Optimization Using Matrix Geometric and

Cutting Plane Methods

Sachin Jayaswal

Beth JewkesDepartment of Management Sciences

University of Waterloo

&

Saibal RayDesautels Faculty of Management

McGill University

2

Outline

• Motivation

• Model Description

• Mathematical Model

• Solution Approach

• Sample Results & Insights

• Further Research

3

Motivation

• A firm selling 2 substitutable products

• Market sensitive to price and time

• How to price the two products?

4

Real-Life Situations

Courier Service– FedEx Ground– FedEx Custom Critical

5

Real-Life Situations

Online shopping– Express Delivery– Priority Delivery

6

Real-Life Situations

Call Centers– Ordinary Calls– Priority Calls

7

Problem Statement

• A firm selling 2 substitutable products:– 1: priority product– 2: normal product

• Market sensitive to price and time

• Shared production capacity

• Industry standard delivery time for product 2

• Decisions:– Delivery time guarantee for product 1? – Prices for product 1 and product 2?

8

Model Description

• A 2-class pre-emptive priority queue• Class 1 served in priority over class 2 and

charged a premium for shorter guaranteed delivery time

9

Notations

• pi : price for class i• Li : delivery time for class i• λi : demand rate (exponential) for class i • µ : service rate (exponential)• m : unit operating cost• Π : profit per unit time for the firm• A : marginal capacity cost• Wi: waiting time (in queue + service) of class i• Si : delivery time reliability level, P(Wi <= Li)

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Model Description• Demand:

– Exponential with rates λ1 and λ2

– price and delivery time sensitive

1211211 LLLpppa LLpp

2122122 LLLpppa LLpp

demanditivity ofprice sensp :

e differencards pricehovers towy of switcsensitivitp :

differencevery time nteed deliards guarahovers towy of switcsensitivitL :

emandivity of dime sensitdelivery tL :

productze for the market si potentiala :

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Mathematical Model

Aμλmpλmp

, μ, μ, L, ppMaximize Π

2211

21121

00

0

1

212121

21

222

111111

, λ, λLm, Lm, p p

μλ λ

αLW P S

α eLWP S

:subject toLλμ

How to express this constraint analytically? This can be evaluated numerically using matrix-geometric method

(MGM).

How to use the numerical results in

mathematical model for optimization?

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Solution Approach: Literature Review

• Atalson, Epelman & Henderson (2004): Call center staffing with simulation and cutting plane methods

• Henderson & Mason (1998): Rostering by integer programming and simulation

• Morito, Koida, Iwama, Sato & Tamura (1999): Simulation based constraint generation with applications to optimization of logistic system design

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Solution Approach

Aμλmpλmp

, μ, μ, L, ppMaximize Π

2211

21121

00

0

1

212121

21

222

111111

, λ, λLm, Lm, p p

μλ λ

αLW P S

α eLWP S

:subject toLλμ

Relaxing the complicating constraint reduces the problem to a simple quadratic program with linear constraints (for a given value of L1). The resulting values of the decision variables can be used in MGM to evaluate the service level of low priority customers (relaxed constraint).

1

1

1ln

L

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Matrix Geometric Method for service level of low priority customers

• State Variables:– N1(t): Number of high priority customers in the

system (including the one in service)

– N2(t): Number of low priority customers in the system (including the one in service)

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Matrix Geometric Method for service level of low priority customers

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Rate Matrix

17

Rate Matrix

18

Rate Matrix

19

Matrix Geometric Method

20

Matrix Geometric Method

21

Matrix Geometric Method

22

Service level of low priority customers

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Solution Approach

Aμλmpλmp

, μ, μ, L, ppMaximize Π

2211

21121

00

0

1

212121

21

222

111111

, λ, λLm, Lm, p p

μλ λ

αLW P S

α eLWP S

:subject toLλμ

If the relaxed constraint function is concave, it can be linearized by using an infinite set of hyper planes

Is it really concave, how do we know?

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Solution Approach

1213

1415

16

7

8

9

10

110.9992

0.9994

0.9996

0.9998

1

ph

pl

P(W

l <= L

l)

Sojourn Time Distribution of low priority customers in a pre-emptive priority queue as a function of p1 and p2

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15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 200.985

0.99

0.995

1

mu

Sl

Sojourn Time Distribution of low priority customers vs. service rate

Solution Approach

Convinced about the joint concavity of the function?Not yet?We will numerically check for concavity assumption in the algorithm.

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Solution Approach

Linear approximation of a concave function

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Solution Algorithm

Solve the relaxed quadratic program (QP)

Using MGM compute service level S2 for the values of p1, p2, and µ obtained from QP

Compute approximate gradient to the curve using finite difference

Add a tangent hyper- plane to the (QP)Is S2 >= α?Stop

yes

No

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Sample Results

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

10

15

20

Lh

Tot

al p

rofit

0.5 1 1.5 2 2.5 3 3.5 4 4.5 513.4

13.6

13.8

14

14.2

14.4

14.6

14.8

15

Lh

p h

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Sample Results

0.5 1 1.5 2 2.5 3 3.5 4 4.5 59.5

10

10.5

11

11.5

12

12.5

Lh

p l

0.5 1 1.5 2 2.5 3 3.5 4 4.5 56

7

8

9

10

11

12

13

Lh

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Sample Results

0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

Lh

h

0.5 1 1.5 2 2.5 3 3.5 4 4.5 52.2

2.4

2.6

2.8

3

3.2

3.4

3.6

Lh

l

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Managerial Insights (Future Research)

• Impact of L1 on relative pricing and total profit?

• Impact of A on pricing decisions ?

• Impact of a shared production capacity on pricing decisions and total profit?

• Role of market characteristics (βp, βL, θp, θL) on leadtime and pricing decisions?

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