Optimization Using Matrix Geometric and Cutting Plane Methods Sachin Jayaswal Beth Jewkes Department of Management Sciences University of Waterloo & Saibal.

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

16

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

23

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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