Product Pricing, Lead Time and Capacity Selection in Price and Time Sensitive Markets

Transcription

1 Product Pricing, Lead Time and Capacity Seection in Price and Time Sensitive Markets SACHIN JAYASWAL Department of Management Sciences University of Wateroo, Canada joint work wit Eizabet Jewkes¹ and Saiba Ray² INFORMS Annua Meeting 2007, Seatte, USA ¹Department of Management Sciences, University of Wateroo, Canada ²Facuty of Management, McGi University, Canada

2 Motivation Less tan 30% of corporate-customers and ess tan 10% of end-customers base teir purcasing decisions ony on an item s seing price. (Baker et a., 2001) 2

3 Motivation Te rest aso care about oter customer service attributes (Deivery Time, Deivery reiabiity, etc.) 3

4 Price & Time Preferences FedEx Ground FedEx Custom Critica 4

5 Probem Statement Market sensitive to Price and Time Different segments of customers ave different sensitivities to Price and Deivery Time. Decisions: Deivery time guarantee for eac segment? Price carged to eac segment? Ddi Dedicated dvs. Sared capacity? Objective: Maximize profit 5

6 Mode Description Hig priority () customers Low priority () customers A 2-cass pre-emptive priority queue Cass served in priority over cass, and carged a premium for sorter guaranteed deivery time 6

7 Notations p i : price for cass i L i : deivery time for cass i λ i : demand rate (exponentia) for cass i µ : service rate (exponentia) m : unit operating cost Π : profit per unit time for te firm A : margina capacity cost W i : waiting time (in queue + service) of cass i S i : deivery time reiabiity eve, P(W i <= L i ) α : deivery time guarantee 7

8 Mode Description Demand: Exponentia wit rates λ 1 and λ 2 Price and deivery time sensitive λ = a β p + θ ( p p ) β L + θ ( L ) 1 p 1 p 2 1 L 1 L 2 L1 λ = a β p p + θ ( p p ) β L + θ ( L ) 2 p 2 p 1 p 2 L 2 L 1 L 2 a : β p : β p potentia market size for te price sensitivity of demand product β L : deivery time sensitivity of demand θ p : sensitivity of switcovers towards price p difference θ L : sensitivity of switcovers towards guaranteed deivery time difference 8

9 Matematica Mode Maximize Π = subject to: λ p S S ( p, p, L, μ) ( p m) λ + ( p m) λ Aμ (1) = P W = P W ( μ λ ) ( L ) = 1 e ( L ) α + λ μ < 0 m, p m, L L 1 α L, λ 0, λ 0 (2) (3) (4) () (5) 9

10 Soution Approac μ λ Maximize Π = subject to: ( 1 α ) S L ( p, p, L, μ) ( p m) λ + ( p m) λ Aμ (1) n ( μ λ ) ( ) S λ p = = P W = 1 e ( L ) α (3) P W L + λ μ < 0 m, p m, L L, λ L 0, λ α (2) (4) 0 (5) If S is concave, it can be approximated using a set of tangent yperpanes, and te resuting probem can be soved using a cutting pane metod d(keey 1990) 10

11 Soution Approac 1 1 P(W <= L ) p p mu ( p p ) S vs, vs μ S S Linear approximation of constraint (3) at (6) 11

12 Markov Cain Representation State Variabes: N (t): Number of ig priority customers in te system N (t): Number of ow priority customers in te system - 2-dimensiona i Markov process State space: 12

13 Rate Matrix 13

14 Matrix Geometric Metod Obtain stationary distribution of using Matrix Geometric Metod (Neuts, 1981) Uniformize Markov Cain to obtain waiting time distribution of ow priority customer (Latouce and Ramaswami, 1999) Obtain gradients of using finite difference Obtain a tangent yperpane: 14

15 Cutting Pane Agoritm 15

16 Market Scenarios Price and time difference sensitive (PTD) market: Time and price difference sensitive (TPD) market: NitrPTD Neiter nor rtpd market: rkt 16

17 Sared or Dedicated Capacity? SC DC π L Profits vs. Lead Time of ig priority Compared to dedicated capacities, saring capacities resuts in a iger profit, atoug it may not necessariy resut in a iger tota revenue. Tis gain in profit margins is maximum under a TPD market. 17

18 Sared or Dedicated Capacity? Prices vs. Lead time of ig priority Compared to dedicated capacities, saring capacities resuts in a generay iger optima price for ig priority customers and a ower price for ow priority customers, for a given time differentiation. 18

19 Sared or Dedicated Capacity? Hig and ow priority Demand vs. Lead Time Compared to dedicated capacities, saring capacities aows a firm to capture a greater tota market sare, if te deivery time differentiation aready estabised by te industry is sufficienty arge. Atasmaer time differentia, asiftto sared capacity environment may cause a partia oss of its sare of tota demand. Tota demand vs. Lead Time 19

20 Sared or Dedicated Capacity? Market Type SC DC L-L* p*-p* L-L* p*-p* PTD Neiter TPD Price and Time Differentiation vs. Market Conditions Compared to dedicated capacities, saring capacities aows for a greater price and deivery time differentiation if a firm can seect its optima deivery time. 20

21 Market Conditions Price differentiation vs. Lead time of ig priority A PTD market aows for te greatest price differentiation, wie a TPD aows forte east. es. 21

22 Capacity Costs Lead Time Differentiation Price Differentiation Capaity Cost Capacity Cost Time and Price Differentiation vs. Capacity Costs As te capacity gets more expensive, firm soud differentiate ess on ead times and more on prices. 22

23 Future Researc Extend te mode to a competitive setting Computing Nas equiibrium i a major caenge 23

24 24

25 Soution Approac: Literature Review Atason, Epeman & Henderson (2004): Ca center staffing wit simuation and cutting pane metods Henderson & Mason (1998): Rostering by integer programming and simuation Morito, Koida, Iwama, Sato & Tamura (1999): Simuation based constraint generation wit appications to optimization of ogistic system design 25

26 Rate Matrix 26

27 Matrix Geometric Metod Stationary probabiity vector x: Partition x by eves into sub vectors xi, were x can be obtained using te baance equations, given in matrix form by: 27

28 Matrix Geometric Metod Te above set of equations can be soved recursivey using te reation (Neuts, 1981): R is te minima non-negative soution to te matrix quadratic equation: Te matrix R can be computed using we known metods (Latouce and Ramaswami, 1999): 28

29 Matrix Geometric Metod Te probabiities are determined from te boundary condition: subject to te normaization condition: 29

30 Matrix Geometric Metod for service eve of ow priority customers Time spent in te system by a ow priority customer is te time unti absorption in a modified Markov process, obtained by setting. 30

31 Matrix Geometric Metod for service eve of ow priority customers can be computed more convenienty by uniformizing te Markov process wit a Poisson process of rate γ, were 31

32 Service eve of ow priority customers For given prices and service rate priority customers can be computed as:, service eve for ow 32