Dealing with forecast uncertainty in inventory models

Dealing with forecast uncertainty in inventory models 19th IIF workshop on Supply Chain Forecasting for Operations Lancaster University Dennis Prak Supervisor: Prof. R.H. Teunter June 29, 2016 Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 1 / 29

Introduction Separation of forecasting and decision making Cost function Model of future demand Distribution and parameters Inventory models contain demand parameters In practice, parameters unknown Textbooks: forecasting separated from decision making Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 2 / 29

Introduction Current practice vs. optimal approach Fundamental assumption is violated: substituted estimates are not equal to the true parameters. Current practice: non-optimal Choose demand distribution and estimate parameters Substitute into cost function Optimize as if parameters are known Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 3 / 29

Introduction Current practice vs. optimal approach Fundamental assumption is violated: substituted estimates are not equal to the true parameters. Current practice: non-optimal Choose demand distribution and estimate parameters Substitute into cost function Optimize as if parameters are known Optimal approach Choose demand distribution and estimate parameters Model the estimation error Take expectation of cost function w.r.t. this error Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 3 / 29

Introduction The effect of misestimating Future forecast errors are correlated Mean-stationary demand D t = µ + error (i.i.d.) µ = 10, ˆµ = 8 Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 4 / 29

Introduction Literature Quotes The relationship between the forecast error during the lead time and that during the forecast interval depends in a complicated fashion on the specific underlying demand model, the forecast updating procedure and the values of the smoothing constant used. (E.A. Silver, D.F. Pyke, R. Peterson, 1998) We should never see forecasting as an isolated task, carried out for its own sake. (P. Goodwin, 2009) The interactions between forecasting and stock control are not yet fully understood. (M.M. Ali, J.E. Boylan, and A.A. Syntetos, 2012) Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 5 / 29

Introduction Research questions How can we construct a general approach to incorporating estimation uncertainty in any inventory decision model? What are the benefits of the improved approach compared to the naïve approach? Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 6 / 29

Approach and model General approach Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 7 / 29

Approach and model General approach How to incorporate estimation uncertainty? (universally applicable) TC(S; θ): Cost function to be minimized S: order policy (frequently: order-up-to level) θ: parameter vector of demand distribution ˆθ: point estimate of θ κ θ : θ ˆθ TC depends on θ, which is unknown We observe ˆθ (based on chosen consistent estimator) We should model κ θ Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 8 / 29

Approach and model General approach How to model the estimation error? Normal approximation (main approach) Parameter estimators have an asymptotic variance AVar(ˆθ) Asymptotically, κ θ θ ˆθ MVN(0, AVar(ˆθ)) Estimate AVar(ˆθ) and denote the estimated pdf of κ θ by f κθ Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 9 / 29

Approach and model General approach How to model the estimation error? Normal approximation (main approach) Parameter estimators have an asymptotic variance AVar(ˆθ) Asymptotically, κ θ θ ˆθ MVN(0, AVar(ˆθ)) Estimate AVar(ˆθ) and denote the estimated pdf of κ θ by f κθ Pro: Readily available, easily applicable, invariant to the demand distribution Con: Only valid asymptotically Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 9 / 29

Approach and model General approach After having decided on f κθ, take the expectation of the cost function w.r.t. κ θ. New objective function: ] TC(S; ˆθ) = E κθ [TC(S; ˆθ + κ θ ) =... TC(S; ˆθ + κ θ )f κθ ( κ θ )dκ θ1... dκ θk. Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 10 / 29

Approach and model General approach After having decided on f κθ, take the expectation of the cost function w.r.t. κ θ. New objective function: ] TC(S; ˆθ) = E κθ [TC(S; ˆθ + κ θ ) Example: θ = f κθ = ( ) µ. σ... is the joint distribution of κ µ and κ σ Two-dimensional integral, w.r.t κ µ and κ σ TC(S; ˆθ + κ θ )f κθ ( κ θ )dκ θ1... dκ θk. Compare with the naïve approach: TC(S; ˆθ) Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 10 / 29

Approach and model General approach So, for any inventory model: Choose estimation method Choose error modeling approach Take expectation of cost function w.r.t. errors Minimize w.r.t. S Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 11 / 29

Approach and model Model choice Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 12 / 29

Approach and model Model choice Mean-stationary Normal demand Fundamentals h > 0 holding costs p/u p/t p > h shortage costs p/u p/t L = 1, 2,..., order lead time n = 1, 2,..., historical demands S order-up-to level Order of events 1. Demand occurs 2. Outstanding orders arrive 3. New orders can be placed (free) Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 13 / 29

Approach and model Model choice Mean-stationary Normal demand Fundamentals h > 0 holding costs p/u p/t p > h shortage costs p/u p/t L = 1, 2,..., order lead time n = 1, 2,..., historical demands S order-up-to level Order of events 1. Demand occurs 2. Outstanding orders arrive 3. New orders can be placed (free) IL(t): Inventory on hand at time t, after demand, before order arrival C(t) = h(il(t)) + + p(il(t)) Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 13 / 29

Approach and model Model choice C(t) = h(il(t)) + + p(il(t)) Order placed at n arrives at the end of n + L Next possible order arrives at the end of n + L + 1 So, order-up-to level at n affects costs at n + L + 1 Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 14 / 29

Approach and model Model choice C(t) = h(il(t)) + + p(il(t)) Order placed at n arrives at the end of n + L Next possible order arrives at the end of n + L + 1 So, order-up-to level at n affects costs at n + L + 1 IL(n + L + 1) = S D [n+1,n+l+1] Assumption: optimal S can always be achieved model decomposable Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 14 / 29

Approach and model Model choice TC(S, n) = he(s D [n+1,n+l+1] ) + + pe(s D [n+1,n+l+1] ) Cost function: TC(S, n) Model of future demand: D [n+1,n+l+1] Distribution: Normal, i.i.d. Parameters Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 15 / 29

Approach and model Optimizing under uncertainty of parameters Example: Mean-stationary demand, µ and σ unknown Estimate µ and σ by Maximum Likelihood/OLS Leads to intuïtive estimators: sample mean and standard deviation Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 16 / 29

Approach and model Optimizing under uncertainty of parameters Example: Mean-stationary demand, µ and σ unknown Estimate µ and σ by Maximum Likelihood/OLS Leads to intuïtive estimators: sample mean and standard deviation ˆµ and ˆσ are asympt. uncorrelated and Var(ˆµ) = ˆσ2 n and Var(ˆσ) = ˆσ2 2n Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 16 / 29

Approach and model Optimizing under uncertainty of parameters Example: Mean-stationary demand, µ and σ unknown Estimate µ and σ by Maximum Likelihood/OLS Leads to intuïtive estimators: sample mean and standard deviation ˆµ and ˆσ are asympt. uncorrelated and Var(ˆµ) = ˆσ2 n and Var(ˆσ) = ˆσ2 2n ( [ ]) ˆσ 2 approx. So, κ θ MVN 0, n 0 ˆσ 0 2 2n Define f κθ, take expectation of TC, minimize w.r.t. S Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 16 / 29

Results Results What are the benefits of the improved approach compared to the naïve approach? Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 17 / 29

Results Simulation study Fix parameters Draw historical demands Estimate parameters Calculate S via both approaches Repeat 5000 times Evaluate costs via true cost function Classical approach: S Min TC(S, n; ˆθ) Results in suboptimal S Improved approach: S ] Min E κθ [TC(S, n; ˆθ + κ θ ) Results in optimal S Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 18 / 29

Results Simulation study mean-stationary demand, µ = 10 σ n p h L Exp. dem. S S 100 TC(S ) TC( S) TC( S) 2 3 10 1 5 60 66 71-12% 2 5 10 1 5 60 66 69-9% 2 10 10 1 5 60 66 68-4% Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 19 / 29

Results Simulation study mean-stationary demand, µ = 10 σ n p h L Exp. dem. S S 100 TC(S ) TC( S) TC( S) 2 3 10 1 5 60 66 71-12% 2 5 10 1 5 60 66 69-9% 2 10 10 1 5 60 66 68-4% 2 3 10 1 10 110 118 128-19% 2 5 10 1 10 110 118 125-16% 2 10 10 1 10 110 119 123-8% Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 19 / 29

Results Simulation study mean-stationary demand, µ = 10 σ n p h L Exp. dem. S S 100 TC(S ) TC( S) TC( S) 2 3 10 1 5 60 66 71-12% 2 5 10 1 5 60 66 69-9% 2 10 10 1 5 60 66 68-4% 2 3 10 1 10 110 118 128-19% 2 5 10 1 10 110 118 125-16% 2 10 10 1 10 110 119 123-8% 2 3 20 1 5 60 67 74-22% 2 5 20 1 5 60 68 72-16% 2 10 20 1 5 60 68 70-8% Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 19 / 29

Results Simulation study mean-stationary demand, µ = 10 σ n p h L Exp. dem. S S 100 TC(S ) TC( S) TC( S) 2 3 10 1 5 60 66 71-12% 2 5 10 1 5 60 66 69-9% 2 10 10 1 5 60 66 68-4% 2 3 10 1 10 110 118 128-19% 2 5 10 1 10 110 118 125-16% 2 10 10 1 10 110 119 123-8% 2 3 20 1 5 60 67 74-22% 2 5 20 1 5 60 68 72-16% 2 10 20 1 5 60 68 70-8% 3 3 10 1 5 60 69 76-12% 3 5 10 1 5 60 69 74-9% 3 10 10 1 5 60 70 72-4% Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 19 / 29

Results Simulation study mean-stationary demand, µ = 10 σ n p h L Fill rate p p+h S S 100 TC(S ) TC( S) TC( S) 2 3 10 1 5 66 71-12% 2 5 10 1 5 91% 66 69-9% 2 10 10 1 5 66 68-4% 2 3 10 1 10 118 128-19% 2 5 10 1 10 91% 118 125-16% 2 10 10 1 10 119 123-8% 2 3 20 1 5 67 74-22% 2 5 20 1 5 95% 68 72-16% 2 10 20 1 5 68 70-8% 3 3 10 1 5 69 76-12% 3 5 10 1 5 91% 69 74-9% 3 10 10 1 5 70 72-4% Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 20 / 29

Results Violin plot of costs for order-up-to levels S and S Base case, n = 5 Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 21 / 29

Results Violin plot of costs for order-up-to levels S and S Base case, n = 5 Probability of costs > 30 twice as large under classical approach Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 21 / 29

Modeling alternatives Modeling alternatives Different options for modeling the error distribution Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 22 / 29

Modeling alternatives Exact distribution Given demand distribution and point estimate, model its error exactly Depends heavily on Normal distribution Error in regression coefficient vector: multivariate student s T Error in σ 2 : inverse χ 2 (multiplicative) Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 23 / 29

Modeling alternatives Bayesian analysis Prior knowledge of parameters data posterior parameter distribution Depends on choice of prior Uninformative prior results identical to our exact approach Parameters are considered random variables Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 24 / 29

Modeling alternatives Others Bootstrap Jack-knife Invariant to specific distribution Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 25 / 29

Conclusion & discussion Conclusion & discussion Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 26 / 29

Conclusion & discussion Conclusion General approach to incorporating uncertainty in inventory modeling Freedom in choice of estimator and approximation of error Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 27 / 29

Conclusion & discussion Conclusion General approach to incorporating uncertainty in inventory modeling Freedom in choice of estimator and approximation of error Safety stock mark-up Twofold benefit: Reduction in average costs Reduction in cost variance Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 27 / 29

Conclusion & discussion Conclusion General approach to incorporating uncertainty in inventory modeling Freedom in choice of estimator and approximation of error Safety stock mark-up Twofold benefit: Reduction in average costs Reduction in cost variance Largest benefit: n small, L large, p large Benefit invariant to demand parameterization Cost benefits of 30-50% realistic Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 27 / 29

Conclusion & discussion Exact expressions under mean-stationary, Normal demand Prak, Dennis, Ruud Teunter and Aris Syntetos, On the calculation of safety stocks when demand is forecasted, European Journal of Operational Research, forthcoming. Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 28 / 29

Conclusion & discussion Dealing with forecast uncertainty in inventory models 19th IIF workshop on Supply Chain Forecasting for Operations Lancaster University Dennis Prak Supervisor: Prof. R.H. Teunter June 29, 2016 Dennis Prak (RuG) Forecast uncertainty in inventory models June 29, 2016 29 / 29