A stochastic multi-item lot-sizing problem with bounded number of - PowerPoint PPT Presentation

www.argon-consult.com A stochastic multi-item lot-sizing problem with bounded number of setups Séminaire des doctorants February 8th, 2017 Etienne de Saint Germain joint work with Frédéric Meunier and Vincent Leclère

Outline • Business problem and model • Deterministic model • Stochastic model • Numerical experiments

Business problem • Context: production function of the Supply Chain for one assembly line • Objective: reduction of holding costs • Main constraints: industrial flexibility and “high service level” • Typical horizon: 10 to 15 weeks • Typical time step: 1 week • Input data: ◮ a set of references r ∈ R ◮ capacity of the line ◮ demand for each reference and each week 2 / 20

Classical problem: Capacitated Lot-Sizing Problem (CLSP) T � � ( h r t s r t + c r t x r min t ) t = 1 r ∈R s r t = s r t − 1 + q r t − d r s.t. ∀ t , ∀ r t � q r t ≤ 1 ∀ t (CLSP) r ∈R q r t ≤ x r ∀ t , ∀ r t x r t ∈ { 0 , 1 } ∀ t , ∀ r q r t , s r t ≥ 0 ∀ t , ∀ r • with ( r ∈ R for references, t ∈ [ T ] for weeks): input data variables h r s r holding cost inventory level t t c r q r setup cost produced quantity t t d r x r demand setup variable t t 3 / 20

Disadvantage of CLSP formulation (according to Argon Consulting) • Hard to compare holding costs and setup costs • Number of setups is a “technical” constraint ◮ Given by operational level ◮ Represent scheduling constraints (which are neglected at tactical level) 4 / 20

Multi-item lot-sizing problem with bounded number of setups T � � ( h r t s r c r t x r min t + ✟✟ ❍❍ ✟ t ) ❍ t = 1 r ∈R s r t = s r t − 1 + q r t − d r s.t. ∀ t , ∀ r t � q r t ≤ 1 ∀ t (P) r ∈R q r t ≤ x r ∀ t , ∀ r t � x r t ≤ N ∀ t r ∈R x r t ∈ { 0 , 1 } ∀ t , ∀ r q r t , s r t ≥ 0 ∀ t , ∀ r • Bounded number of setups per week ( N ) • Easier for industrials to quantify holding costs and N 5 / 20

Outline • Business problem and model • Deterministic model • Stochastic model • Numerical experiments

The deterministic model is hard • (P) is NP -hard (reducing 3-PARTITION) ◮ Decide if there is a solution when N = 1 is polynomial ◮ Cases N = 1 and N = 2 still open • Continuous relaxation of (P) does NOT depend on N • 2 natural extended formulations: � R � ◮ 1 binary variable x p , t where p ∈ per possible plan for a week N ◮ 1 binary variable y q , r where q ∈ 2 [ T ] per possible plan for a reference references y q , r weeks x p , t ◮ Same continuous relaxations than compact formulation 6 / 20

Outline • Business problem and model • Deterministic model • Stochastic model • Numerical experiments

Need for backlog • Mathematical reason: ◮ In general, there is no feasible solution ◮ Simple example: • bounded capacity C • demand = Gaussian noise around forecast probability of realization infeasible region demand forecast capacity • Industrial reason: ◮ Negative inventories are commercial constraints ⇒ “soft” constraints = ◮ Firms can deliver late 7 / 20

Stochastic model � T � � � ( h r s r t + γ b r min t ) E t ˜ t = 1 r ∈R s r s r t − b r s.t. t = ˜ ∀ t , ∀ r t s r t = s r t − 1 + q r t − d r ∀ t , ∀ r t � q r t ≤ 1 ∀ t (S) r ∈R q r t ≤ x r ∀ t , ∀ r t � x r t ≤ N ∀ t r ∈R x r t ∈ { 0 , 1 } ∀ t , ∀ r q r s r t , b r t ≥ 0 ∀ t , ∀ r t , ˜ � � ( d r 0 , . . . , d r σ ( q r t ) , σ ( x r t ) ⊂ σ ∀ t , ∀ r t ) r ∈R s r with: backlog penalization, ˜ inventory level, γ t b r backlog quantity t 8 / 20

Stochastic model: size difficulty • Extensive formulation leads to a huge number of variables 0 t = 0 1 2 t = 1 1 2 1 2 t = 2 1 2 1 2 1 2 1 2 t = 3 ◮ example: for each references, 2 independent possibilities for demand � 2 T � |R| = ⇒ number of variables multiplied by ◮ for T = 10, |R| = 10, numbers of variables ≈ 10 30 • Need for heuristics to solve (S) ◮ lot-size and cover-size ◮ open-loop feedback approach ◮ repeated two-stage stochastic programming approach 9 / 20

Aside: computing lot-size and cover-size • Simplified model: ◮ Constant demand for each reference over time ◮ Aggregated demand Inventory level of reference r: Corresponding program to solve: s r = 1 2 d r T 2 ¯ inventory level r 1 � min 2 h r d r T r − d r r ∈R lot-size 1 � s.t. T r = N time T r 2 T r r ∈R cover-size T r > 0 ∀ r • Closed-form expressions of solutions √ � 2 �� 1 N h r d r 1 ν ∗ � r = r = and Cost = h r d r T ∗ � 2 N � h p d p p ∈R r ∈R • Closed-form expressions of solutions for stochastic case 10 / 20

Strategy: lot-size and cover-size • Heuristic parameters: safety stock for each reference foreach r ∈ R do Compute cover-size T r / lot size ℓ r = d r T r ; for week from 1 to T do foreach r ∈ R do Observe inventory level of r ; if current inventory level < safety stock then case lot-size: produce quantity ℓ r ; case cover-size: produce the cumulated expected demand for the T r next weeks; • Example: ◮ For a reference r , T r = 2 weeks ◮ Expected demand is: week t 1 2 3 4 5 expected demand f t 2 3 5 4 1 ◮ If current inventory level < safety stock at week 1, we must produce: f 1 + f 2 = 2 + 3 = 5 units of reference r 11 / 20

Strategy: open-loop feedback approach • At week t : ◮ Observe current inventory level ◮ Solve deterministic version of (S) where the random variable d r t is replaced by the deterministic expected demand • It is a Mixed Integer Program • Almost program (P) but with backlog ◮ Set production decisions for week t 12 / 20

Strategy: repeated two-stage stochastic programming approach • At week t : ◮ Observe current inventory level ◮ Construct a fan of demand scenarios to approximate the tree of scenarios in (S) 0 t = 0 0 1 2 3 t = 1 1 2 3 t = 2 t = 3 Complete tree of scenarios Fan of scenarios ◮ Solve (S) ◮ Set production decisions for week t 13 / 20

There is a lot of possible forecasts • Static deterministic forecast ◮ Expectation, median... • Adaptative deterministic forecast ◮ Autoregressive process, time series... • Stochastic forecast ◮ Tree of scenario, fan of scenarios... Every strategy works even if we do not know distribution laws. We just need a forecast function! 14 / 20

Outline • Business problem and model • Deterministic model • Stochastic model • Numerical experiments

Simulations Demand realization Simulator Key performance Strategy indicators: • Holding costs • Start at week t = 0 Forecast • Backlog costs • For each week t: • Cycle service current Data past level time • Fill rate service level ◮ Observe inventory level ◮ Use strategy and forecast to • . . . compute the production planning ◮ Set production for current week • Return KPI 15 / 20

Data of the instances • We use historical data from industrial • Numerical values: ◮ Horizon T = 13 weeks ◮ |R| = 30 references ◮ Demands 0 ≤ d r t ≤ 4000 units ◮ Weekly capacity C ≈ 13000 units ◮ Weekly number of setups N = 10 ◮ Holding costs 50 ≤ h r t ≤ 80 per units 16 / 20

Building the distribution of the demand • Autoregressive Process (AR1). α = 50 % For each reference r , σ = 80 % d t + 1 = f t + 1 + α e t + ( 1 − α ) ǫ t + 1 forecast � �� � e t + 1 where (at week t ): demand ◮ d t is the demand ◮ f t is the forecast ◮ e t is the forecast error ◮ ǫ t ∼ N ( 0 , σ f t ) is a white noise • 2 parameters: ◮ α ∈ [ 0 , 1 ] proportion error/noise ◮ σ is the volatility. time 17 / 20

Results: holding costs for several realizations of demand Cover-size Open-loop feed back Lot-size Two-stages stochastic 4 · 10 6 α = 50 % σ = 80 % Holding costs 3 · 10 6 2 · 10 6 1 · 10 6 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 Fill rate service shortage 18 / 20

Results: holding costs for several values of volatility 3 · 10 6 0.8 2 1.5 2 . 5 · 10 6 0.5 0 0.2 Holding costs 2 · 10 6 0.8 0.8 2 2 1.5 2 0 1.5 0.8 0.5 1 . 5 · 10 6 1.5 0.2 0.5 0.2 0.5 0 1 · 10 6 Cover-size Open-loop feed back 0.2 0 Lot-size Two-stages stochastic 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 Fill rate service shortage 19 / 20

Results: holding costs for several values of volatility Thanks for your attention! 20 / 20