Optimal Inventory Policies for Assembly Systems under Random Demands

Optimal Inventory Policies for Assembly Systems under Random Demands Main Result • Remodel an assembly system as a series system • Simple re-order policies are optimal Main Result • Assembly system: - Ordered amounts available after a fixed lead time - Random customer demands only for the end product - Assumptions on cost parameters • Under assumptions and restriction on initial stock levels, assembly system can be treated as a series system • Optimal inventory policy – can be calculated by approach in Clark and Scarf’s paper (1960)* Relevant Literature • Clark and Scarf (1960) – derive optimal ordering policy for pure series system • Fukuda (1961) – include disposal of items in stock • Federgruen and Zipkin (1984) – generalize Clark and Scarf approach to stationary infinite horizon case Model • • N items (components, subassemblies, the end item) Each non-end item has exactly one successor - product networks forms a tree rooted in the end item • Exactly one unit of each item required for the end item • Notation: s (i ) = unique immediate successor of item i=1…N; s(1) = 0 A(i ) = the set of all successors of item i P (i ) = the set of immediate predecessors of item i B (i ) = the set of all predecessors of item I l i = number of periods (lead-time) for assembly (or delivery) of item i Model • At the beginning of a time period: 1. Outstanding orders arrive and new ordering decisions made 2. Old backlogs fulfilled and customer demands occur (for the end period) 3. Backlog and inventory holding costs incurred Notation ξt λ = iid demand in period t for the end item with density φ (⋅) and distribution Φ (⋅) = E[ξ t ], expected value of ξ t X it = echelon inventory position of item i in period t before ordering decision are made ( = inventory on hand + units in assembly/order - units backlogged) Yit = echelon inventory position of item i in period t after ordering decisions are made; Yit ≥ X it Yit − X it = amount ordered for item i in period t; arrives after l i periods Notation: contd.. • X itl = echelon inventory on hand of item i in period t before ordering decisions are made but after assembles arrive t −1 Y − = ∑ k = t − li ξ k i ,t − li • Ykt ≤ X itl if i ∈ P ( k ) cannot order more than at hand (no intermediate shortage) Model: Cost parameters H i = unit installation holding cost per period of item i hi = unit echelon holding cost per period of item i hi = H i − ∑ k∈P (i ) H k p = unit backlogging cost per period of the end item α = period discount factor 0 < α ≤ 1 • Cost in period t N ∑ H i ( X itl − X sl (i )t ) + H1 ⋅ Max (0, X1lt − ξt ) + p ⋅ Max (0, ξt − X1lt )  i =2  Holding cost for end item/ Backlogging cost Installation Holding Cost Model: Cost • Alternate Formulation: (see page 567 of Rosling paper, left hand column) • Using X il,t +li − ξ t +li = Yit − ∑s =1i ξ s t +l • Total Expected Cost over an infinite horizon: (see equation 1 on page 567) Φ 1l +1 (⋅) : convolution of Φ (⋅) over (l1 + 1) periods Long-Run Inventory Position • M i : total lead-time for item i and all its successors M i = l i + ∑k∈A(i ) l k Long-Run Balance • Assembly system is in long-run balance in period t iff for i=1,..,N-1 X M −µ it ≤X M −µ i +1,t for µ = 1,..., M i − 1 • Inventory positions equally close to the end item increase with total lead-time • Satisfied trivially if (i + 1) ∈ P(i) Assumptions on Cost parameters • hi > 0 for all i – All echelon holding costs positive N • ∑ hi ⋅ α i =1 −M s (i ) < p + H1 – Better to hold inventory than incur a backlog Long-run Inventory position Theorem 1: “Any policy satisfying Lemmas 1 and 2 leads the system into long-run balance and keeps it there. This will take not more than MN+1 periods after accumulated demand exceeds Maxi Xi1.” Proof: Outline – X it ≤ X iL+ 1, t for all t ≥ q ( i ) by Lemma 1 M −µ = Yiq − – X it t −1 ∑ξ r =q t −1 r ≤ X iL+1,q −∑ ξ r = X iM+1−,tµ r =q for t = q(i ) + M i − µ – long-run balance for i for – Upper bound q(i) t ≥ q(i) + M i Equivalent Series System Theorem 2: If the Assumptions hold and system is initially in long-run balance, then optimal policies of the assembly system are equivalent to those of a pure series system where: – i succeeds item i+1 – lead-time of item i is Li l −L – holding cost hi ← hi ⋅ α Proof: Cost function i i ∞ ⎧⎪ ∞ ⎛ N L ⎞⎫⎪ l L − L t −1 ⎜ L +1 i i i 1 Min E ⎨∑ α ⋅ ∑ α ⋅ (hiα )Yit +α ⋅ ( p + H 1 ) ∫ (ξ − Yit )φ1 (ξ )dξ ⎟⎬ Y ⎜ i =1 ⎟⎪ ⎪⎩ i =1 Yit ⎝ ⎠⎭ + Constant Equivalent Series System Easy to show, using Theorem 1,: X it ≤ X ktM − M i for all i, t and k ∈ P(i ) Hence, using Lemma 1, X it ≤ Yit* ≤ X iL+1,t Use this constraint in Problem P. New Formulation for P ∞ N ⎧⎪ ∞ ⎛ ⎞⎫⎪ Li li − Li L1 t −1 ⎜ L +1 Min E ⎨∑ α ⋅ ∑ α ⋅ (hiα )Yit +α ⋅ ( p + H 1 ) ∫ (ξ − Yit )φ1 (ξ )dξ ⎟⎬ Y ⎜ i =1 ⎟⎪ ⎪⎩ i =1 Yit ⎝ ⎠⎭ + Constant such that X it ≤ Yit ≤ X iL+1,t for all i, t where X L i +1,t = Yi +1,t − L − and X it = Yi ,t −1 − ξ t −1 t −1 ∑ξ s =t − L s Equivalent Series System Corollary 2: There exist Si’s such that the following policy is optimal for all i and t Yit* = Min( S i , X iL+1,t ) if X it ≤ S i Yit* = X it if X it ≥ S i Si – obtained from Clark and Scarf’s (1960) procedure • Critically dependent on initial inventory level assumption (long-run balance initial inventory levels) • Generally optimal policy by Schmidt and Nahmias (1985) Equivalent Series System Corollary 3: If Li=0, then – Optimal order policy with Si = Si-1 – i and i-1 can be aggregated – lead time of aggregate Li-1 – holding cost coefficient hi −1 +hiα − L i −1 General Assumption on Costs • Generalized Assumption • Allowed to have hi ≤ 0 for some i • Examples: Meat or Rubber after cooking/ vulcanization – Components more expensive to store than assemblies – May have negative echelon holding costs Generalized Assumption • Aggregation procedure to eliminate i for which hi ≤ 0 • Leads to an assembly system satisfying the original assumption Practical Necessity of Generalized Assumption 1. If hi ⋅ α −M s (i ) +∑k∈B (i ) hk ⋅α −M s(k ) < 0 for some i – Minimal cost of P is unbounded 2. If hi ⋅ α − M s (i ) +∑k∈B ( i ) hk ⋅α −M s(k ) = 0 for some i – item i is a free good, hence predecessors of i may be neglected 3. If assumption (ii) not satisfied – production eventually ceases Summary • Assembly system: - Ordered amounts available after a fixed lead time - Random customer demands only for the end product - Assumptions on cost parameters • Under assumptions and restriction on initial stock levels, assembly system can be treated as a series system • Simple optimal inventory policy – can be calculated by approach in Clark and Scarf’s paper (1960)* Comments • Series analogy does not work for: – Non stationary holding/production costs – Non-zero setup costs