University:
Massachusetts Institute of Technology
Course:
15.992 | S-Lab: Laboratory for Sustainable Business
Academic year:

2023
Views:

328

Pages:

31
Author:

Jurazeitkr

Managing Demand and Sales Dynamics in New Product Diffusion Under Supply Constraint Motivation (Newspaper clipping about Apple Computer’s worldwide launch of the iPod Mini to strong demand) • How will limited supply affect overall demand? • How will the competition’s reaction affect demand? • How long should Apple delay the global launch? • Should Apple have delayed original launch? Agenda • • • • Introduction Overview Related Literature Methodology – – – – Model Formulation Optimal Sales Plan Supply-Constrained New Product Diffusion Optimal Supply Decisions • Discussion • Critique • Questions / Discussion Introduction Operations Literature – Capacity sizing when launching a new product – Specify an exogenous demand trajectory – Assumes: Capacity does not affect Demand Marketing Research – Characterization of the demand process – Social Diffusion Process including internal and external factors (price, advertising, population, etc.) – Assumes: Unconstrained Supply How does a new product diffuse in the presence of a supply constraint? Overview Joint Analysis of supply-related decisions and demand dynamics • Improved characterization of the constrained demand and sales dynamics – Back-ordering vs. Lost Customers – Generalized diffusion model distinguishes between the demand process and sales process (min of demand and available supply) • Improved Operational Planning – Capacity planning (Cost of backordering and lost customers vs. overcapacity) – Launch decision (MTS Æ delayed launch = preproduction) – Should we sell less than is currently demanded (given sufficient supply)? Related Literature • Analysis builds on the traditional Bass Model of new product diffusion (Bass, 1969) – – – Widely used in marketing to forecast demand New product demand follows patterns of social diffusion processes similar to those in epidemiology and the natural sciences (e.g. SIR epidemic model) Bass diffusion Model Basics: Potential Adopters subject to two means of communication: • External Influence (mass-media communication Æ advertising) • Internal Influence (word-of-mouth) • Related Research – Jain, et al., 1991: Diffusion of telephone service in Israel • • • – Kurawarwala and Matsuo, 1998 • • – No competition = no customer losses Capacity grows with backorders (assumed short lead time for capacity expansion) Supply constraint is always binding Æ sales trajectory mirrors capacity Model of procurement with Bass-like demand process with known parameters of internal and external influence, unknown market size Extended newsvendor model Fine and Li, 1988 • • • Conditions for switching from one supply process to another during product life-cycle Assume demand with symmetrical growth and decline stages Assume that process switching will not influence the underlying demand dynamics (i.e. they assume that demand is exogenous to the model) Methodology • • • • Model Formulation Optimal Sales Plan Supply-Constrained New Product Diffusion Optimal Supply Decisions Model Formulation The Firm: • Introducing a new product (e.g. Mini-IPOD) – Short Lifecycle – Long lead times • Key Decisions: – Capacity sizing (Assumes constant c throughout the product life cycle) – Time to Market (tl >= 0) – Sales Plan s(t) Customer Diffusion D(t ) = S (t ) + W (t ) + L(t ) Unconstrained Supply: W(t)=0, L(t)=0 Æ D(t) = S(t) Else: dL(t ) = lW (t ) dt Demand Process Arrival of customers orders follows Bass-like dynamics: dD(t ) q = p[m − D(t )] + S (t )[m − D(t )] m dt External Influence: • Innovation Dynamics • Advertising • Internal Influence: •Interaction Dynamics •Word-of-Mouth Assumes a uniqueness of the new product – New brand, new product category (movies, video game console, Pentium III) • Allows for customer losses (i.e. Does not require monopoly) – Cross-brand or cross-category substitution Production Connecting Demand to the Supply Process: Total Production: Production Rate: R(t ) = I (t ) + S (t ) ⎧ c, t < t * ⎪ r (t ) = ⎨ dD(t ) * ≥ , t t ⎪⎩ dt t = min(t dD(t ) / dt < c, d D(t ) / dt < 0) * 2 2 Choosing a Sales Rate: s(t) Objective: maximize life-cycle discounted profits, with c and tl fixed Profit Function: ⎛ +∞ ⎞ − θ t P (c, tl ) = max⎜ ∫ (a (t ) s (t ) − hI (t ))e dt {I (tl ) = ctl }⎟, a (t ) > 0 s ( t )≥0 ⎜ ⎟ t ⎝ l ⎠ Simplified by shifting the time origin to tl: ⎛ +∞ P (c, tl ) = max⎜⎜ ∫ (a(t ) s (t ) − h I (t ))e −θt dt I (0) = ctl s ( t )≥0 ⎝0 { ⎞ ⎟ ⎟ ⎠ } a (t ) = a (t + tl ), s (t ) = s (t + tl ), I (t ) = (t + tl ) tl ≥ 0 Choosing a Launch Time: tl>0 Given: the optimal selling plan s*(t) Discounted pre-launch inventory costs tl hc ⎛ 1 −θt −θt ⎞ h ∫ cte dt = ⎜ (1 − e ) − tl e ⎟ θ ⎝θ ⎠ 0 −θt Objective: Maximize Profits ⎛ hc ⎛ 1 −θt −θt ⎞ ⎞ P (c) = max⎜⎜ P(c, tl ) − ⎜ (1 − e ) − tl e ⎟ ⎟⎟ tl ≥ 0 θ ⎝θ ⎠⎠ ⎝ * Choosing Production Capacity: c Given: the optimal selling plan s*(t) and the launch time tl ( max = P (c) − Hc c * ) Optimal Sales Plan Tactical Decision: choosing the sales rate s(t) to maximize profits with c and tl fixed Profit Function: ⎛ +∞ ⎞ −θt P (c, tl ) = max⎜⎜ ∫ (a (t ) s (t ) − hI (t ))e dt ⎟⎟ s ( t )≥0 ⎝0 ⎠ Proposition 1. For any profit margin a(t)>0, holding cost h>0, and launch time tl >= 0, the optimal sales rate is given by: ⎧ r (t ), W * (t ) > 0 ⎪ * s (t ) = ⎨min(r (t ), d * (t )), I * (t ) = 0,W * (t ) = 0 * * ⎪ ( ), (t ) > 0 d t I ⎩ Where: d*(t), I*(t) and W*(t) are optimal values I*(t)W*(t) = 0 Bottom Line: The firm should always favor the immediate sale. Supply-Constrained New Product Diffusion Given the optimal sales plan: • Specify the demand D(t) and sales S(t) dynamics and compare to the unconstrained Bass demand dynamics • Obtain an expression for discounted profits IOT determine the optimal capacity c and time to market tl Consider two cases: • Patient Customers (l = 0) • Impatient Customers (l > 0) Patient Customers, L(t)=0 D(t ) = S (t ) + W (t ) R(t ) + ctl = S (t ) + I (t ) Solve this set of equations given: dD(t ) q = p[m − D(t )] + S (t )[m − D(t )] m dt * ⎧ , < c t t dR(t ) ⎪ = ⎨ dD(t ) * , ≥ t t dt ⎪⎩ dt D(0) = S (0) = R (0) = 0 ⎧ ⎪ c, W (t ) > 0 dS (t ) ⎪⎪ ⎛ dD(t ) ⎞ I (t ) = 0, = ⎨min⎜ c, ⎟, dt dt ⎠ W (t ) = 0 ⎝ ⎪ dD(t ) ⎪ I (t ) > 0 , ⎪⎩ dt t * = min(t dD(t ) / dt < c, d 2 D(t ) / dt 2 < 0) Analyzing the diffusion process for any chosen capacity c and launch time tl, produces three different regimes: Regime 1: Unconstrained Diffusion (UD) Regime 2: Initially Unconstrained Diffusion (IUD) Regime 3: Initially Constrained Diffusion (ICD) Unconstrained Diffusion (UD) c and tl are high enough to ensure that W(t)=0 for all t. Even with tl = 0, capacity c could be sufficient to ensure unconstrained Bass diffusion is preserved • What is the smallest capacity level required? or • Given c, what is the earliest launch time? Unconstrained Diffusion (UD) Determining the smallest capacity required to sustain UD. τ + = max(τ c = d Bass (τ )) pm(q + p ) 2 exp(( p + q )τ + ) c= (q + p exp(( p + q )τ + )) 2 ⎛ c ⎞ ⎜1+ 1− * ⎟ co ⎟ ⎜ q 1 1 ln( ) + ln⎜ τ+ = ⎟ p+q p p+q c ⎜ 1− 1− * ⎟ ⎟ ⎜ c o ⎠ ⎝ co* = m( p + q ) 2 / 4q m( q − p ) m( p + q ) c + DBass (τ + ) = 1− * co 2q 2q For tl = 0, the smallest capacity required cs* ( p, q, m) is determined as the capacity c where cτ + = DBass (τ + ) It follows: cs* ( p, q, m) < co* ( p, q, m) Unconstrained Diffusion (UD) Determining the earliest launch time that sustains UD. Lemma 1. For a given c, unconstrained diffusion is sustained IFF: tl > tl* (c) The critical launch time is a nonincreasing function of c: ∂t (c) / ∂c ≤ 0 * l Provides the level of preproduction that avoids any supply shortages over the entire life cycle. ⎧ 0, ⎪ ⎪ ⎪ m( q − p ) m( p + q ) + ⎪ 2qc ⎪ 2qc ⎪ ⎪− 1 ln⎛⎜ q ⎞⎟ * t l (c ) = ⎨ ⎜ p⎟ p q + ⎝ ⎠ ⎪ ⎪ ⎛ c ⎜1+ 1− * ⎪ co 1 ⎜ ⎪ ln ⎪− p + q ⎜ ⎜ 1 − 1 − c* ⎪ ⎜ co ⎪⎩ ⎝ c ≥ cs* 1− c co* c < cs* ⎞ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠ Initially Unconstrained Diffusion (IUD) For a given c and 0 < tl < tl* pre-launch inventory is insufficient to support Bass diffusion over the entire life cycle of the product. Given a finite amount of inventory at t = 0: • It is possible to sustain an UD for a finite duration • The diffusion process goes through 3 phases: – Initial Unconstrained Bass Diffusion (UP1) – Constrained Diffusion (CP) – Second Unconstrained Bass Diffusion (UP2) Initially Unconstrained Bass Diffusion (UP1) Demand and Sales are identical and increasing: s(t)=d(t), ds(t)/dt>0 ⎡ exp(( p + q )t ) − 1 ⎤ D(t ) = S (t ) = pm ⎢ ⎥ ⎣ q + p exp(( p + q )t ) ⎦ W (t ) = 0 This phase lasts until production + inventory can no longer sustain the unconstrained diffusion. We can define the ending time of this phase: ⎛ ⎞ ⎛ ⎞ + q p ⎟⎟ ⎟ τ 1 = min⎜τ c(τ + tl ) = m⎜⎜1 − ⎜ ⎝ q + p exp(( p + q )τ ) ⎠ ⎟⎠ ⎝ Constrained Bass Diffusion (CP) Customers are waiting and sales rate is equal to capacity: W(t) > 0, dS/dt = c Given : D(τ 1 ) = S (τ 1 ) = D1 Note: ⎡ ⎛⎛ D1 ⎞ qc (t − τ 1 ) 2 ⎞⎤ ⎟⎟⎥, D(t ) = m − (m − D1 ) exp ⎢− ⎜⎜ ⎜ p + q ⎟(t − τ 1 ) + 2 ⎠⎦ m⎠ m ⎣ ⎝⎝ S (t ) = D1 + c(t − τ 1 ), s (t ) = c s (t ) ≠ d (t ) ⎛ ⎛ ⎛ D1 ⎞ qc(t − τ 1 ) 2 ⎞ ⎞ ⎟⎟ ⎟ W (t ) = −c(t − τ 1 ) + (m − D1 )⎜⎜1 − exp⎜⎜ − ⎜ p + q ⎟(t − τ 1 ) − ⎟ 2 m m ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ This phase lasts until there are no customers waiting. Ending time of this phase: τ 2 = min (t t > τ 1 , W (t ) = 0 ) Second Unconstrained Bass Diffusion (UP2) Demand and Sales are identical and decreasing: s(t)=d(t), ds(t)/dt<0 (m − D2 )( p + q) D(t ) = S (t ) = m − q q q − D2 + ( p + D2 ) exp(( p + q )t − τ 2 )) m m W (t ) = 0 D2 = D (τ 2 ) Lemma 2. Peak Demand and Sales Rates τ S max ≤τ D max for all values of production capacity τ τ D max S max ⇒ d (τ ⇒ s (τ D max S max ) ) Initially Constrained Diffusion (ICD) tl = 0 and c < initial inflow of potential adopters (pm). Initial constrained diffusion later replaced by unconstrained Bass process. Behaves much like the 2nd and 3rd phase of the IUD regime. W (t ) > 0, 0 < t < τ 2 W (t ) = 0, t ≥τ2 Lemma 3. Demand and Sales Dynamics in ICD Regime τ D max ⇒ d (τ D max ) Maximum sales rate is equal to c. 2 ⎛ ⎡ ⎛ ⎞⎤ ⎞⎟ qc t ⎜ ⎟⎟⎥ , D(t ) = m 1 − exp ⎢− ⎜⎜ pt + ⎜ ⎟ m 2 ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ S (t ) = ct , 2 ⎛ ⎛ ⎛ ⎞ ⎞ ⎞⎟ qct ⎜ ⎟⎟ ⎟ , W (t ) = −ct + m 1 − exp⎜⎜ − ⎜⎜ pt + ⎟⎟ ⎜ m 2 ⎝ ⎠ ⎝ ⎠⎠ ⎝ τ 2 = min(t t > 0, W (t ) = 0) Impatient Customers, l > 0 c W (t , l ) = − (1 − exp(−l (t − τ 1 ))) l + (m − D1 ) exp(−l (t − τ 1 )) 2 ⎛ ⎛ ⎛~ ⎞ ⎞ ⎞⎟ − τ ( ) qc t 1 ⎜ ⎜ ⎟⎟ ⎟ × 1 − exp⎜ − ⎜⎜ p (t − τ 1 ) + ⎟⎟ ⎜ 2 m ⎝ ⎠ ⎝ ⎠⎠ ⎝ + (m − D1 ) exp(−l (t − τ 1 )) ⎛ 2πm m~ p2 × ⎜⎜ l exp qc 2qc ⎝ ⎛ ⎛ qc m ~⎞ p ⎟⎟ × ⎜ Φ⎜⎜ (t − τ 1 ) + ⎜ qc ⎠ ⎝ ⎝ m ⎛ m ~ ⎞ ⎞ ⎞⎟ p ⎟⎟ ⎟ − Φ⎜⎜ ⎟ ⎝ qc ⎠ ⎠ ⎟⎠ D(t , l ) = m − (m − D1 ) ⎡ ⎛⎛ D1 ⎞ qc (t − τ 1 ) 2 ⎞⎤ ⎟⎟⎥, × exp ⎢− ⎜⎜ ⎜ p + q ⎟(t − τ 1 ) + m m⎠ 2 ⎠⎦ ⎣ ⎝⎝ S (t , l ) = D1 + c(t − τ 1 ), L(t , l ) = D(t , l ) − S (t , l ) − W (t , l ) Proposition 2. NP diffusion dynamics subject to customer loss behave as outlined by Lemmas 1-3. Unconstrained phases unchanged, constrained phases described: Enables firm to track the fraction of lost customers at any time. Impatient Customers (cont) Proposition 3. The length of the constrained phase is given by: Tc (l ) = τ 2 (l ) − τ 1 and is a decreasing function of l: ∂Tc (l ) <0 ∂l Suggests that the duration of the constrained phase decreases as customer impatience increases. Impatience = fn (Competition) Proposition 4. The fraction of customers lost is given by: ⎛ ⎡ ⎛ qc Tc2 (l ) ⎞⎤ ⎞⎟ ⎜ ⎟⎟⎥ − cTc (l ) (m − D1 ) 1 − exp ⎢− ⎜⎜ pTc (l ) + ⎜ m 2 ⎠⎦ ⎟⎠ ⎣ ⎝ ⎝ f (l ) = m and is an increasing function of l: ∂f (l ) >0 ∂l Optimal Supply Decisions Given: characterizations of the demand and sales dynamics Tactical Decision: choose the optimal capacity and time to launch +∞ Profit Function: P (c, tl ) = ∫ (as (t ) − hI (t ))e −θt dt 0 Proposition 5. Performs the integration resulting in a “seemingly” complex equation for profit function which can be used for computing the optimal capacity and time to market. Discussion • • • It is important to include supply constraints in the estimation of diffusion parameters Increase in pre-production (delaying product launch) can act as a substitute for capacity Shows how optimal time to market and capacity vary with diffusion parameters – Timing and capacity are more sensitive to imitation (q) than innovation (p) – Optimal capacity exhibits saturation effect as the speed of diffusion increases • • Value of endogenizing demand in supply-related decisions can be substantial Informs operational decision-making – Develop more accurate forecasts of demand – Challenges assumption that demand forecasts merely serve as inputs to operations planning processes and are not affected by supply decisions – Suggests it is optimal to pre-produce • Future Research – Estimation of diffusion parameters – Using marketing mix variables to influence diffusion – Waiting time dynamics Major Contributions (to the Operations and Marketing literature) • Developed closed-form expressions of demand and sales dynamics in a Bass-like diffusion environment with a supply constraint • Integrated capacity, time to market, and sales plan into a unified decision hierarchy • Endogenized demand dynamics in determining the optimal capacity in a constrained diffusion environment Critique • Aggressive agenda, but… – Needs a more robust application • 52 equations in ~18 pages – Didn’t provide the intuition for some of the math • Proof of key “finding” s*(t) not included • Missing graphs (3a-3b) referred to in paper • Relies on many constant parameters that would not be constant in real life (e.g. loss rate) • Many other factors “rolled up” in innovation parameter (e.g. price, competition, etc.) • Offers some operationally useful ideas