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

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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

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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

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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.)

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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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