we receive the second equation 2 1 exp − (T − µˆ t ) / σˆ t 2 =t , µˆ t − σˆ t 2pΦ (T − µˆ t ) / σˆ t
where t =
Ψ ( h) + F ( h) , τΨ (h) + F (h)
σˆ (xt ) = Q 2 (1 − τ )
We consider the sale period’s durations as a normal random variable with the mean E {t} = µt = Q a1λ and the variance
Var {t} = σt2 = a2 λQ (a1λ)3 .
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1 n−m ∑ ti . n − m i =1
We can rewrite (2) as 2 1 exp − (1 − νˆ ) / sˆ 2 = τ, νˆ − sˆ ˆ ˆ 2pΦ (1 − ν ) / s
(2)
We replicate the procedures eleven times for n = 300 and n = 50. Tables 1−3 report the mean absolute percent errors (MAPE) of corresponding estimates in percentages; m n − m (t ) m 2 n − m (t )2 µˆ = µˆ x + µˆ x and σˆ = are σˆ x + σˆ x n n n n weighted estimates. In Figures 1−3 and 4−6 the numerical results for exponential batch size’s distribution are given for n = 300 and n = 50 m is equal respectively. For Q = 540 the ratio n−m approximately to 3, i.e. the probability of leftovers is m ≈ 1 , i.e. the approximately 0.75, for Q = 500 the ratio n−m m 1 ≈ , lot size is close to optimal, for Q = 455 the ratio n−m 3 i.e. the probability of stock-out is approximately 0.75. Table 1. Uniform distribution, a1 λT = 625 ,
a2 λT = 59.16
where τ = t T .
MAPE(∙)% Q
We get the system of equations
µˆ x
µˆ (xt )
µˆ
σˆ x
σˆ (xt )
σˆ
n = 300
νˆ + sˆΨ (h) = 1, ˆ (h) = τ. νˆ − sF The estimates 1425
585
0.70
0.45
0.48
8.70
2.76
2.93
625
0.68
0.70
0.69
7.72
4.85
4.93
665
0.61
1.40
0.80
4.84
9.27
3.37
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1360
n = 50 585
1,93
0,98
1,08
14,93
12,74
10,68
625
1,12
1,27
1,18
10,93
14,89
6,27
665
1,26
1,65
1,30
7,43
13,74
8,36
Table 2. Exponential distribution, a1 λT = 500 ,
a2 λT = 63.25 MAPE(∙)% Q
µˆ x
µˆ (xt )
µˆ
σˆ x
σˆ (xt )
σˆ
n = 300
Fig. 2. Exponential distribution, Q = 500, n = 300.
455
0.92
1.02
0.90
9.72
2.86
3.47
500
0.69
0.64
0.66
6.65
4.92
3.64
540
1.01
1.65
1.05
3.25
13.13
4.31
n = 50 455
2,60
2,10
2,12
16,94
10,88
9,14
500
1,37
1,39
1,36
10,78
11,21
8,05
540
1,50
2,35
1,63
6,57
15,66
5,40
Table 3. Beta distribution, a1 λT = 75, MAPE(∙)% Q
µˆ x
µˆ (xt )
70
1.27
0.68
75
0.89
80
a2 λT = 7.91
σˆ x
σˆ (xt )
σˆ
n = 300 0.79
9.79
4.81
3.78
0.85
0.87
7.65
7.31
6.07
0.45
0.86
0.54
4.75
8.70
4.14
70
1,69
0,73
n = 50 0,89
15,56
8,70
6,30
75
1,74
1,53
1,56
17,85
15,45
12,15
80
1,04
1,81
1,23
11,25
14,63
8,95
µˆ
Fig. 3. Exponential distribution, Q = 455, n = 300.
Fig. 4. Exponential distribution, Q = 540, n = 50.
Fig. 1. Exponential distribution, Q = 540, n = 300.
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of Application. In Dudin A. et al. (eds.), Lecture Notes in Computer Science, 189–196, Springer, Switzerland. Lau, H.-S., and Lau, A.H.-L. (1996). Estimating the demand distributions of single period items having frequent stockouts. European Journal of Operational Research, 92, 254–265. Nahmias, S. (1994). Demand estimation in lost sales inventory systems. Naval Research Logistics, 41, 739– 757. Rossia, R., Prestwichb, S., Tarimc, S.A., and Hnich, B. (2014). Confidence-based optimisation for the newsvendor problem under binomial, Poisson and exponential demand. European Journal of Operational Research, 239, 674–684. Silver, E.A., Pyke, D.F., and Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. Wiley, New York.
Fig. 5. Exponential distribution, Q = 500, n = 50.
Fig. 6. Exponential distribution, Q = 455, n = 50. So we recommend to use weighted estimates µˆ and σˆ . It allows us to increase significantly the accuracy of demand’s standard deviation’s estimation which is comparatively low, and the accuracy of µˆ does not differ significantly from the best one proposed. REFERENCES Agrawal, N., and Smith, S.A. (1996). Estimating negative binomial demand for retail inventory management with unobservable lost sales. Naval Research Logistics, 43, 839–861. Arrow, K.J., Harris, Th.E., and Marschak, J. (1951). Optimal Inventory Policy. Econometrica, 19 (3), 205–272. Hill, R.M. (1992). Parameter estimation and performance measurement in lost sales inventory systems. International Journal of Production Economics, 28, 211–215. Kitaeva, A.V. (2014). Stabilization of Inventory System Performance: On/Off Control. Proceedings of the 19th IFAC World Congress, 10748–10753, Cape Town, South Africa, 24-29 August 2014. Kitaeva, A., Subbotina, V., and Zmeev, O. (2014). Diffusion Appoximation in Inventory Management with Examples 1427