Stochastic (Random) Demand Inventory Models

Transcription

1 Stocastic (andom) Demand Inventory Models George Liberopoulos 1 Assumptions/notation Single-period orizon Uncertain demand in te period: D (parts) assume continuous random variable Density function and cumulative distribution function of D: f(x) and F(x) a df( x) Fa ( ) PD ( a) f( x ) f( a) x0 Infinite production/replenisment rate (instantaneous replenisment) Zero lead time Overage cost rate: cost per unit of positive inventory remaining at te end of te period: c o ( per left-over part) Underage cost rate: cost per unit of unsatisfied demand (negative ending inventory) : c u ( per missing part or unsatisfied demand) No fixed setup production/order cost Decision Order quantity at te beginning of te period: (parts) xa 1

2 Definitions (Positive) inventory remaining at te end of te period: I + Unsatisfied demand (negative inventory) at te at te end of te period: I I ( D) max( D,0) I ( D) max( D,0) Total overage and underage cost at te end of te period: G(, D) Expected cost: G() GD (, ) ci ci c( D) c( D) o u o u G ( ) EGD [ (, )] Gx (, ) f( x ) D x0 o u x0 x0 c ( x) f( x) c ( x) f( x) c ( x) f( x) c ( x) f( x) o x0 u x 3 I ( D) max( D,0) f ( x) I ( D ) max( D,0) c o x0 ( x) f ( x) μ = Ε[D] c u x ( x ) f ( x) D 4

3 Problem Minimize G ( ) First derivative of cost function dg( ) d co ( x) f( x) cu ( x ) f( x) d d x0 x d co ( x) f( x) 1( x) f( x) 0( x) f( x) x x0 d x0 d cu ( x) f( x) 0( x) f( x) 1( x) f( x) x x d x c 1 f( x) c ( 1) f( x) o x0 u x cf ( ) c [ 1 F ( )] o u 5 In deriving te previous formula, we used Leibnitz s rule for taking te derivative of an integral wole limits are functions of te variable wit respect to wic te derivative is taken. Leibnitz s rule is: b( y) b( y) d df( x, y) db( y) da( y) f( x, y) f( b( y), y) f( a( y), y) dy dy dy dy a( y) a( y) 6 3

4 Second derivative dg ( ) d cf o c u F c o c u f d ( ) [1 ( )] ( ) ( ) 0 d First-order condition for minimization dg( ) 0 cf ( ) c[1 F ( )] 0 ( c c) F ( ) c d 0 o u o u u c : F( ) F * * u * 1 u co cu co cu Note: F() is te fill rate, i.e. te probability tat a demand will be satisfied! c ecall: EO model wit backorders: F b * b 7 Special case: D Normal(μ, σ) D F ( ) PD ( ) P Normal(0,1) standardized Normal cumulative distribution function F ( ) z, z z and ence F( ) can be evaluated from standardized Normal tables z ( z) ( z)

5 Special case: D Normal(μ, σ) cont d * * * * cu cu 1 cu : F( ) co cu co cu co cu * zc u ( cocu) z c ( c c ) u o u Example D Normal(10, 45) buying price c = 30 selling price S = 110 salvage price s = 10 cu S c co cs cu z c c 0 80 o u * z Extension: Discrete demand Uncertain demand in te period: D (parts) assume discrete random variable Probability mass function and cumulative distribution function of D: p(x) and F(x) Fa ( ) PD ( a) px ( ) pa ( ) Fa ( ) Fa ( 1) xa Expected cost: G() Problem: Minimize G ( ) First-order difference 1 o u x x0 x G ( ) EGD [ (, )] Gx (, ) px ( ) c ( x) px ( ) c ( x) px ( ) D G ( 1) G ( ) c px ( ) c px ( ) cf ( ) c[1 F ( )] o u o u x0 x1 First-order condition for minimization * :smallest suc tat G( 1) G( ) 0 cof( ) cu[1 F( )] 0 * * cu :smallest suc tat F( ) c c o u 10 5

6 Extension: Staring inventory y > 0 Still want to be at * after ordering, because * is te minimizer of G() Order quantity: U Optimal policy now depends on starting inventory: * y,if y U ( y) 0, if y * * Note: U * optimal order quantity * optimal order-up-to point inventory target level base stock level * 11 Interpretation of c o and c u for te single-period model S = selling price ( per part) c = variable cost ( per part) = olding cost ( per part per period) p = loss-of-goodwill cost ( per part sort per period) GD (, ) c ( D) pd ( ) Smin( D, ) order cost leftover inventory cost sortage cost sales sales revenue G( ) E[ G(, D)] c ( x) f( x) p ( x) f( x) S[ xf( x) f( x) ] D 0 0 c ( x) f ( x) ( p S) ( x ) f ( x) S xf ( x) 0 dg( ) 0 cf ( ) ( ps)(1 F ( )) 0 ( x) f ( x) E[( D) ] d * ps c : F( ) cu ps c, co c ps 1 6

7 Extension: infinite periods (infinite orizon) wit backorders Same assumptions as single-period model except tat: D t = demand in period t; D 1, D, D 3, are i.i.d. wit distribution f(x), F(x) U t = amount ordered in period t Optimal policy in eac period is order-up-to In te long-run, te inventory can never be iger tan In steady-state (long run): U t = D t 1 Inventory/ backorders Steady state Transient U t = D t 1 time t 13 Extension: infinite periods (infinite orizon) wit backorders (cont d) Total cost in a period wit demand D Expected average cost per period First-order condition for minimizing G() * * p : F( ) Newsventor formula: cu p, co p Note: c and S play no role in determining *, because in te long run, all demands are satisfied regardless of ; terefore, te expected average ordering cost minus revenue per period is (c S)μ regardless of. GD (, ) ( c SD ) ( D) pd ( ) order cost sales revenue inventory olding cost backorder cost G( ) E[ G(, D)] ( c S) E[( D) ] pe[( D ) ] D dg( ) 0 F( ) p(1 F( )) 0 d 14 7

8 Extension: infinite periods (infinite orizon) wit lost sales Same assumptions as infinite orizon wit backorders except tat: Unmet demand is not backordered but is lost In steady-state (long run): U t = min(, D t 1 ) Inventory/ lost sales Transient U t = min(, D t 1 ) time t 15 Extension: infinite periods (infinite orizon) wit lost sales (cont d) Total cost in a period wit demand D GD (, ) ( cs) min( D, ) ( D) pd ( ) Expected average cost per period G( ) E[ G(, D)] ( c S) E[( D ) ] E[( D) ] pe[( D ) ] D First-order condition for minimizing G() dg( ) 0 F( ) ( p S c)(1 F( )) 0 d * * ps c : F( ) Newsvendor formula: cu ps c, co ps c Note: c and S now play a role in determining *, because in te long run, te demand satisfied and te orders are min(, D), so tey depend on ; terefore, te expected average ordering cost minus revenue per period is (c S)E[min(, D)]. 16 8

9 Lot size eorder point (, ) model Assumptions Infinite orizon Continuous review (as opposed to periodic review) D t : random stationary demand per unit time (e.g., daily demand) mean λ Ε[D t ], variance σ t = E[(D t λ) ] Unmet demand is eiter backordered or lost τ: Fixed replenisment order lead time Costs: Variable unit production/order cost: c ( per part) Fixed setup production/order cost: K ( per production run/order) Inventory olding cost rate: ( per part per unit time) Stock-out (sortage/penalty) cost rate ( cases / 4 situations: see next) Order policy (, ) policy: order wen inventory position falls below Decision variables : lot size (reorder quantity) : reorder point 17 (, ) model Assumptions on stock-outs and stock-out cost rate Case 1: Backordered demand p 1 ( per stock-out occasion) p ( per part sort ) p 3 ( per part sort per unit time) Case : Lost sales p L ( per lost sale) 18 9

10 (, ) model: Backordered demand + Inventory position On-and inventory time τ Backorders 19 (, ) model: Backordered demand Analysis D: demand during lead time τ Density function and cumulative distribution function of D: f(x) and F(x) D = D 1 + D + + D τ Mean: μ E[D] = Ε[D 1 + D + + D τ ] = τ Ε[D t ]= τ λ Variance: σ Var[D] = Ε[(D μ) ] = Var[D 1 + D + + D τ ] = τ Var[D t ]= τ σ t Inventory / backorders τ μ f(x) x time 0 10

11 (, ) model: Backordered demand Inventory olding cost Safety stock ss μ = λτ Expected average inventory approximation Ī ss + / = λτ+ / (underestimates true value) Expected average inventory old cost = Ī = ( λτ+ /) Inventory / Backorders ss + = λτ+ ss = λτ τ λ μ = λτ time 1 (, ) model: Backordered demand To see wy expected average inventory approximation underestimates true expected average inventory: I E[ D] xf( x) appr 0 I E[( D) ] ( x) f( x) F( ) xf( x) true I I (1 F( )) xf( x) appr true f( x) xf( x) f ( x) xf ( x) ( x) f( x) E[( D) ] 0 I I appr true

12 (, ) model: Backordered demand Setup cost Expected average order frequency = λ/ Expected average setup cost per unit time = K λ/ Stock-out (penalty) cost Assumption: τ << /λ stock-out per cycle depends only on B() : Expected stock-out cost per cycle (depends on definition of stockout cost rate) Expected average stock-out cost per unit time = B() λ/ Total expected average cost per unit time G(, ) K B( ) 3 (, ) model: Backordered demand Optimization problem Minimize G(, ) K B( ), Optimality conditions G(, ) K B( ) [ K B( )] 0 [ K B( )] (1) G (, ) db( ) 0 d db( ) () d 4 1

13 (, ) model: Backordered demand Optimality condition (): 3 cases Case 1: Stock-out cost p 1 per stock-out occasion db( ) B ( ) p[1 F ( )] pf( ) 1 1 d probabilty of stock-out per cycle Condition () : pf 1 ( ) f( ) p Case : Stock-out cost p per part sort db( ) d n( ) expected number x of stock-outs per cycle n( ) B ( ) p E[( D) ] p ( x) f ( x ) p[1 F ( )] Condition () : p[1 F( )] F( ) 1 p 1 5 (, ) model: Backordered demand Case 3: Stock-out cost p 3 per part sort per unit time Average waiting time per backorder wen te demand is D: ( D ) Average total waiting time of all backorders wen te demand is D: ( D) [( D) ] ( D) ( x ) p3 B ( ) p3 f( x ) ( x ) f( x) x x expected total waiting time of all backorders per cycle db( ) p3 p3 p3 ( x ) f( x) E[( D ) ] n( ) d x p n n p 3 Condition () : ( ) ( )

14 (, ) model: Backordered demand Simultaneous solution of conditions (1) and () Solve by fixed-point iteration: 1. Given, solve (1) to find. Given, solve () to find 3. epeat until convergence 7 (, ) model: Backordered demand Illustration for case : Stock-out cost p per part sort Optimality conditions for case [ n ( )] (1 K p ) F ( ) 1 () p Assumption: D~Normal(μ, σ) Use standardized cumulative distribution function (cdf) Φ(z) to compute F() D F( ) P( D ) P Normal(0,1) F ( ) z, z Φ(z) and ence F() can be evaluated from standardized Normal cdf tables 8 14

15 (, ) model: Backordered demand Use standardized loss function L(z) to compute n() Y ~Normal(0,1) L( z) E[( Y z) ] ( yz) ( y) dy L(z) and ence n() can be evaluated from standardized loss function tables It can be sown tat Lz ( ) ( z) z[1 ( z)] n ( ) Lz ( ) ( z) ( )[1 ( z)], z yz Normal(0,1) density function D n ( ) E[( D) ] E L Normal(0,1) n ( ) Lz ( ), z 9 (, ) model: Backordered demand Under te assumption D~Normal(μ, σ), te optimality conditions become [ K pl( z)] ( z) 1 () p z (3) (1) 30 15

16 (, ) model: Backordered demand Fixed point iteration algoritm for case under te assumption D~Normal(μ, σ) K 1 0 0, z0 1, 0 z0, n 1 p [ K pl( zn 1)] Step 1: n Step Step 3: n z p z 1 : n 1 n n Step 4: O nn1, GOTO Step 1 n n1 n n (, ) model: Lost sales Inventory position On-and inventory Lost sales τ time 3 16

17 (, ) model: Lost sales Modify total expected average cost per unit time function G (, ) n( ) K pl n ( ) expected lost Optimality conditions sales per cycle expected lost sales per unit time [ K pln( )] (1) (same as case of backordered demands) G (, ) dn( ) pl dn ( ) pl dn ( ) d d d pl 1 1 [1 F( )] 0 pl F ( ) 1 () (Newsvendor formula: co, cu pl) pl pl ( p same as case of backordered demands) L 33 (, ) model: Lost sales Note G(, ) is an approximation, because it overestimates te order frequency In te lost sales model, not all demand is met; on average, demands are met and n() demands are not met Average number of demands per cycle = + n() parts per cycle Number of demands per unit time = λ parts per unit time Order frequency = λ/[ + n()] instead of λ/ A more accurate expression for G(, ) is: G(, ) ( ) L ( ) n K n( ) p n( ) n Usually, n() <<, so te order frequency can still be approximated by λ/ 34 17

18 Eq. (1): (, ) model: Summary [ K B( )] z Situation p 1 ( per stockout occasion) p ( per part sort) p 3 ( per part sort per unit time) p L ( per lost sale) 1 B( ) p [1 F ( )] B( ) D~Normal(μ,σ) Eq. () f( ) p 1 1 pn ( ) p3 ( ) ( ) x x f x pn ( ) L p [1 ( z )] p L( z) p L( z) L F( ) 1 p 3 n ( ) p pl F( ) p L Eq. () D~Normal(μ,σ) ( z) p 1 ( z) 1 p Lz ( ) p pl ( z) p 3 L 35 (, ) model: Service Levels Service levels in (, ) systems Type 1 Service (replaces stock-out cost p 1 per stock-out occasion) S 1 Probability of not stocking out during te lead time S 1 = P(D ) = F() Optimization problem Minimize G(, ) K, subject to F( ) (i.e., subject to S ) Solution * K EO = minimum suc tat ( ) * F D F * 1 continuous r.v. ( )

19 (, ) model: Service Levels Type Service (replaces stock-out cost p per part sort) S Proportion of demands met from stock S = 1 n()/ Optimization problem Minimize G(, ) K, n( ) subject to 1 (i.e., subject to S ) Note: Now te constraint depends on bot and 37 (, ) model: Service Levels Type Service (cont d) Approximate solution * K EO * * = minimum suc tat n( ) (1 ) D n * * continuous r.v. ( ) (1 ) D n L z * * * ~Normal(, ) ( ) ( ) (1 ) (1 ) * * * * 1 z, z L 38 19

20 (, ) model: Service Levels Type Service (cont d) More accurate solution Consider first-order conditions (1) and () for case [ K pn( )] (1), F( ) 1 () p () p imputed stock-out cost [1 F ( )] { K n( ) [1 F( )] } (1) quandratic function in n ( ) K n ( ) positive root: (3) 1 F ( ) 1 F( ) (1 ) n( ) (4) 39 (, ) model: andom Lead Time Extension: andom lead-time L: random lead time Mean: τ Ε[L], variance σ L Ε[(L τ) ] D: demand during lead time L Density function and cumulative distribution function of D: f(x) and F(x) D = D 1 + D + + D L, were L is a random variable It can be sown (see next page) tat: Mean: μ E[D] = τ λ Variance: σ Var[D] = Ε[(D μ) ] = τ σ t + λ σ L Everyting else olds!! 40 0

21 (, ) model: andom Lead Time Derivation of μ and σ Mean: ED [ ] EEDL [ [ ]] EL [ ] L DL L Variance: Var[ D] E[( D ) ] E[ D D ] E[ D ] E[ D] E[ ] were we used: [ ] [ DL D L EED [ [ L]] L DL t L t L t L ED L EVarD [ L] ED [ L] ] L L t EED [ [ L]] EL [ L ] ( VarL [ ] ) ( ) t t t L t L L DL L 41 1