Upload
aleesha-robbins
View
216
Download
2
Tags:
Embed Size (px)
Citation preview
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?
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)
10
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 :
11
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?
12
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
13
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
14
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)
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?
24
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
25
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.
27
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
28
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
29
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
30
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
31
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?