PART III. INVENTORY THEORY

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1 PART III Part I: Markov chains PART III. INVENTORY THEORY is a research field within operations research that finds optimal design of inventory systems both from spatial as well as temporal point of view to minimize costs. It studies the decisions faced by firms and the military in connection with manufacturing, warehousing, spare part allocation etc. It provides mathematical basis for logistics. Soňa Kilianová spring semester 2013 Lectures on SMOR 56 / 77

2 Basic deterministic problems. EOQ. LECTURE 9 Basic deterministic models. - warehouse: the ware is consumed continuously (driving fuel, draught beer,...) - costs: delivery, storage, deficit - task: when and how much to restock in order to minimize costs Soňa Kilianová spring semester 2013 Lectures on SMOR 57 / 77

3 Deterministic model without deficit Deterministic model without deficit Assumptions of the simplest model: the good/ware is infinitely divisible consumption of λ units per unit of time (known, given, uniform) warehouse capacity unconstrained, height of order unconstrained, goods don t get old costs of one delivery are independent of the amount no deficit allowed Soňa Kilianová spring semester 2013 Lectures on SMOR 58 / 77

4 Deterministic model without deficit Notation: c s - storage costs for 1 unit of good and 1 unit of time c d - costs (price) for 1 order (delivery), regardless of its size Q S(t) - size of inventory at time t [0, T ], i.e. S(t) = Q λt for t [0, T ] = [0, Q/λ] Soňa Kilianová spring semester 2013 Lectures on SMOR 59 / 77

5 Deterministic model without deficit Storage costs: for good left from the warehouse in the interval [t, t + t]: c s [S(t) S(t + t)]t + o( t) overall storage costs between 2 deliveries (generally): c s T 0 S(t)dt c ss(t )T in the simple case with S(t) = Q λt, T = Q/λ, S(T ) = 0: λc s T 2 /2 Q52. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 60 / 77

6 Deterministic model without deficit Overall costs: overall costs for 1 period: c s Q 2 2λ + c d number of deliveries per unit of time: v = λ Q overall costs per unit of time: C(Q) = (c s Q 2 2λ + c d) λ Q = c sq/2 + c d λ/q Q53. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 61 / 77

7 Deterministic model without deficit Optimization: we look for ˆQ such that C( ˆQ) = min Q C(Q) results: ˆQ = 2λc d /c s ˆT = ˆQ/λ, C( ˆQ) = 2λc d c s Q54. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 62 / 77

8 Deterministic model with deficit, type 1 LECTURE 10 Deterministic model with deficit (1) Deficit costs proportional to its volume and duration: Notation: c - deficit costs for 1 unit of good and 1 unit of time T - interval between deliveries (unknown) T 1 - duration of deficit Q - overall demand on interval [0, T ] S - unsatisfied demand Q S - satisfied demand Goal: C(T, T 1 ) min or C(Q, S) min Soňa Kilianová spring semester 2013 Lectures on SMOR 63 / 77

9 Deterministic model with deficit, type 1 Soňa Kilianová spring semester 2013 Lectures on SMOR 64 / 77

10 Deterministic model with deficit, type 1 Costs as function of Q, S for period T between deliveries: storage: c s (Q S) 2 2λ deficit: c S2 2λ delivery: c d overall costs per unit of time: C(Q, S) = [c s (Q S) 2 2λ Q55. Derive. + c S2 2λ + c d] λ Q Soňa Kilianová spring semester 2013 Lectures on SMOR 65 / 77

11 Deterministic model with deficit, type 1 Optimization: we look for Q, S such that C( Q, S) = min Q,S C(Q, S) results: S = Q cs c s+c (< Q) Q = ˆQ ( c +c s c ) 1/2 where ˆQ is optimal size in problem without deficit Q56. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 66 / 77

12 Deterministic model with deficit, type 2 Deterministic model with deficit (2) Deficit costs proportional to its volume only: Costs as function of Q, S for period T between deliveries: storage: c s (Q S) 2 2λ deficit: c S delivery: c d overall costs per unit of time: (Q S) C(Q, S) = [c 2 s 2λ + c S + c d ] λ Q Q57. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 67 / 77

13 Deterministic model with deficit, type 2 Optimization: we look for Q, S such that C(Q, S ) = min Q,S C(Q, S) results: { Q 2λcd /c = s if λ 2c d c s /(c ) 2, otherwise { 0 if λ S 2cd c = s /(c ) 2, otherwise Q58. Derive. Homework: 37, 38, 42. Soňa Kilianová spring semester 2013 Lectures on SMOR 68 / 77

14 Stochastic problem with signaling Notation: LECTURE 11 Stochastic models Stochastic problem with signaling λ - random consumption per unit of time λ = E(λ) r - signaling threshold (at which order is made) δ - duration needed for delivery ρ - average amount of inventory at the moment of delivery Q - size of order (delivery) Goal: E(C(Q, r)) min Soňa Kilianová spring semester 2013 Lectures on SMOR 69 / 77

15 Deterministic model with deficit, type 1 Soňa Kilianová spring semester 2013 Lectures on SMOR 70 / 77

16 Stochastic problem with signaling Computations: average volume of inventory at delivery: ρ = r δ λ E(C(Q, r)) = c s ( Q 2 + r δ λ) λ + c d Q + D(r, Q) where D(r, Q) is lost due to deficit Q59. Derive. Other notation: λ δ - random consumption on interval of length δ. Deficit appears when λ δ > r. f δ - probabilistic density of λ δ. Soňa Kilianová spring semester 2013 Lectures on SMOR 71 / 77

17 Types of deficit costs Two options for deficit: 1. proportional to number of deficits (how many orders per unit of time ends up in deficit, but independent of its size) D(r, Q) = c P(λ δ > r) λ λ Q = c Q r f δ (η)dη 2. proportional to average size of deficit D(r, Q) = c P(λ δ > r) λ Q E(λ δ r λ δ r > 0) = c λ Q r (η r)f δ (η)dη Q60. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 72 / 77

18 Stochastic problem with periodic control Stochastic problem with periodic control The size of inventory is controlled in regular time intervals T. Depending on the current state of inventory we re-stock to level R. Computations: overall expected costs per unit of time: E(C(R, T )) = c s (R λδ λ T 2 ) + c d 1 T + D(R, T ) Q61. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 73 / 77

19 The newsvendor problem LECTURE 12 The newsvendor problem - one-shot order Notation: D - random demand with density f Q - size of order (delivery) if D > Q, q is the lost profit per unit of good if Q > D, the unsold goods is sold with loss r Soňa Kilianová spring semester 2013 Lectures on SMOR 74 / 77

20 The newsvendor problem Computations: vendor s loss: q(d Q) of D > Q r(q D) of D < Q expected loss: E(C(Q)) = r Q (Q D)f (D)dD + q Q (D Q)f (D)dD Q62. Derive. Goal: E(C(Q)) min Homework: 48, 50, 52, 56, 57, 58. Soňa Kilianová spring semester 2013 Lectures on SMOR 75 / 77

21 A multi-item model Part I: Markov chains A multi-item model Notation: λ i - demand for goods i per unit of time c si - storage cost for unit of good i per unity of time c d - cost of order (delivery), independent of its size Q i - size of order of i-th good T - interval between deliveries Soňa Kilianová spring semester 2013 Lectures on SMOR 76 / 77

22 A multi-item model Part I: Markov chains Computations: overall costs (for delivery and storage) per unit of time: 1 C(T ) = c d T + N i=1 c λ i T s i 2 Optimization: we look for ˆT such that C( ˆT ) = min T C(T ) results: ( ˆT = 2c d N i=1 cs i λ i ) 1/2 Q63. Derive. Soňa Kilianová spring semester 2013 Lectures on SMOR 77 / 77