Optimization Using Matrix Geometric and Cutting Plane Methods Sachin Jayaswal Beth Jewkes Department of Management Sciences University of Waterloo & Saibal Ray Desautels 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 p i : price for class i L i : delivery time for class i λ i : 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 W i : waiting time (in queue + service) of class i S i : delivery time reliability level, P(W i <= L i )

10 Model Description Demand: –Exponential with rates λ 1 and λ 2 –price and delivery time sensitive

11 Mathematical Model 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 Relaxing the complicating constraint reduces the problem to a simple quadratic program with linear constraints (for a given value of L 1 ). The resulting values of the decision variables can be used in MGM to evaluate the service level of low priority customers (relaxed constraint).

14 Matrix Geometric Method for service level of low priority customers State Variables: –N 1 (t): Number of high priority customers in the system (including the one in service) –N 2 (t): Number of low priority customers in the system (including the one in service)

15 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 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 Sojourn Time Distribution of low priority customers in a pre-emptive priority queue as a function of p 1 and p 2

25 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.

26 Solution Approach Linear approximation of a concave function

27 Solution Algorithm Solve the relaxed quadratic program (QP) Using MGM compute service level S 2 for the values of p 1, p 2, and µ obtained from QP Compute approximate gradient to the curve using finite difference Add a tangent hyper- plane to the (QP) Is S 2 >= α? Stop yes No

28 Sample Results

29 Sample Results

30 Sample Results

31 Managerial Insights (Future Research) Impact of L 1 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 (β p, β L, θ p, θ L ) on leadtime and pricing decisions?

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