optimal stock allocation
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optimal stock allocation for a capacitatedsupply system
Francis de Véricourt, Fikri Karaesmen, Yves Dallery26.1.2015
Presentation by Sönke Matthewes and Ingo Marquart
Outline
1. Introduction
2. Model Set-up & Problem Definition3. Exploring the Optimal Solution3.1 What Value Functions make sense?3.2 Multilevel Rationing (ML) Policies3.3 Induction Proof Sketch
4. Illustration: Inventory Pooling w/ & w/o Optimal StockAllocation
1
Introduction
· What is stock allocation?common stock → different customer types· differentiated wrt the costs of not satisfying their demand (e.g. varyingimportance, supply contracts)
· Limited resources⇒ nontrivial optimisation problem· production constraints· holding costs· backorder costs
2
Introduction (cont.)
· Why common stocks?· delayed product differentiation· centralisation of inventories
· Net gains > 0? → Stock Allocation Problem is key!· Previous literature:· uncapacitated supply· two-class demand· lost sales instead of backordering
⇒ Research question: What is the optimal stock allocation policyto a single-server, single-product, make-to-stock queue withmultiple demand classes?
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Example
4
Components for our example
1. State vector x = (x1, x2, x3) ∈ RxR2−
2. Controls to be defined below, with policy functions π
3. Linear Costs per timec(x(t)) = hx+1 (t) + b1x
−1 (t)− b2x2(t)− b3x3(t)
4. Demand arrival rates λ1, λ2, λ3
5. Production rate µ ≥∑
i λi
6. This is a homogeneous rate, continuous DP problem Pn (heren = 3)
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Controls, or what we can do
For policy π, controls depend on which event happens. Production(C0) or arrival of demand k (Ck)
Cπ0 =
0, Produce nothing
1, Produce item and assign it to stock or demand 1
k, Produce and assign to demand k
(1)
Cπk =
1, Satisfy demand k
k, Backorder demand k(2)
Cπ1 = 1,Satisfy demand or backorder if inventory is empty (3)
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The Optimization Problem
DM seeks to find policies so
minπ
E[∫ ∞
0e−αt(hx+1 (t) + b1x
−1 (t)− b2x2(t)− b3x3(t))dt
]As in lecture this leads to
v∗(x) = c(x) + µT0v∗(x) + λ1T1v
∗(x) + λ2T2v∗(x) + λ3T3v
∗(x)
With the overall T operator s.th. Tv(x) is equal to the RHS. The Tk
operators code the controls, ex.
T2v(x) =
min [v(x− e2), v(x− e1)] if x1 > 0
v(x− e2), ifx1 ≤ 0
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What value functions make sense?
Define Un as follows:
1. v(x+ ei) < v(x) if xi < 0
⇔ better to satisfy demands; you have to produce2. v(x+ ei) < v(x+ ej) if 1 ≤ i < j
⇔ satisfying more expensive demands saves more costs⇒ if i is backordered, then i+ 1, ... , n have to be backordered aswell
3. v(x+ e1 − ej) < v(x+ e1 − ei) if 1 < i < j
⇔ backordering more expensive demands will cost you more⇒ if you satisfy j, then you have to satisfy j − 1, ... , 1 as well
Lemma 1. If v ∈ Un then Tv ∈ Un.8
Multilevel Rationing (ML) Policies
Definition 1. An ML policy π, is a policy characterized by the (n+1)dimensional rationing level vector z, wherez1 = 0 ≤ z2 ≤ ... ≤ zn+1, such that:
Cπ0 =
0, if x1 ≥ zn+1 and m(x) = n+ 1
k, if x1 ≥ zk and m(x) = k < n+ 1
1, if x1 < zk and m(x) = k
(4)
Cπk =
1, if x1 > zk and m(x) ≥ k .
k, if x1 ≤ zk(5)
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Multilevel Rationing (ML) Policies
Scenario 1: z4 > x1 > z3 > z2 > z1 = 0
⇒ all arriving demands are satisfied⇔ Cπ
k = 1, ∀k
Scenario 2: z4 > z3 > z2 > x1 > z1 = 0
⇒ classes 2 & 3 are backordered, only 1 immediately satisfied⇔ Cπ
k = k, for k = 2, 3 and Cπ1 = 1
(⇒ production for inventory: Cπ0 = 1 as x1 < zk until x1 = z2)
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Main Result 1
Theorem:
1. The optimal policy for Pn is a ML policy with rationing levels z
2. z is such that the projection (subvector) zk and corresponding MLpolicy also solves a transformed P̂n−1, k < n
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Sketch of proof
Define Vn ⊂ Un, for v(x) ∈ Vn have that
1. Costs ratios are preserved from solution to Pn−1 subproblem
2. Better to sell than accumulate if stock is big enough3. Conditions if only backorder is in n or none at all
· Benefit of selling vs stocking inc in stock, dec in demand· Benefit of stock decreases with n-type backorders· Returns are decreasing
4. Base level zn+1: (Future cost) optimal stock when no demand.If stock less than zn+1, production decreases costs
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Part 1 of Theorem
· Step 1. Assume v ∈ Vn, and optimal policies π∗k for k < n are
indeed ML· For v, actions pertaining to events k < n come from π∗
k →ML· Vn membership implies ML for event n, so ⇒ π is ML
· Step 2. Show that operator T preserves Vn (Lemma 2)· Take v ∈ Vn, verify conditions 1-4 for Tv· Optimal value f. exists in Vn by iteration with T (Banach FP)
· Step 3. Finish proof by induction on n· For n = 1, optimal policy is in Vn and ML· Assume true for n− 1, by above Tv for n is ML and in Vn
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Part 2 of Theorem
· Consider transformed problem P̂n−1 to Pn
· Event of n is uniformized out via transformation of the cost c → d,eliminating n and such that d2(d3) = d2
· Dynamics for the transformed subproblem are same as the generalPn−1
· Knowing that πn is ML, recursively move back via dn−1(dn) etc.
· For each step, use part 1 of theorem, where zn−1 is ML, for Pn−1
and hence P̂n−1
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Illustration
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Illustration (cont.)
· g0 = initial costs under standard base-stock policy
· g1 = costs after redesign under standard but suboptimal allocation
· g2 = costs after redesign under optimal ML policy
· Let:∆1 =
g0−g1g0
and ∆2 =g0−g2g0
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Illustration (cont.)
17
Thank you
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Appendix 1: Definitions and distributions
· Time to demand arrival τλiis exponentially distributed: rate λi,
expected time to arrival 1λi
· Similarly µ ≥∑
i λi is the production rate parameter
· These events can happen in each state
· Time to transition τ thus distributed ν =∑
i λi + µ sinceτ = min{(τλi
)i, τµ}
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Appendix 2: T-Operators
T0v(x) = min1<i≤n
[v(x), v(x+ e1), v(x− ei Ixi<0)]
T1v(x) = v(x− e1)
Tkv(x) =
min [v(x− ek), v(x− e1)] if x1 > 0
v(x− ek), ifx1 ≤ 0
for k s.t. 1 < k ≤ n.
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