Yale School of Management · DO NOT DISTRIBUTE Dynamic Release Management of Product Features Daniel Adelman, Angelo J. Mancini University of Chicago Booth School of Business, Chicago,

DO NOT DISTRIBUTE

Dynamic Release Management of Product Features

Daniel Adelman, Angelo J. ManciniUniversity of Chicago Booth School of Business, Chicago, IL 60637

[email protected], [email protected]

We consider a release manager who sequentially releases new versions of her product by drawing from a fixedand non-replenishable finite set of features while facing an exogenous, stochastically evolving marketplace.In the absence of fixed costs, we provide conditions under which there exists a quasi-open-loop optimalpolicy, i.e., an optimal policy that depends only on the set of available features and not the state of theexternal market. Computing such a policy amounts to solving a related deterministic release-sequencingproblem, and we apply an exchange argument to obtain an index condition necessary for optimality. In thesimplest case we consider, a heuristic based on this condition reduces to an optimal ordering of Gittins indicesand is equivalent to the “weighted discounted shortest processing time first” stochastic scheduling rule.However, we prove it is suboptimal by an arbitrarily large margin in other settings. We apply approximatedynamic programming (ADP) to address the case with positive fixed costs by making a novel value functionapproximation motivated by the case without fixed costs. The resulting policy can provably outperform anintuitive certainty-equivalent heuristic by an arbitrarily large margin, and performs within 3.5% of optimalityacross a range of numerical trials.

Key words : product development; release management; semi-Markov decision processes; approximatedynamic programming

History : First version: May 30, 2014; Second version: October 13, 2014

1. Introduction A profit-maximizing firm must regularly release new versions of its productto provide new functionality to consumers. The managers issuing these releases must balance severaltradeoffs. Ideally, the firm would instantaneously issue releases with all available functionality. Inreality, features take time to implement, and implementation time often increases in the desirabilityof the feature. Longer development cycles not only postpone the realization of revenues, but alsoincrease the likelihood that changes in the marketplace will diminish the release’s value. In addition,managers must consider the fixed cost associated with issuing a release, e.g., testing, training andmarketing expenses. The goal of this paper is to provide release managers with a tactical modelthat negotiates these tradeoffs. Our motivation comes from the software industry, but extends toany sector in which firms must continually release udpated products.

The existing product development literature typically models product quality as a real-valuedscalar, multi-dimensional vector, or with a binary “high/low” classification. However, in softwaredevelopment releases consist of sets of discrete features that often can not be easily prioritized. Wetherefore represent the release manager’s problem as a combinatorial release-sequencing problemembedded within a semi-Markov decision process (SMDP) that accounts for a stochastic market-place. To the best of our knowledge, no such formulation appears in either the product developmentor project scheduling literatures.

By casting the release management problem as a special case of the SMDP in Adelman andMancini (2014), we decompose the problem into the exogenous market dynamics over which therelease manager exerts no influence, and the endogenous release-sequencing dynamics entirelywithin her control. This approach has yet to appear in the literature, and illustrates how to enrich asequencing problem by accounting for an exogenous marketplace without sacrificing tractability. Inorder to obtain this decomposition, we characterize the external marketplace in terms of its overall

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intensity. As the market becomes more intense over time, the release manager’s product gener-ates lower revenues unless new functionality is added to compensate. Possible drivers of marketintensity include changing consumer preferences, increases in the utility of consumers’ best outsideoption, changing macroeconomic factors, or some combination of these forces. Formally, marketintensity evolves as a nondecreasing scalar-valued multiplicative compound Poisson processes thatis uninfluenced by the release manager’s product or actions. While the compound Poisson modeladmittedly oversimplifies market dynamics, its parsimony eases the burden of applying our results.Based on discussions with managers at a large software firm, it is also consistent with the amountof market information normally available to tactically-focused release managers. If market shocksinduce geometric decay in the rate at which revenues accumulate to the release manager’s firm,and there are no fixed development costs, Adelman and Mancini (2014) reduces the SMDP to adeterministic dynamic program that accounts for market stochasticity with a modified discountfactor. This discount factor depends only on the statistical properties of the external market, andincreases in both the average frequency and expected magnitude of market shocks. The solution tothis dynamic program is an optimal release sequence that, without loss of optimality, the releasemanager can implement without monitoring the evolution of the marketplace. In this paper, weleverage the combinatorial structure of the release management problem to analyze the reduceddynamic program furnished by Adelman and Mancini (2014) for the case with no fixed costs. Wethen apply our insights to the case with positive fixed costs, which lies outside the scope of Adelmanand Mancini (2014).

When there are no fixed costs, we show that the endogenous dynamics of the release-sequencingproblem and the exogenous dynamics of the stochastic marketplace are coupled via a set of option-ality multipliers. For any subset of undeveloped features, the optionality multiplier encodes thevalue of continuing to issue new releases as opposed to permanently stopping development. Weprove that these multipliers decrease in the intensity of the marketplace, and that the multiplierassociated with a set of undeveloped features exceeds that of any of its subsets when the incremen-tal value of new features decreases in the amount of functionality already present in the releasemanager’s product.

We then analyze the optionality multipliers to obtain an index ordering condition necessary forthe optimality of a release sequence. This index is of interest even in the absence of an externalmarketplace, as it exhibits two unconventional properties: path dependence and nesting of featuresubsets. Path dependence requires knowledge of the full release history when calculating the indexto determine the next release in the sequence. Nesting refers to the fact that once a feature isdeveloped, the release manager can not issue any releases containing that feature in the future.While this index coincides with the Gittins index in a specialized version of our problem, we showthat a greedy heuristic based on this index can perform arbitrarily poorly in other settings.

When the release manager incurs positive fixed costs for issuing a release, much of our analysisbreaks down. Instead, we take a novel approach to approximate dynamic programming (ADP) andapproximate the value function with a decomposition motivated by the optionality multipliers iden-tified in the case without fixed costs. After inserting this approximation into an infinite dimensionallinear program representing the SMDP, we reduce the linear program to a finite dimensional linearprogram by exploiting the same assumptions applied in the case without fixed costs. By solvingthis linear program, we obtain a policy bound and an appoximate policy that performs within 3.5%of optimality across all of our numerical trials. Finally, we also prove that the approximate policycan outperform a certainty-equivalent heuristic by an arbitrarily large margin. This paper is thefirst to apply the linear programming approach to ADP in the context of product development,and our technique illustrates how to extend the methods from Adelman and Mancini (2014) tosequencing problems with fixed costs.

We begin with a brief literature review in §2. We formulate our model in §3 and present assump-tions in §4. Section 5 considers the case with no fixed costs, including the key value function

Adelman and Mancini: Dynamic Release ManagementDO NOT DISTRIBUTE 3

Generations # Rivals Quality Demand Model PricingCohen et al. (1996) Single One-Static Scalar Logit Exog.

Bayus (1997) Single One-Static Scalar Logit Exog.

Morgan et al. (2001) Multiple One-Static Scalar Logit Exog.

Gjerde et al. (2002) Multiple Zero Vector Generic Exog.

Souza et al. (2004) Multiple One-Static Binary Generic Exog.

Klastorin and Tsai (2004) Single One-Strategic Scalar Diffusion Endog.

Savin and Terwiesch (2005) Single One-Strategic Not Modeled Diffusion Exog.

Krankel et al. (2006) Multiple Zero Scalar Diffusion Exog.

Arslan et al. (2009) Multiple One-Strategic Not Modeled Linear in price w/time decay Endog.

Lobel et al. (2013) Single Zero Scalar Strategic Consumers Endog.

Table 1. Sample Product Development Literature

decomposition (Theorem 2) and quasi-open-loop results (Theorem 3), as well as the novel option-ality results and index-based analysis of the optimal release sequence. In §6 we apply approximatedynamic programming to the case with fixed costs. Extensions and avenues for further researchare discussed in §7.

2. Literature Review The release management problem most closely resembles work in thenew product development literature. Table 1 categorizes a sample of ten relevant articles along thefollowing five dimensions: the number of simultaneous product generations allowed in the market,the number and strategic nature of the firm’s rivals, the method for parameterizing product quality,the type of consumer demand model employed by the authors, and whether or not product pricing isexogenous or endogenous. Several of these articles allow for the coexistence of multiple generationsof the same product in the marketplace, whereas we omit this possibility. When interpreted as theconsumer utility associated with a rival product, our model of market intensity as an exogenousshock process is a compromise between the “one-static” and “one-strategic” formulations displayedin Table 1, and provides a reasonable tradeoff between tractability and realism given the novelcombinatorial complexity in our model. Regarding consumer demand and pricing, we require onlya generic mapping that calculates the firm’s revenue accrual rate as a function of market intensityand the features in the release manager’s product. Our key departure from the product developmentliterature is that we mdoel the firm’s product as a collection of discrete features.

In the project scheduling literature, a firm aims to schedule a set of projects with either known orrandom duration times in order to maximize the net present value of payments received upon com-pletion. Examples include Elmaghraby and Herroelen (1990), Herroelen and Gallens (1993), Reyckand Herroelen (1998), and Vanhoucke et al. (2003). In certain cases, project scheduling correspondsto the single machine stochastic scheduling problem with total weighted discounted completion timeobjective. In fact, the “weighted discounted shortest processing time first” (WDSPT) rule foundin the stochsatic scheduling literature (Pinedo (2012)) solves a specialized instance of our problem.We contribute to the project scheduling and stochastic scheduling literatures by accounting for anexogenous, stochastically evolving marketplace that impacts the firm’s revenues.

The linear programming approach to approximate dynamic programming was introduced bySchweitzer and Seidmann (1985) and later revisited by de Farias and Van Roy (2003). Applicationsinclude stochastic inventory management and routing (Adelman (2004), Topaloglu and Kunnumkal(2006)), price-directed control of remnant inventory systems (Adelman and Nemhauser (1999)),network revenue management (Adelman (2007), Farias and Roy (2007), Zhang and Adelman (2009),Tong and Topaloglu (2012)), the valuation of natural resources (Nadarajah et al. (2011)), and thetraveling salesman problem (Toriello (2014)).


Finally, we have adapted the SMDP formulation from Adelman and Mancini (2014) to therelease management problem. The exogeneity assumptions in §4.2 and the multiplicative sepa-rability assumption in §4.3 come directly from the more general paper. Those assumptions leaddirectly to the modified discount factor identified in Proposition 1, and form the foundation forthe value function decomposition derived in Theorem 2 and associated quasi-open-loop optimalpolicy identified in Theorem 3. However, all analysis of the optionality multipliers in §5.1 and theoptimal release sequence in §5.2 and §5.3 is novel.

3. Semi-Markov Decision Process Formulation In this section, we describe the releasemanagement problem, formalize its transition structure, and develop the optimality equations ofthe associated SMDP.

3.1. Description At the beginning of the horizon, let F 0I denote the features already included

in the release manager’s product, F 0A the finite set of all features available for development in future

releases, and x0 ∈R+ the initial intensity of the marketplace. We define the state space of our SMDP

as S :=

(x,FA)∣∣∣x∈ [x0,∞) ,FA ⊆ F 0

A

. For each state (x,FA)∈S , the set of actions φ available

to the release manager consists of all subsets of features that could be included in the next release.

We define the set of permissible state-action pairs as K :=

((x,FA) , φ)∣∣∣ (x,FA)∈S , φ⊆ FA

,

and take as given a deterministic mapping τ : 2F0A→R++ that determines the required development

time for any collection of features. Our assumption that development times are deterministic easesexposition and does not restrict our analytical results. However, the time required to develop abundle of features could depend on the set of features already in the existing product. For instance,features that simplify the software’s code may accelerate development of future features. We discussthis extension in §7.

Suppose that the release manager is in state (x,FA) and takes the action φ⊆ FA. She immediatelyincurs a fixed, finite development cost c ∈ R+ if φ 6= ∅, and otherwise incurs no fixed cost. Whilethe bundle φ is under development, market intensity increases stochastically in continuous-time.If FI denotes the set of features already included in the release manager’s product, then at anytime u ∈ [0, τ (φ)) after initiating development of the bundle φ, revenue accumulates at the ratee−βug (X (u) ,FI) for some measurable function g :R×2F

0I ∪F

0A→R+ and discount factor β > 0. The

next decision epoch occurs when development of the bundle φ is complete, at which time the setof available features reduces to FA\φ and the release manager immediately replaces her existingproduct with the set FI ∪φ. The release manager seeks to implement a policy that maximizes herexpected discounted profits over an infinite horizon.

We emphasize five key properties of our model. First, the set F 0A contains all features available

for development over the entire horizon, and no new features arrive over time. Second, the releasemanager can not remove previously released features from her product. These two assumptionsimply that if the set of available features is FA ⊆ F 0

A, then the set of features already included inthe release manager’s product must be FI = F 0

I ∪ (F 0A\FA). We use the notation FI throughout the

paper, with its definition clear from context. Similarly, we sometimes take the set FI as given andinstead infer that FA = F 0

A\FI . Third, the release manager can not pause development and changethe set of features slated for the next release. Once she chooses to develop a bundle of features, shemakes no further decisions until that bundle is completed and released to the market. In addition,we do not allow multiple generations of the release manager’s product to exist simultaneously inthe market. Finally, our model embeds the release manager’s pricing strategy in the instantaneousrevenue function g and implicitly assumes that price is a deterministic function market intensityand the features in the release manager’s product.


3.2. Formal Transition Structure As much of our analysis depends on the inter-epochevolution of the marketplace and its interaction with the release manger’s product and decisions,we now formalize the transition structure of the release management SMDP.

Recall that the local index u measures time relative to the most recent decision epoch. Weintroduce the following objects:• a probability space (Ω,Σ, P ),• a R+-valued stochastic process X (u) : u≥ 0 representing the intensity of the external market,• a 2F

0A-valued stochastic process FA (u) : u≥ 0 representing the set of features available for

development,• a 2F

0I ∪F

0A-valued stochastic process FI (u) : u≥ 0 representing the set of features already

included in the release manager’s product, and• a 2F

0A-valued random variable A representing the release manager’s most recent action.

The space Ω is local in the sense that if the decision maker takes an action φ in state(x,FA), her observations until the next decision epoch consist of a realization of the ran-dom variable (X (u) ,FA (u)) : 0≤ u≤ τ (φ) subject to the conditional probability measureP(·∣∣ (X (0) ,FA (0)) = (x,FA) ,A= φ

). The local time index u is reset to zero at each decision epoch,

and all objects defined over Ω are reinstantiated. As discussed in §3.1, the release manager’s prod-uct only changes at decision epochs. Hence, if the release manager takes the action φ while in state(x,FA), then FA (u) = FA and FI (u) = FI for all u∈ [0, τ (φ)). Furthermore, FA (τ (φ)) = FA\φ andFI (τ (φ)) = FI ∪ φ, which represents the completion of the bundle φ after a development cycle ofduration τ (φ).

3.3. Optimality Equations We apply the contraction mapping approach in Puterman(2005) to develop the optimality equations for our SMDP and characterize an optimal station-ary deterministic policy. We require two technical assumptions. The first prohibits instantaneoustransitions between decision epochs, and identifies the action φ= ∅ with the decision to wait for afixed amount of time. The second assumption requires that, in addition to being nonnegative, theinstantaneous revenue function g is also bounded.

Assumption 1. There exists a scalar w ∈ (0,∞) such that τ (∅) =w≤ τ (φ)<∞ for all featurebundles φ⊆ F 0

A.

Assumption 2. The measurable function g is nonnegative on R× 2F0I ∪F

0A, and there exists a

constant M ∈ [0,∞) such that g≤M .

We denote the value function for our SMDP by V ∗. For any measurable mapping d : S → 2F0A

such that d (x,FA) ⊆ FA for all (x,FA) ∈ S , we denote the associated stationary deterministicpolicy by d∞ and its value by V d∞ . Without loss of generality, we now focus on characterizing anoptimal stationary deterministic policy, i.e., a policy d∞∗ such that V d∞∗ = V ∗.

Let Vb denote the space of real-valued, bounded, measurable functions on S equipped with thesupremum norm ‖ · ‖∞. We define the inter-epoch reward r as follows: for all ((x,FA) , φ)∈K ,

r ((x,FA) , φ) := −c ·1φ6=∅+

E

[∫ τ(φ)

0

e−βug (X (u) ,FI)du∣∣ (X (0) ,FA (0)) = (x,FA) ,A= φ

], (1)

with E denoting expectation with respect to P on Ω. Consider the following operator L definedon Vb: for all v ∈Vb and all (x,FA)∈S ,

Lv (x,FA) := maxφ⊆FA

r ((x,FA) , φ)+

e−βτ(φ)E[v (X (τ (φ)) ,FA\φ)

∣∣ (X (0) ,FA (0)) = (x,FA) ,A= φ]. (2)


Our assumption that F 0A has finite cardinality justifies the use of the max operator, and ensures

that Lv is measurable on S for all v ∈ Vb (Proposition D.5 in Hernandez-Lerma and Lasserre(1996)). Furthermore, Lv is bounded on S since c is finite and g is bounded by Assumption 2.Hence, the range of L is Vb. Assumption 1 and the fact that β > 0 therefore imply that L is acontraction operator on Vb. The following standard result thus follows from Puterman (2005):

Theorem 1.(i) The value function V ∗ is the unique solution in Vb of the optimality equations v=Lv.

(ii) For any v ∈Vb, if Ln+1v :=L (Lnv) for all n∈N, then ‖Lnv−V ∗‖∞→ 0.(iii) If there exists a mapping d∗ : S → 2F

0A such that, for all (x,FA)∈S ,

d∗ (x,FA) ∈ arg maxφ⊆FA

r ((x,FA) , φ)+

e−βτ(φ)E[V ∗ (X (τ (φ)) ,FA\φ)

∣∣ (X (0) ,FA (0)) = (x,FA) ,A= φ], (3)

then the stationary deterministic policy d∞∗ is optimal.

By part (iii) of Theorem 1, the assumption that F 0A has finite cardinality immediately implies the

existence of a decision rule d∗ that satisfies V d∞∗ = V ∗.

4. Assumptions The following assumptions hold throughout the paper.

4.1. Initial Parameters and Revenues The first assumption is self-evident, as a productwith no features generates no revenues:

Assumption 3. For all x∈ [x0,∞), g (x,∅) = 0.

In order to avoid division by zero later in the paper, we impose the following condition:

Assumption 4. The release manager’s initial product is nonempty, i.e., F 0I 6= ∅.

The next assumption describes the interaction between market intensity and the revenues earnedby the release manager’s firm.

Assumption 5. For any pair of sets F 1I and F 2

I such that F 1I ⊂ F 2

I ⊆ (F 0I ∪F 0

A), the differenceg (·,F 2

I )− g (·,F 1I ) is strictly positive and nonincreasing on [x0,∞).

This assumption formalizes the intuition that a product with more functionality should generatemore revenue than a product with less functionality. However, the marginal value of additionalfunctionality presumably decreases as the marketplace becomes more intense due to improvementsin rival products, increased consumer disaffection with the product category, etc. Observe thattaking F 1

I = ∅ in Assumption 5 implies that g (·,FI) is both strictly positive and nonincreasing on[x0,∞) for all nonempty subsets FI of F 0

I ∪F 0A.

4.2. The Market Intensity Process We now impose structure on the market intensityprocess, and describe its relationship to the release manager’s product and actions. Relaxation ofthese assumptions are discussed in §7.

Assumption 6. There exists a Poisson process N with rate λ∈ (0,∞) and a sequence of i.i.d.shocks ξii∈N, both defined on (Ω,Σ, P ), such that ξi : Ω→ [1,∞) for all i∈N and

X (u) :=X (0)

Nu∏i=1

ξi ∀u≥ 0. (4)


The events of the process N correspond to increases in marketplace intensity, and the shocks ξii∈Nrepresent the magnitudinal changes in intensity. For instance, if intensity represents the quality ofa rival product, a shock of magnitude 1.05 suggests a five percent increase in the consumer utilityassociated with a new release of that product. Since the range of the shock variables is [1,∞), weare implicitly assuming that the market only becomes more intense over time. To avoid a trivialscenario, we also impose two additional conditions. The first avoids a market that never evolves,and the second ensures that market intensity is not fixed at zero given the multiplicative structurein (4).

Assumption 7. For all i∈N, P (ξi = 1)< 1.

Assumption 8. The initial market intensity level is strictly positive, i.e., x0 > 0.

We now describe the interaction between the shock-time process N and the shocks ξii∈N.

Assumption 9. For all u≥ 0 and all i∈N, the following conditions hold on (Ω,Σ, P ):(i.) Nu and X (0) are independent,

(ii.) ξi and X (0) are independent, and(iii.) ξi and Nu are independent.

Under Assumption 9, neither the shocks nor the process N can depend on the market intensity atthe beginning of a decision epoch. Furthermore, part (iii) implies that the magnitude of a shockcan not depend on the length of time between successive shocks.

The next assumption characterizes the interaction between market intensity and the releasemanager’s state and actions.

Assumption 10. For all i∈N and all u≥ 0, the following conditions hold on (Ω,Σ, P ):(i.) ξi and A are independent,

(ii.) ξi and FA (u) are independent,(iii.) Nu and A are independent, and(iv.) Nu and FA (u) are independent.

Parts (i) and (iii) imply that market intensity evolves independently of the release manager’sactions. These conditions reflect the fact that the external market is likely not privy to the releasemanager’s internal prouct development decisions. However, conditions (ii) and (iv) also requirethat the market intensity process is not impacted by the release manager’s existing product. Wediscuss the role of these conditions in §7. Finally, Assumption 10 permits us to condition on onlythe intensity level at each epoch in both the definition (1) of the inter-epoch reward r and theoptimality equations: for all ((x,FA) , φ)∈K ,

r ((x,FA) , φ) := −c ·1φ6=∅+E

[∫ τ(φ)

0

e−βug (X (u) ,FI)du∣∣X (0) = x

], (5)

and, for all (x,FA)∈S ,

V ∗ (x,FA) := maxφ⊆FA

r ((x,FA) , φ) + e−βτ(φ)E

[V ∗ (X (τ (φ)) ,FA\φ)

∣∣X (0) = x]. (6)

4.3. Multiplicative Separability Motivated by Assumption 6, we propose a multiplicativeseparability condition on the instantaneous revenue function g:

Assumption 11 (Multiplicative Separability). There exists some measurable, nonincreas-ing function ϑ∗ : [1,∞)→ (0,1] such that

g

(x0

j∏i=1

ξi,FI

)= g (x0,FI)

j∏i=1

ϑ∗ (ξi) ∀FI ⊆ F 0I ∪F 0

A ∀j ∈N a.e.-P.


Under the multiplicative separability assumption, successive increases in market intensity inducegeometric decay in the rate at which the release manager’s revenues accumulate. For instance, ifintensity increases by 10%, then the release manager observes a (1−ϑ∗ (1.1))×100% decline in herinstantaneous revenue rate. This approach is parsimonious, and allows managers to conceptualizethe impact of market shocks in terms of elasticities. By appropriately specifying the function ϑ∗,a manager can also model complex relationships between market intensity and revenues. The keylimitation of Assumption 11 is that the percentage change in the revenue rate g in response to amarket shock does not depend on the release manager’s product FI or the current level of marketintensity.

When combined with the independence assumptions in §4.2, multiplicative separability leadsto two crucial results. The first result permits a simplification of the inter-epoch reward r byintroducing a modified discount factor that encodes the impact of market intensity on the releasemanager’s revenues. It follows immediately from Proposition 1 and Theorem 3 in Adelman andMancini (2014).

Proposition 1 (Modified Discount Factor). For all FA ⊆ F 0A and all φ⊆ FA,

r ((X (0) ,FA) , φ) =−c ·1φ 6=∅+

[1− e−γ∗τ(φ)

γ∗

]g (X (0) ,FI) a.e.-P, (7)

with γ∗ := β+λ (1− ζ∗) and ζ∗ :=E [ϑ∗ (ξ1)].

Proposition 1 indicates that the expected value of the stochastic revenue streame−βug (X (u) ,FI) : 0≤ u< τ (φ)

coincides with the value of a deterministic revenue stream

e−γ∗ug (X (0) ,FI) : 0≤ u< τ (φ)

that depends only on the intensity level at time u = 0 and

accounts for market stochasticity with the modified discount factor γ∗. Since ζ∗ ∈ (0,1], γ∗ ≥ β,implying that intensity induces additional decay. Furthermore, the fact that γ∗ increases in thevalues of λ and (1− ζ∗) confirms the following intuition: a market with frequent and/or significantshocks is more intense than one in which shocks are rare and/or minor.

The second implication of multiplicative separability is technical in nature and follows fromAssumption 11 by inspection. We formally present it here given its importance throughout ouranalysis. Let Xj := x0

∏j

i=1 ξi for all j ∈N.

Proposition 2 (Ratio Invariance). If FI and F ′I are arbitrary subsets of F 0I ∪F 0

A with F ′I 6=∅, then

g (Xj,FI)

g (Xj,F ′I)=g (x0,FI)

g (x0,F ′I)a.e.-P ∀j ∈N.

Since F ′I 6= ∅, Assumptions 3 and 5 eliminate the possibility of division by zero in Proposition 2.To ease notation, we let

b (FI ,F′I) :=

g (x0,FI)

g (x0,F ′I)(8)

for arbitrary subsets FI and F ′I of F 0I ∪F 0

A with F ′I 6= ∅.Before proceeding, we provide an example of a function that satisfies Assumptions 3, 5, and 11.

Consider the following specification for g:

g (x,FI) :=z (FI)

xα∀x∈ [x0,∞) ∀FI ∈ F 0

I ∪F 0A, (9)

with z : 2F0I ∪F

0A→R+. If z (∅) = 0 and z is strictly increasing on its domain with respect to the set-

containment partial ordering, then the specification (9) satisfies Assumptions 3 and 5. Regardlessof the structure of the shock random variables ξii∈N, Assumption 11 is satisfied with ϑ∗ (y) := 1

yα

for all y ∈ [1,∞). If the distribution of the shocks is known, then other specifications for g mayalso satisfy Assumptions 3, 5, and 11 (Adelman and Mancini (2014)).


5. The Case with No Fixed Costs In this section, we analyze the combinatorial dynamicsof the release management problem in the absence of fixed costs. For the remainder of this section,we impose the condition that c= 0.

5.1. Value Function Decomposition Suppose that the release manager is in state (x,FA)at some decision epoch and decides to permanently cease development. Letting τ (φ)→∞ in Propo-sition 1 suggests that the value of this decision is g(x,FI )

γ∗ . However, this decision may be suboptimalas it foregoes the potential benefits of future development. The following result indicates thatthe value of an optimal policy scales this guaranteed revenue stream by an optionality multiplierrepresenting the value of the undeveloped features FA.

Theorem 2. The value function V ∗ decomposes as follows:

V ∗ (Xj,FA) =g (Xj,FI)

γ∗Λ∗ (FA) a.e.-P ∀FA ⊆ F 0

A ∀j ∈N, (10)

with the values Λ∗ (FA) : FA ⊆ F 0A determined by the equations:

Λ∗ (FA) = maxφ⊆FA

(1− e−γ

∗τ(φ))

+ e−γ∗τ(φ)b (FI ∪φ,FI)Λ∗ (FA\φ)

∀FA ⊆ F 0

A. (11)

Since the decomposition (10) holds almost everywhere with respect to the probability measureP , it applies both at each decision epoch, as well as between decision epochs.

Proof. By Theorem 4 in Adelman and Mancini (2014),

V ∗ (Xj,FA) =g (Xj,FI)

g (x0,FI)V ∗ (x0,FA) a.e.-P ∀FA ⊆ F 0

A ∀j ∈N.

If we define

Λ∗ (FA) :=V ∗ (x0,FA)[

g(x0,FI )

γ∗

] ∀FA ⊆ F 0A, (12)

then (10) immediately follows.To verify (11), we combine the optimality equations for our SMDP with the decomposition (10).

For arbitrary FA ⊆ F 0A,

V ∗ (x0,FA) = maxφ⊆FA

[1− e−γ∗τ(φ)

γ∗

]g (x0,FI) + e−βτ(φ)E

[V ∗ (X (τ (φ)) ,FA\φ)

∣∣∣X (0) = x0

]by (6) and Proposition 1

= maxφ⊆FA

[1− e−γ∗τ(φ)

γ∗

]g (x0,FI)+

e−βτ(φ)

∞∑j=0

E[V ∗ (Xj,FA\φ)

∣∣∣Nτ(φ) = j]P(Nτ(φ) = j

)by Assum.’s 6 and 10

= maxφ⊆FA

[1− e−γ∗τ(φ)

γ∗

]g (x0,FI)+

e−βτ(φ)

∞∑j=0

E

[g (Xj,FI ∪φ)

γ∗Λ∗ (FA\φ)

∣∣∣Nτ(φ) = j

]P(Nτ(φ) = j

)by (10)

= maxφ⊆FA

[1− e−γ∗τ(φ)

γ∗

]g (x0,FI)+

e−βτ(φ) g (x0,FI ∪φ)

γ∗Λ∗ (FA\φ)

∞∑j=0

(ζ∗)jP(Nτ(φ) = j

)by Assum.’s 9 and 11

= maxφ⊆FA

[1− e−γ∗τ(φ)

γ∗

]g (x0,FI) + e−γ

∗τ(φ) g (x0,FI ∪φ)

γ∗Λ∗ (FA\φ)

,


with the final line following from the fact that N is Poisson by Assumption 6. Dividing both sidesby g(x0,FI )

γ∗ yields (11). As the next result shows, optionality can not make the release manager worse off.

Proposition 3. For all FA ⊆ F 0A, Λ∗ (FA)≥ 1, with equality if the empty set maximizes (11).

Proof. Recall that τ (∅) =w > 0. For any subset FA of F 0A, evaluating the argument of the max

operator in (11) at φ= ∅ implies that:

Λ∗ (FA) ≥(

1− e−γ∗w)

+ e−γ∗wb (FI ,FI)Λ∗ (FA)

=(

1− e−γ∗w)

+ e−γ∗wΛ∗ (FA) by (8)

⇐⇒ Λ∗ (FA) ≥ 1.

The same derivation trivially holds with equality for any set FA ⊆ F 0A for which the empty set is a

maximizer of (11). However, increased market intensity diminishes the value of undeveloped functionality.

Proposition 4. The option value Λ∗ (FA) of any nonempty subset FA of F 0A is nonincreasing

in γ∗.

Proof. First, Proposition 3 and equation (11) indicate that for all nonempty subsets FA of F 0A,

Λ∗ (FA) = max

1, maxφ⊆FA, φ6=∅

(1− e−γ

∗τ(φ))


. (13)

We now proceed by induction on the cardinality of FA. If FA = f for some feature f ∈ F 0A, then

(13) yields:

Λ∗ (f) = max

1,(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f)b (FI ∪f ,FI)Λ∗ (∅)

= max

1,(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f)b (FI ∪f ,FI)

by Prop. 3. (14)

Since b (FI ∪f ,FI)> 1 by Assumption 5, both arguments of the max operator in (14) are non-increasing functions of γ∗. The result follows because the pointwise maximum of nonincreasingfunctions is itself nonincreasing.

Assume that the result holds for all cardinalities i∈ 1,2, . . . , n and let FA ⊆ F 0A be an arbitrary

subset of cardinality n+ 1. Since∣∣∣FA\φ∣∣∣< n+ 1 for all nonempty subsets φ of FA, our induction

hypothesis together with the fact that Λ∗ (FA\φ) ≥ 1 by Proposition 3 implies that each of thearguments of the max operator in (13) is nonincreasing in γ∗. The result immediately follows.

Finally, we examine the intuition that, in some sense, more optionality should be “better” thanless.

Proposition 5. Let h (FI , φ) := g (x0,FI ∪φ) on

(FI , φ)∣∣∣FI ⊆ F 0

I ∪F 0A, φ⊆ FA

, and suppose

that h is submodular on its domain with respect to the set-containment partial ordering. For anyFA ⊆ F 0

A, Λ∗ (FA)≥Λ∗ (F ′A) for all F ′A ⊆ FA.

Proof. We proceed by induction on the cardinality of FA. If∣∣FA∣∣ = 0, the statement trivially

follows from Proposition 3 since Λ∗ (∅) = 1. Suppose now that the statement holds for all FA ⊆ F 0A

with∣∣FA∣∣≤ n. Let FA be an arbitrary subset of F 0

A of cardinality n+1, and F ′A an arbitrary subset ofFA. If the empty set maximizes (11) for F ′A, then Proposition 3 implies that 1 = Λ∗ (F ′A)≤Λ∗ (FA).

Next, consider the case when the empty set does not maximize (11) for F ′A. Since F ′A ⊆ FA, itfollows that F ′I ⊇ FI . For any nonempty subset φ of F ′A, submodularity of h thus implies:

g (x0,F′I ∪φ)− g (x0,F

′I)≤ g (x0,FI ∪φ)− g (x0,FI) . (15)


Furthermore, Assumption 5 ensures that both sides of (15) are strictly positive and that g (x0,F′I)≥

g (x0,FI). Therefore,

g (x0,F′I ∪φ)− g (x0,F

′I)

g (x0,F ′I)≤ g (x0,FI ∪φ)− g (x0,FI)

g (x0,FI)⇒ b (F ′I ∪φ,F ′I)≤ b (FI ∪φ,FI) .

In addition,∣∣FA\φ∣∣ ≤ n since φ 6= ∅, and so our induction hypothesis implies that Λ∗ (FA\φ) ≥

Λ∗ (F ′A\φ). Hence,

Λ∗ (FA) ≥ maxφ⊆F ′

A,φ6=∅

(1− e−γ

∗τ(φ))


since F ′A ⊆ FA

≥ maxφ⊆F ′

A,φ6=∅

(1− e−γ

∗τ(φ))

+ e−γ∗τ(φ)b (F ′I ∪φ,F ′I)Λ∗ (F ′A\φ)

= Λ∗ (F ′A) .

Since FA and F ′A were arbitrary, we are done. The validity of the submodularity requirement depends on the relationship between the set of

features available for development and the existing product. If there are significant complemen-tarities, then (15) may be inappropriate since the incremental benefit of adding features thatcomplement existing functionality could exceed the benefit of adding the same functionaliy to aproduct lacking complimentary features. In order to preserve the fidelity of our model, we do notrequire submodularity elsewhere in our analysis.

5.2. Quasi-Open-Loop Optimal Policy We now turn our attention to identifying an opti-mal policy for the release management problem in the absence of fixed costs. The next result followsimmediately from Theorem 4 in Adelman and Mancini (2014), and Theorem 2 in §5.1.

Theorem 3. If π = d∞ is a stationary deterministic policy such that d (x,FA) = dx0 (FA) on

S for any function dx0 : 2F0A→ 2F

0A satisfying the following condition:

dx0 (FA)∈ arg maxφ⊆FA

(1− e−γ

∗τ(φ))


∀FA ⊆ F 0

A, (16)

then the policy π is optimal with probability one, i.e., V π (Xj,FA) = V ∗ (Xj,FA) a.e.-P for allFA ⊆ F 0

A and all j ∈N.

We refer to the policy π as quasi-open-loop because the function dx0 ignores the current level ofmarket intensity, but still depends on the set of available features. Computing the policy π thusreduces to evaluating the finite set of conditions (16), as opposed to an uncountably infinite set ofconditions on the full state space S . Furthermore, the release manager can implement the policy πknowing that, with probability one, she will only transition to states at which the policy is optimal.

Theorem 3 suggests that constructing the policy π amounts to identifying a release sequenceφ∗i i∈N that solves a deterministic dynamic program. Applying (11) recursively, we obtain:

φ∗1 := dx0(F 0A

)φ∗i+1 := dx0

(F 0A\(∪in=1φ

∗n

))∀i≥ 1.

(17)

As the next result shows, the release manager has no incentive to wait between releases sinceAssumption 5 implies that adding functionality to the existing product always leads to strictlyhigher revenues.

Proposition 6. If FA ⊆ F 0A and FA 6= ∅, then dx0 (FA) 6= ∅.


Proof. If dx0 (FA) = ∅, then (11) implies that Λ∗ (FA) = 1. However, for any f ∈ FA,

Λ∗ (FA) ≥(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f)b (FI ∪f ,FI)Λ∗ (FA\f)

≥(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f)b (FI ∪f ,FI) since Λ∗ (F ′A)≥ 1 by Prop. 3

>(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f) since b (FI ∪f ,FI)> 1 by Assumption 5

= 1 which is a contradiction.

Proposition 6 indicates that the release manager will only take the waiting action once all fea-tures have been implemented. Hence, there must exist some j∗ ≥ 1 such that φ∗i 6= ∅ for all i ∈1,2, . . . , j∗, and φ∗i = ∅ for all i > j∗. Furthermore, j∗ <∞ since F 0

A has finite cardinality, and⋃j∗

i=1 φ∗i = F 0

A.For each i ∈ 1,2, . . . , j∗, the incremental revenue generated by the ith release in the sequence

Φ∗ equals:

E

[∫ ∞0

e−βu[g(X (u) ,F 0

I ∪(∪in=1φ

∗n

))− g

(X (u) ,F 0

I ∪(∪i−1n=1φ

∗n

))]du∣∣∣X (0) = x0

],

which reduces to the following by the definition (5) of the inter-epoch reward r and Proposition 1:

g (x0,F0I ∪ (∪in=1φ

∗n))− g

(x0,F

0I ∪(∪i−1n=1φ

∗n

))γ∗

.

The following result thus suggests that the sequence Φ∗ optimally manages the tradeoff betweenthe timing and magnitude of the increased revenues resulting from new releases by solving adeterministic sequencing problem with a static market and discount factor γ∗. A proof is includedin Appendix A.

Proposition 7. If Φ∗ := φ∗i i∈N is the optimal release sequence defined by (17), then

V ∗(x0,F

0A

)=g (x0,F

0I )

γ∗+

j∗∑i=1

e−γ∗[∑in=1 τ(φ∗n)]

[g (x0,F

0I ∪ (∪in=1φ

∗n))− g

(x0,F

0I ∪(∪i−1n=1φ

∗n

))γ∗

]. (18)

Observe that the determinism of (18) does not imply that the release manager can entirely ignorethe revenue impact of market intensity. She must note the initial intensity level and account forfuture changes by adjusting her discount factor.

5.3. Characterizing the Optimal Release Sequence We now derive structural propertiesof the optimal release sequence φ∗i i∈N defined in (17) by applying an exchange argument motivatedby (18). To ease notation, we define the function Υ as follows: for any release sequence Φ = φii∈Nsuch that there exists some j ≥ 1 for which φi 6= ∅ for all i∈ 1,2, . . . , j, and φi = ∅ for all i > j, let

Υ(Φ) :=g (x0,F

0I )

γ∗+

j∑i=1

e−γ∗[∑in=1 τ(φn)]

[g (x0,F

0I ∪ (∪in=1φn))− g

(x0,F

0I ∪(∪i−1n=1φn

))γ∗

]. (19)

The function Υ calculates the value of any given release sequence.


Proposition 8. If the release sequence φ∗i i∈N is optimal and 2≤ j∗ <∞, then for all m ∈1,2, . . . , j∗− 1:

e−γ∗τ(φ∗m) [g (x0,F

0I ∪ (∪mn=1φ

∗n))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]1− e−γ∗τ(φ∗m)

≥

e−γ∗τ(φ∗m+1)

[g(x0,F

0I ∪(∪m−1n=1 φ

∗n ∪φ∗m+1

))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]1− e−γ∗τ(φ∗m+1)

. (20)

Proof. Let m ∈ 1,2, . . . , j∗− 1 be arbitrary and consider the permuted sequence Φ′ :=φ∗1, . . . , φ

∗m−1, φ

∗m+1, φ

∗m, φ

∗m+2, . . .

. The optimality of Φ∗ implies that Υ(Φ∗)−Υ(Φ′)≥ 0, which

by (19) requires:e−γ

∗[∑mn=1 τ(φ∗n)]

[g (x0,F

0I ∪ (∪mn=1φ

∗n))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))γ∗

]+

e−γ∗[∑m+1n=1 τ(φ

∗n)]

[g(x0,F

0I ∪(∪m+1n=1 φ

∗n

))− g (x0,F

0I ∪ (∪mn=1φ

∗n))

γ∗

]−

e−γ∗[∑m−1n=1 τ(φ∗n)+τ(φ∗m+1)]

[g(x0,F

0I ∪(∪m−1n=1 φ

∗n

)∪φ∗m+1

)− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))γ∗

]+

e−γ∗[∑m+1n=1 τ(φ

∗n)]

[g(x0,F

0I ∪(∪m+1n=1 φ

∗n

))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

)∪φ∗m+1

)γ∗

]≥ 0.

Multiplying through by γ∗, canceling the term e−γ∗[∑m+1n=1 τ(φ

∗n)]g

(x0,F

0I ∪(∪m+1n=1 φ

∗n

)), and dividing

through by e−γ∗[∑m−1n=1 τ(φ∗n)] yields:e−γ

∗τ(φ∗m) [g (x0,F0I ∪ (∪mn=1φ

∗n))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]−

e−γ∗[τ(φ∗m)+τ(φ∗m+1)]g

(x0,F

0I ∪ (∪mn=1φ

∗n))−


[g(x0,F

0I ∪(∪m−1n=1 φ

∗n

)∪φ∗m+1

)− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]−

e−γ∗[τ(φ∗m)+τ(φ∗m+1)]g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

)∪φ∗m+1

)≥ 0.

By collecting terms, we obtain:

e−γ∗τ(φ∗m)

(1− e−γ

∗τ(φ∗m+1))g(x0,F

0I ∪ (∪mn=1φ

∗n))− e−γ

∗τ(φ∗m)g(x0,F

0I ∪(∪m−1n=1 φ

∗n

))≥


(1− e−γ

∗τ(φ∗m))g(x0,F

0I ∪(∪m−1n=1 φ

∗n

)∪φ∗m+1

)− e−γ

∗τ(φ∗m+1)g(x0,F

0I ∪(∪m−1n=1 φ

∗n

)),

which, after dividing both sides by(

1− e−γ∗τ(φ∗m)

)(1− e−γ

∗τ(φ∗m+1))

, further simplifies to:[e−γ

∗τ(φ∗m)

1− e−γ∗τ(φ∗m)

]g(x0,F

0I ∪ (∪mn=1φ

∗n))−

e−γ∗τ(φ∗m)g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))(1− e−γ∗τ(φ∗m))

(1− e−γ∗τ(φ∗m+1)

) ≥[e−γ

∗τ(φ∗m+1)

1− e−γ∗τ(φ∗m+1)

]g(x0,F

0I ∪(∪m−1n=1 φ

∗n

)∪φ∗m+1

)−e−γ

∗τ(φ∗m+1)g(x0,F

0I ∪(∪m−1n=1 φ

∗n

))(1− e−γ∗τ(φ∗m))

(1− e−γ∗τ(φ∗m+1)

) . (21)

Consider the left-hand side of (21). Adding and subtracting the term[e−γ∗τ(φ∗m)

1−e−γ∗τ(φ∗m)

]g(x0,F

0I ∪(∪m−1n=1 φ

∗m

))yields:

e−γ∗τ(φ∗m) [g (x0,F

0I ∪ (∪mn=1φ

∗n))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]1− e−γ∗τ(φ∗m)

−


e−γ∗[τ(φ∗m)+τ(φ∗m+1)]

(1− e−γ∗τ(φ∗m))(

1− e−γ∗τ(φ∗m+1))g (x0,F

0I ∪(∪m−1n=1 φ

∗n

)). (22)

Now, consider the right-hand side of (21). Adding and subtracting the term[e−γ∗τ(φ∗m+1)

1−e−γ∗τ(φ∗m+1)

]g(x0,F

0I ∪(∪m−1n=1 φ

∗m

))yields:


[g(x0,F

0I ∪(∪m−1n=1 φ

∗n ∪φ∗m+1

))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]1− e−γ∗τ(φ∗m+1)

−

e−γ∗[τ(φ∗m)+τ(φ∗m+1)]

(1− e−γ∗τ(φ∗m))(

1− e−γ∗τ(φ∗m+1))g (x0,F

0I ∪(∪m−1n=1 φ

∗n

)). (23)

The result immediately follows from substituting (22) and (23) into (21). According to the ordering (20), feature bundles with higher incremental revenues and lower

development times will appear earlier in the release sequence. Condition (20) also resembles anordering of Gittins indices for a traditional project selection problem in which each nonemptysubset of F 0

A is an available project, and the release manager must complete a project beforebeginning her next project (Gittins et al. (2011)). However, in the release management problemthere are dependencies between projects. Once the release manager selects a release φ, all subsetsof F 0

A containing any features in φ are no longer available. In addition, the index in (20) exhibitspath dependence, i.e., calculating the index for the mth release requires knowledge of all previousreleases.

Without additional conditions, Proposition 8 says nothing about the number of nonemptyreleases j∗ in the optimal release sequence, or the composition of each release. However, considerthe case when the development time mapping τ is superadditive across disjoint subsets:

τ (φ∪φ′)≥ τ (φ) + τ (φ′) ∀φ,φ′ ⊆ F 0A φ∩φ′ = ∅.

In software development, newly added features are typically tested jointly to ensure that theyinteract appropriately, and such testing can lead to superadditive development times. As shown bythe next result, this condition imposes significant structure on the optimal release sequence in theabsence of fixed development costs.

Proposition 9. If τ is superadditive, then every nonemtpy release selected by the policy πconsists of a single feature, i.e.,

∣∣dx0 (FA)∣∣= 1 for all nonempty FA ⊆ F 0

A.

Proof. If∣∣FA∣∣= 1, then by Proposition 6 we are done. Suppose now that

∣∣FA∣∣> 1. Let φ be anarbitrary subset of FA such that

∣∣φ∣∣> 1, and let f be an arbitrary element of φ. We proceed byshowing that the value of the action f, as measured by the right-hand side of (11), is no lessthan the value of the action φ.

Consider the value of the action f:(1− e−γ

∗τ(f))

+ e−γ∗τ(f)b (FI ∪f ,FI)Λ∗ (FA\f)≥(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f)b (FI ∪f ,FI)[(

1− e−γ∗τ(φ\f)

)+

e−γ∗τ(φ\f)b (FI ∪φ,FI ∪f)Λ∗ (FA\φ)

]=(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f)b (FI ∪f ,FI)(

1− e−γ∗τ(φ\f)

)+

e−γ∗[τ(f)+τ(φ\f)]b (FI ∪φ,FI)Λ∗ (FA\φ) =(

1− e−γ∗τ(f)

)+(e−γ

∗τ(f)− e−γ∗[τ(f)+τ(φ\f)]

)b (FI ∪f ,FI)+(

e−γ∗[τ(f)+τ(φ\f)]− e−γ

∗τ(φ))b (FI ∪φ,FI)Λ∗ (FA\φ) + e−γ

∗τ(φ)b (FI ∪φ,FI)Λ∗ (FA\φ) . (24)


However,(e−γ

∗τ(f)− e−γ∗[τ(f)+τ(φ\f)]

)b (FI ∪f ,FI)>

(e−γ

∗τ(f)− e−γ∗[τ(f)+τ(φ\f)]

)by Assumption 5 since b (FI ∪f ,FI)> 1, and(

e−γ∗[τ(f)+τ(φ\f)]− e−γ

∗τ(φ))b (FI ∪φ,FI)Λ∗ (FA\φ)≥

(e−γ

∗[τ(f)+τ(φ\f)]− e−γ∗τ(φ)

)since b (FI ∪φ,FI)> 1 by Assumption 5, Λ∗ (FA\φ)≥ 1 by Proposition 3, and(e−γ

∗[τ(f)+τ(φ\f)]− e−γ∗τ(φ))≥ 0 by superadditivity. Therefore, (24) implies:(

1− e−γ∗τ(f)

)+ e−γ

∗τ(f)b (FI ∪f ,FI)Λ∗ (FA\f)>(1− e−γ

∗τ(f))

+(e−γ

∗τ(f)− e−γ∗[τ(f)+τ(φ\f)]

)+(e−γ

∗[τ(f)+τ(φ\f)]− e−γ∗τ(φ)

)+

e−γ∗τ(φ)b (FI ∪φ,FI)Λ∗ (FA\φ) =(

1− e−γ∗τ(φ)

)+ e−γ

∗τ(φ)b (FI ∪φ,FI)Λ∗ (FA\φ) . (25)

Since the right-hand side of (25) is the value of the action φ, we are done. .When there are no fixed development costs, the release manager only needs to consider the

tradeoff between development time and revenues. Proposition 9 thus states that the release managerhas no incentive to issue multi-feature releases when τ is superadditive, since bundling featurestogether can only lengthen the release cycle. Proposition 9 also simplifies the computation of thepolicy π as follows: for any set FA ⊆ F 0

A, it suffices to evaluate (11) over the set of individualfeatures f ∈ FA instead of the collection of all subsets of FA.

We now combinine Propositions 8 and 9 to identify a setting in which the condition (20) is bothnecessary and sufficient for optimality. Our result stipulates that the function g (x0, ·) is additiveon 2F

0I ∪F

0A , i.e.,

g(x0,F

1I ∪F 2

I

)= g

(x0,F

1I

)+ g

(x0,F

2I

)∀F 1

I ,F2I ⊆ F 0

I ∪F 0A s.t. F 1

I ∩F 2I = ∅.

Corollary 1. If τ is superadditive and g (x0, ·) is additive on 2F0I ∪F

0A, then it is optimal to

release features individually and in the order f∗1 , f∗2 , . . . , f

∗j∗ indicated by the following ranking:

e−γ∗τ(f∗1 )g (x0,f∗1 )1− e−γ∗τ(f∗1 )

≥ e−γ∗τ(f∗2 )g (x0,f∗2 )1− e−γ∗τ(f∗2 )

≥ · · · ≥e−γ

∗τ(f∗j∗)g(x0,f∗j∗)

1− e−γ∗τ(f∗j∗) , (26)

with j∗ =∣∣F 0

A

∣∣.Proof. By Proposition 9, superadditivity of τ implies that every nonempty release consists of a

single feature. Additivity of g (x0, ·) thus immediately converts (20) into (26). By construction, anysequence f1, f2, . . . , fj∗ satisfying (26) must also satisfy:

fi ∈ arg maxf∈(F0

A\f1,f2,...,fi−1)

e−γ∗τ(f)g (x0,f)1− e−γ∗τ(f)

∀i∈ 1,2, . . . , j∗ . (27)

If there is a unique maximizer in (27) for each i∈ 1,2, . . . , j∗, then there exists only one sequencesatisfying the necessary condition (26). This sequence must therefore be optimal. Alternatively,multiple sequences satisfying (26) exist if multiple maximizers exist in (27) for at least one indexi ∈ 1,2, . . . , j∗. Let Φ∗ =

f∗1 , . . . , f

∗j∗

and Φ′ =f ′1, . . . , f

′j∗

be two sequences satisfying (26).Condition (27) indicates we can convert Φ′ into Φ∗ via a finite sequence of pairwise permutationsof features that have the same index value. By applying the exchange argument from the proof ofProposition 8, however, it follows that the value of a feature sequence does not change when twofeatures with the same index value are permuted. Applying this logic to each of the permuationsrequired to convert Φ′ to Φ∗ indicates that the sequences Φ′ and Φ∗ have the same value.


0

g/τ

τ

Ranking Analysis With No Hostility

(τi, gi/τi)

Dominates fifor γ∗ = β

Dominated by fifor γ∗ = β

0

g/τ

τ

Ranking Analysis With No Hostility

(τi, gi/τi)

Dominates fifor γ∗ = β

Dominated by fifor γ∗ = β

β-IC

0

g/τ

τ

Ranking Analysis With Hostility

Dominates fifor all γ∗ > 0

Dominated by fifor all γ∗ > 0

0

g/τ

τ

Ranking Analysis With Hostility

Dominates fifor all γ∗ > 0

Dominated by fifor all γ∗ > 0

β-ICγ∗-ICOptimisticPessimistic

Figure 1. Bang-for-buck indifference curve analysis for feature fi

When all releases consist of a single feature, the fact that implementing any individual feature

removes all supersets of that feature from future consideration becomes irrelevant. The super-

additivity condition on τ in Corollary 1 thus eliminates the dependencies between projects that

distinguishes the release management problem from the traditional project selection problem.

Furthermore, the additivity condition on g (x0, ·) eliminates the path-dependence inherent in the

exchange condition (20). As a result, the ordering index in (26) exactly corresponds to the Gittins

index for the project selection problem without preemption (Gittins et al. (2011)). It also coincides

with the WDSPT rule found in the stochastic scheduling literature (Pinedo (2012)).

In the context of Corollary 1, the release manager can determine the relative ordering of any two

features in F 0A simply by comparing the indices for each feature. We now consider the stability of

such pairwise orderings with respect to the value of the modified discount factor γ∗. The following

result takes a feature fi as given and compares its development time τ (fi) and “bang-for-buck”

g (x0,fi)/τ (fi) to that of an alternative feature fj to identify scenarios in which feature fi should

precede fj regardless of the value of γ∗, as well as instances in which the relative ranking is sensitive

to the value of γ∗.

Proposition 10. Let τi = τ (fi, ) and gi = g (x0,fi) for any feature fi in F 0A. Suppose τ is

superadditive and g (x0, ·) is additive on 2F0I ∪F

0A. For any pair of distinct features fi and fj in F 0

A,

the following two statements hold:

(i.) if τi ≤ τj and giτi≥ gj

τj, then it is optimal to release feature fi before feature fj for any value

γ∗ ∈ (0,∞), and

(ii.) if τi ≤ τj and giτi<

gjτj

, then there exists a unique value γi,j ∈ (0,∞) such that it is optimal

to release feature fj before feature fi if γ∗ ∈ (0, γi,j) and optimal to release feature fi before

feature fj if γ∗ ∈ [γi,j,∞).


Proof. Let features fi and fj be arbitrary, and define the function qi,j (γ) on (0,∞) as eγτi−1eγτj−1

.By Corollary 1, it is optimal to release feature fi before feature fj if

e−γ∗τigi

1− e−γ∗τi≥ e−γ

∗τjgj1− e−γ∗τj

⇐⇒ gigj≥ eγ

∗τi − 1

eγ∗τj − 1

= qi,j (γ∗) . (28)

We prove statement (i) by analyzing the function qi,j to determine when the second inequality in(28) can or can not hold.

The function qi,j is clearly continuous on (0,∞), and L’Hopital’s rule indicates thatlimγ→0+ qi,j (γ) = τi

τj. Differentiation implies that on the domain (0,∞),

∂qi,j∂γ

< 0 ⇐⇒ (eγτj − 1) (τieγτi)− (eγτi − 1) (τje

γτj )< 0 ⇐⇒ 1− e−γτj1− e−γτi

<τjτi, (29)

with the statement also holding with all inequalities reversed.Suppose now that τi ≤ τj. If τi = τj, then (i) follows trivially from (28). Consider the case when

τi < τj. The function 1− e−γx is strictly concave in x on [0,∞) for any fixed γ ∈ (0,∞), and so

1− e−γτi > 0 ·(

1− τiτj

)+ (1− e−γτj ) ·

(τiτj

)because τi

τj< 1. Hence,

1− e−γτj1− e−γτi

<τjτi,

which indicates that (29) holds for all γ ∈ (0,∞) when τi < τj. The function qi,j is thereforemonotonically decreasing on (0,∞) and thus never exceeds its limit τi

τjat 0. Statement (i) follows

from (28) because

giτi≥ gjτj⇒ gigj≥ τiτj⇒ gigj> qi,j (γ) ∀γ ∈ (0,∞) , (30)

with the final inequality following from the fact that qi,j is monotonically decreasing in γ.Next, we prove statement (ii) by establishing the existence of a value γi,j ∈ (0,∞) that satisfies

(28) with equality. We require the following result, obtained from L’Hopital’s rule:

limγ→∞

qi,j (γ) =

∞, τi > τj

1, τi = τj

0, τi < τj

.

If τi = τj, then statement (ii) follows trivially by reversing the inequalities in (28). Suppose thatτi < τj. As argued earlier, the function qi,j then approaches

τjτi

at 0 and tends to 0 as γ →∞.

If giτi<

gjτj

, then gigj<

τjτi

, and so the intermediate value theorem ensures the existence of a value

γi,j ∈ (0,∞) such that gigj

= qi,j (γi,j). Uniqueness follows from the earlier argument that qi,j is

strictly decreasing when τi < τj, as do the following statements:

gigj< qi,j (γ) ∀γ ∈ (0, γi,j) and

gigj≥ qi,j (γ) ∀γ ∈ [γi,j,∞) .

Statement (ii) follows immediately from (28). Figure 1 illustrates the conclusions of Proposition 10. Given the feature fi, the first panel plots

all development time and bang-for-buck values for an alternate feature fj such that fi and fj havethe same index value when γ∗ = β. We refer to these points as the β-indifference curve (β-IC)through fi, since the release manager has no preference over the sequencing of features on thiscurve. Features above the β-IC curve have a strictly higher index value than feature fi, and thus


precede fi in any optimal release sequence. Analogously, features below β-IC should appear afterfi when γ∗ = β. In the second panel in Figure 1, we include a second indifference curve γ∗-ICthrough fi for some value γ∗ >β. Features to the southeast of fi take longer to develop and offerlower bang-for-buck than fi. As indicated by part (i) of Proposition 10, fi should precede all suchfeatures regardless of the value of γ∗. Similarly, features to the northwest of fi dominate fi for allvalues of γ∗ > 0. These two regions represent “error-free” scenarios in which the release managercan misspecify the market intensity process, or entirely ignore it, and still correctly rank featuresrelative to fi. As indicated by part (ii), however, more care is required when considering a featurefj that compensates for a longer development time than fi by offering a higher bang-for-buck. Ifthe release manager ignores marketplace intensity by setting γ∗ = β instead of using the correctvalue γ∗ > β, she may undervalue the lower development time of fi relative to fj. Because of this“pessimistic” assessment of fi, she will develop fj before fi even though this ordering is actuallysuboptimal. By ignoring or misspecifying market intensity, the release manager may similarlyovervalue the higher bang-for-buck value of fi relative to features fj with lower development timesin the “optimistic” region.

5.4. Evaluating a Greedy Index Heuristic The exchange condition (20) suggests a heuris-tic that greedily constructs a release sequence Φ as follows:

φ1 ∈ arg maxφ⊆F0

A

e−γ

∗τ(φ) [g (x0,F0I ∪φ)− g (x0,F

0I )]

1− e−γ∗τ(φ)

φi ∈ arg maxφ⊆F0

A\(∪i−1

n=1φn)

e−γ∗τ(φ)

[g(x0,F

0I ∪(∪i−1n=1φn

)∪φ)− g

(x0,F

0I ∪(∪i−1n=1φn

))]1− e−γ∗τ(φ)

∀i≥ 2.

By construction, the sequence Φ satisfies (20) and we denote its value by V (x0,F0A) := Υ

(Φ)

However, our next result shows that this heuristic can perform arbitrarily poorly when there areno fixed costs.

Proposition 11. Assume that∣∣F 0

A

∣∣≥ 2. If there exists some f ∈ F 0A such that τ (F 0

A\f) +τ (f) ≤ τ (F 0

A), then for any J ≥ 1 there exists some H ≥ 0 such that if τ (F 0A) −

[τ (F 0A\f) + τ (f)]≥H, then

limg(x0,F0

I∪F0

A)→∞

[V ∗ (x0,F

0A)

V (x0,F 0A)

]≥ J.

Proof. Let J ≥ 1 be arbitrary. As g (x0,F0I ∪F 0

A) tends towards infinity, all of the followingconditions will eventually be satisfied:

e−γ∗τ(F0

A) [g (x0,F0I ∪F 0

A)− g (x0,F0I )]

1− e−γ∗τ(F0A)

≥e−γ

∗τ(φ) [g (x0,F0I ∪φ)− g (x0,F

0I )]

1− e−γ∗τ(φ)

∀φ⊆ F 0

A.

Hence, in the limit the greedy index heuristic opts to develop the entire set F 0A in a single release.

We thus take the value of the sequence Φ to be:

V(x0,F

0A

)= Υ

(Φ)

=g (x0,F

0I )

γ∗+ e−γ

∗τ(F0A)[g (x0,F

0I ∪F 0

A)− g (x0,F0I )

γ∗

]. (31)

Next, consider the release sequence Φ′ defined as follows:

φ′1 = F 0A\f φ′2 = f φ′i = ∅ ∀i > 2.


Applying (19), we obtain the value of the sequence Φ′:

Υ (Φ′) =g (x0,F

0I )

γ∗+ e−γ

∗τ(F0A\f) [g (x0,F

0I ∪F 0

A\f)− g (x0,F0I )]

γ∗+

e−γ∗[τ(F0

A\f)+τ(f)] [g (x0,F0I ∪F 0

A)− g (x0,F0I ∪F 0

A\f)]γ∗

. (32)

By inspection of (31) and (32),

limg(x0,F0

I∪F0

A)→∞

Υ(Φ′)

V (x0,F 0A)

=e−γ

∗[τ(F0A\f)+τ(f)]

e−γ∗τ(F0

A). (33)

Furthermore, basic algebra indicates that

e−γ∗[τ(F0

A\f)+τ(f)]

e−γ∗τ(F0

A)≥ J ⇐⇒ τ

(F 0A

)−[τ(F 0A\f

)+ τ (f)

]≥ ln (J)

γ∗.

Since Υ(Φ′)≤Υ(Φ∗) = V ∗ (x0,F0A), the result follows from (33) with H = ln(J)

γ∗ . In Proposition 11, letting the value of the entire feature set F 0

A approach infinity blinds thegreedy heuristic to the superior two-release sequence. This result stands in stark constrast toCorollary 1, which provides conditions under which the exchange condition (20) leads to an optimalrelease sequence. The key difference is that when development times are superadditive, we knowa priori from Proposition 9 that each nonempty release consists of a single feature. We leveragedthis property to obtain Corollary 1, but it does not necessarily apply in the more general settingconsidered in Proposition 11.

6. The Case with Fixed Costs Before issuing a new release, software firms typically performseveral rounds of testing to ensure that the new functionality is compatible with the existingproduct. Once the product is ready for release, firms will also need to market the product to informboth existing and potential customers of the new functionality. Depending on the type of product,firms may need to organize and host formal education efforts to teach consumers how to use theproduct or expedite its implementation. We accommodate all of these scenarios by allowing for astrictly positive fixed cost, i.e., c > 0. Even if the the external market does not evolve over time,fixed costs raise the profitabilty hurdle for new releases and likely lead to more feature bundlingcompared to the case without fixed costs. Furthermore, the release manager may find herself in asituation in which there is no way to recover the fixed cost of development for a new release giventhe features available for implementation. As a result, she must “leave features on the table” andterminate development. Proposition 6 indicates that this outcome is impossible when there are nofixed costs. A stochastically deteriorating market environment only exacerbates these tensions.

We can not apply the results of Section 5 when c > 0, as Theorem 2 fails when fixed costsare strictly positive. Instead, we approximate the value function of the SMDP with a functionalform inspired by the decomposition (10), and employ the linear programming approach to approx-imate dynamic programming to generate an upper bound on the value function at the intial state(x0,F

0A). This technique leads to a policy that can provably outperform a straightforward certainty-

equivalence heuristic by an arbitrarily large margin, and also performs well in numerical trials.

6.1. Infinite Dimensional Linear Program Consider the following linear program:(LP(x0,F0

A)

)infv∈Vb v

(x0,F

0A

)s.t. v (x,FA)≥ r ((x,FA) , φ) + e−βτ(φ)E

[v (X (τ (φ)) ,FA\φ)

∣∣∣X (0) = x]

∀ ((x,FA) , φ)∈K ,


with r defined as in (1), and optimal value LP ∗(x0,F0

A). The following standard result relates the

linear program to the optimality equations presented in Theorem 1(i):

Proposition 12. The value function V ∗ is an optimal solution to LP(x0,F0A).

Proof. Theorem 1.(i) and the definition (2) of L indicate that V ∗ is a feasible solution toLP(x0,F0

A), and so V ∗ (x0,F0A)≥LP ∗

(x0,F0A)

. However, for any arbitrary feasible v, the constraints of

the linear program require that v ≥ Lv. It follows from Theorem 6.2.2 in Puterman (2005) thatv≥ V ∗ on S , implying that v (x0,F

0A)≥ V ∗ (x0,F

0A). Since v was arbitrary, LP(x0,F0

A) ≥ V∗ (x0,F

0A),

which completes the proof. We aim to exploit the modified discount factor and ratio invariance properties discussed in §4.3 to

simplify the constraints in LP(x0,F0A). Recall, however, that under multiplicative separability these

properties only apply with probability one at every stage market intensity process, not necessarilyon the entire set [x0,∞). To address this issue, we introduce the following definitions:Definition 1. Define the sets Xϑ∗ and Rx0 as follows:

Xϑ∗ :=

x∈ [x0,∞)

∣∣∣g(x j∏i=1

ξi,FI

)= g (x,FI)

j∏i=1

ϑ∗ (ξi) ∀FI ⊆ F 0I ∪F 0

A ∀j ∈N a.e.-P

,

Rx0 :=

x∈ [x0,∞)

∣∣∣g (x,FI)

g (x,F ′I)=g (x0,FI)

g (x0,F ′I)∀FI ,F ′I ⊆ F 0

I ∪F 0A s.t. F ′I 6= ∅

.

We assume that both of these sets are measurable. The set Xϑ∗ consists of all values x ∈ [x0,∞)where multiplicative separability holds relative to the function ϑ∗, and Assumption 11 implies thatx0 ∈Xϑ∗ . Similarly, the set Rx0 contains all intensity levels for which the function g exhibits ratioinvariance with respect to x0, and it follows trivially that x0 ∈Rx0 . In light of Definition 1, we turnour attention to the following relaxation of LP(x0,F0

A):

(LP (x0,F0

A)

)infv∈Vb v

(x0,F

0A

)s.t. v (x,FA)≥ r ((x,FA) , φ) + e−βτ(φ)E

[v (X (τ (φ)) ,FA\φ)

∣∣∣X (0) = x]

∀ ((x,FA) , φ)∈K s.t. x∈Xϑ∗ ∩Rx0 ,

with optimal value LP∗

(x0,F0A). By leveraging the fact that the set Xϑ∗ ∩Rx0 is closed under transi-

tions in the market intensity process (Adelman and Mancini (2014)), the next result indicates thatthis relaxation is without loss of optimality. A proof is included in Appendix B.

Proposition 13. LP ∗(x0,F0

A)= LP

∗

(x0,F0A).

It follows immediately from Propositions 12 and 13 that any feasible solution to LP (x0,F0A) generates

an upper bound on V ∗ (x0,F0A).

Before proceeding, we further simplify LP (x0,F0A). The following result extends the closed-form

expression (7) for the inter-epoch reward r to all intensity levels in the set Xϑ∗ . The proof followsfrom the definition of the set Xϑ∗ and Theorem 3 in Adelman and Mancini (2014).

Proposition 14. For all FA ⊆ F 0A and all φ⊆ FA,

r ((x,FA) , φ) =−c ·1φ 6=∅+

[1− e−γ∗τ(φ)

γ∗

]g (x,FI) ∀x∈Xϑ∗ .


Therefore, the linear program LP (x0,F0A) becomes:(

LP (x0,F0A)

)infv∈Vb v

(x0,F

0A

)s.t. v (x,FA)≥−c ·1φ6=∅+

[1− e−γ∗τ(φ)

γ∗

]g (x,FI) +


∣∣∣X (0) = x]

∀ ((x,FA) , φ)∈K s.t. x∈Xϑ∗ ∩Rx0 .

6.2. A Value Function Approximation and Approximate Policy Depending on thecardinality of the set Xϑ∗ ∩Rx0 , the linear program LP (x0,F0

A) may have an uncountable set of

variables and constraints. Instead of attempting to solve this program directly, we introduce aset of surrogate optionality multipliers Θ(FA) : FA ⊆ F 0

A ⊆ R and the following value functionapproximation motivated by the decomposition (10):

v (x,FA)≈ g (x,FI)

γ∗Θ(FA) ∀ (x,FA)∈S . (34)

Inserting the approximation (34) into LP (x0,F0A), we obtain:

(ALP (x0,F0

A)

)infΘ

g (x0,F0I )

γ∗Θ(F 0A

)s.t.

g (x,FI)

γ∗Θ(FA)≥−c ·1φ6=∅+

[1− e−γ∗τ(φ)

γ∗

]g (x,FI) + (35)

e−βτ(φ)E

[g (X (τ (φ)) ,FI ∪φ)

γ∗Θ(FA\φ)

∣∣∣X (0) = x

]∀ ((x,FA) , φ)∈K s.t. x∈Xϑ∗ ∩Rx0

with optimal value ALP∗

(x0,F0A). By Propositions 12 and 13, the multipliers Θ∗ (FA) : FA ⊆ F 0

Aobtained by solving this linear program provide the tightest upper bound on V ∗ (x0,F

0A) among all

approximations of the form (34). We proceed by showing that the semi-infinite program ALP (x0,F0A)

reduces to the following finite linear program:(ALP (x0,F0

A)

)infΘ

g (x0,F0I )

γ∗Θ(F 0A

)s.t.

g (x0,FI)

γ∗Θ(FA)≥−c ·1φ 6=∅+

[1− e−γ∗τ(φ)

γ∗

]g (x0,FI) + (36)

e−γ∗τ(φ) g (x0,FI ∪φ)

γ∗Θ(FA\φ) ∀FA ⊆ F 0

A ∀φ⊆ FA,

with optimal value ALP∗

(x0,F0A). Unlike ALP (x0,F0

A), the linear program ALP (x0,F0A) only has con-

straints corresponding to the initial intensity level x0.

Proposition 15. There exists an optimal solution Θ∗ (FA) : FA ⊆ F 0A to ALP (x0,F0

A), and

ALP∗

(x0,F0A) = ALP

∗

(x0,F0A).

Proof. By Theorem 2, the optionality multipliers Λ∗ (FA) : FA ⊆ F 0A are a feasible solution to

ALP (x0,F0A) because c > 0. Furthermore, Assumptions 4 and 5 indicate that g (x0,FI) > 0 for all

FA ⊂ F 0A. The constraints corresponding to the set (FA,∅) : FA ⊆ F 0

A thus indicate that for any

feasible solution Θ, Θ(FA)≥ 1 for all FA ⊆ F 0A. The optimal value ALP

∗

(x0,F0A) is therefore finite.


Hence, there exists an optimal solution to the linear program ALP (x0,F0A) since it is finite, feasible,

and bounded.The process N is Poisson by Assumption 6, and so the definition of the set Xϑ∗ together with

the proof of Theorem 2 indicates:

e−βτ(φ)E

[g (X (τ (φ)) ,FI ∪φ)

γ∗Θ(FA\φ)

∣∣∣X (0) = x

]= e−γ

∗τ(φ) g (x,FI ∪φ)

γ∗Θ(FA\φ)

∀ ((x,FA) , φ)∈K s.t. x∈Xϑ∗ ∩Rx0 ,

which converts the constraints (35) into: for all ((x,FA) , φ)∈K such that x∈Xϑ∗ ∩Rx0 ,

g (x,FI)

γ∗Θ(FA) ≥ −c ·1φ6=∅+

[1− e−γ∗τ(φ)

γ∗

]g (x,FI) + e−γ

∗τ(φ) g (x,FI ∪φ)

γ∗Θ(FA\φ) . (37)

Dividing both sides by g(x,FI )

γ∗ , and employing the definition of Rx0 yields the equivalent set ofconstraints:

Θ(FA) ≥ −cγ∗

g (x,FI)·1φ6=∅+

(1− e−γ

∗τ(φ))

+ e−γ∗τ(φ)b (FI ∪φ,FI)Θ(FA\φ)

∀ ((x,FA) , φ)∈K s.t. x∈Xϑ∗ ∩Rx0 . (38)

However, for any FA and φ⊆ FA, all constraints in (38) with x > x0 are redundant since −c≤ 0

and g (x,FI) is nonincreasing in x by Assumption 5. Hence, to solve ALP (x0,F0A) it suffices to solve

the following linear program:

infΘ

g (x0,F0I )

γ∗Θ(F 0A

)s.t.

g (x0,FI)

γ∗Θ(FA)≥−c ·1φ6=∅+

[1− e−γ∗τ(φ)

γ∗

]g (x0,FI) +

e−γ∗τ(φ) g (x0,FI ∪φ)

γ∗Θ(FA\φ) ∀FA ⊆ F 0

A ∀φ⊆ FA,

which we obtain from ALP (x0,F0A) by retaining only those constraints in (37) corresponding to the

intensity level x0. The result follows as this linear program is identical to ALP (x0,F0A).

The linear program ALP (x0,F0A) has 2

∣∣F0A

∣∣variables and 3

∣∣F0A

∣∣constraints. As we show in §6.4, when

FA is sufficiently small we can solve ALP (x0,F0A) numerically to obtain the minimizing values Θ∗

and an upper bound on V ∗ (x0,F0A).

For any (x,FA) ∈S , let Θ∗ (F ′A; (x,FA)) : F ′A ⊆ FA denote the optimal solution to the linear

program ALP (x,FA) constructed by substituting (x,FA) for (x0,F0A) in the definition of ALP (x0,F0

A).

Our approximation technique thus gives rise to the following decision rule:

dΘ∗ (x,FA) ∈ arg maxφ⊆FA

−c ·1φ 6=∅+

(1− e−γ∗τ(φ)

γ∗

)g (x,FI)+

e−γ∗τ(φ) g (x,FI ∪φ)

γ∗Θ∗ (FA\φ; (x,FA))

∀ (x,FA)∈S , (39)

which we evaluate at each decision epoch by re-solving ALP . For the remainder of the paper, weevaluate the performance of the stationary deterministic policy d∞Θ∗ . We denote the value of thispolicy by VΘ∗ .


6.3. Comparison to a Certainty-Equivalent Heuristic As a benchmark, we consider aheuristic corresponding to the solution of a deterministic dynamic program in which the markettransitions along its expected path. The release manager’s inter-epoch reward function becomes

rCE ((x,FA) , φ) := −c ·1φ6=∅+

∫ τ(φ)

0

e−βug(E[X (u)

∣∣∣X (0) = x],FI

)du

∀ ((x,FA) , φ)∈K , (40)

and she constructs a certainty-equivalent heuristic by solving the following optimality equations:

v∗CE (x,FA) = maxφ⊆FA

rCE ((x,FA) , φ) + e−βτ(φ)v∗CE

(E[X (τ (φ))

∣∣∣X (0) = x],FA\φ

)∀ (x,FA)∈S . (41)

While the certainty-equivalent dynamic program (CE-DP) may be easier to solve than our SMDP,we now show that it can perform arbitrarily poorly compared to the approximation heuristic d∞Θ∗ .Throughout the following analysis, we assume that g (x,FI) = z(FI )

xαis defined as in (9) with α= 1.

First, we simplify the inter-epoch reward function (40). Assumption 6 indicates

E[X (u)

∣∣∣X (0) = x]

= xeλ(E[ξ]−1)u for all x∈ [x0,∞). The functional form of g from (9) thus implies

rCE ((x,FA) , φ) = −c ·1φ6=∅+

∫ τ(φ)

0

e−βu[

z (FI)

xeλ(E[ξ]−1)u

]du

= −c ·1φ6=∅+z (FI)

γ′x

(1− e−γ

′τ(φ))∀ ((x,FA) , φ)∈K , (42)

with γ′ := β+λ (E [ξ]− 1). As in our SMDP, the certainty-equivalent dynamic program also utilizesa modified discount factor to account for market intensity. However, the next result shows that thecertainty-equivalent discount factor γ′ is conservative relative to the modified discount factor γ∗.

Proposition 16. The discount factor γ∗ is strictly less than the discount factor γ′.

Proof. It follows from the definitions of γ∗ and γ′ that

γ∗ <γ′ ⇐⇒(

1−E[

1

ξ

])< (E [ξ]− 1) . (43)

Jensen’s inequality yields

1−E[

1

ξ

]< 1− 1

E [ξ],

and so it suffices to prove that 1− 1E[ξ]≤ (E [ξ]− 1). However, (43) thus follows immediately since

1− 1

E [ξ]≤E [ξ]− 1 ⇐⇒ 0≤ (E [ξ])

2− 2E [ξ] + 1 = (E [ξ]− 1)2

and (E [ξ]− 1)2 ≥ 0.

We now exploit the discrepancy in the two discount factors to expose the performance gapbetween the certainty-equivalent heuristic and our approximation policy.

Proposition 17. Assume that c > 0, F 0A = f, and that g (x,FI) = z(FI )

xαis defined as in (9)

with α= 1. For any J ≥ 1, there exists a value H ≥ 0 and intensity level x such that if z (F 0I ∪f)−

z (F 0I )≥H, then

VΘ∗ (x,f)VCE (x,f)

≥ J.


Proof. First, we compute the decision rule dΘ∗ . For arbitrary x∈ [x0,∞), it follows by inspection

that Θ∗ (∅; (x,f)) = 1 for the linear program ALP (x,f). The definition (39) thus implies thatdΘ∗ (x,f) = f if and only if

−c+

(1− e−γ∗τ(f)

γ∗z (F 0

I )

x+e−γ

∗τ(f)

γ∗z (F 0

I ∪f)x

)≥ z (F 0

I )

γ∗x

⇐⇒ −c+e−γ

∗τ(f)

γ∗

[z (F 0

I ∪f)− z (F 0I )

x

]≥ 0

⇐⇒ x≤ xΘ∗ ,

with

xΘ∗ := sup

x∈R

∣∣∣− c+e−γ

∗τ(f)

γ∗

[z (F 0

I ∪f)− z (F 0I )

x

]≥ 0

=

(e−γ

∗τ(f)

cγ∗

)[z(F 0I ∪f

)− z

(F 0I

)]since c > 0. (44)

By Assumption 5, z (F 0I ∪f)− z (F 0

I )> 0 and so xΘ∗ must be positive. For simplicity, we assumethat xΘ∗ >x0. It follows that

VΘ∗ (x,f) =

−c+z(F0

I )γ∗x + e−γ

∗τ(f)

γ∗

[z(F0

I ∪f)−z(F0I )

x

], x0 ≤ x≤ xΘ∗

z(F0I )

γ∗x , x∈ (xΘ∗ ,∞)

VΘ∗ (x,∅) =z (F 0

I ∪f)γ∗x

∀x∈ [x0,∞) .

In Appendix C, we show that the certainty-equivalent heuristic selects the waiting action ifmarket intensity exceeds the threshold

xCE := sup

x∈R

∣∣∣− c+e−γ

′τ(f)

γ′

[z (F 0

I ∪f)− z (F 0I )

x

]≥ 0

=

(e−γ

′τ(f)

cγ′

)[z(F 0I ∪f

)− z

(F 0I

)]because c > 0. (45)

For simplicity, we assume that xCE > x0. If x > xCE, the value of the CE heuristic in the contextof our SMDP is thus

VCE (x,f) =z (F 0

I )

γ∗x∀x∈ (xCE,∞) .

because the heuristic selects the waiting action at every decision epoch.Comparing (44) to (45) and applying Proposition 16 immediately indicates that xCE < xΘ∗ .

Therefore, for any x ∈ (xCE, xΘ∗) the approximate policy opts to develop the feature f while theCE heuristic decides to wait. We now exploit this observation by considering a specific intensitylevel between the two thresholds.

Let J ≥ 1 be arbitrary and define x := (xCExΘ∗)1/2

, the geometric mean of the two thresholds.Since xCE < x< xΘ∗ , we have:

VΘ∗ (x,f)VCE (x,f)

=−c+ 1−e−γ

∗τ(f)

γ∗z(F0

I )x

+ e−γ∗τ(f)

γ∗z(F0

I ∪f)x

z(F0I )

γ∗x


=−cγ∗xz (F 0

I )+e−γ

∗τ(f) [z (F 0I ∪f)− z (F 0

I )]

z (F 0I )

+ 1

=−cγ∗

z (F 0I )

z (F 0I ∪f)− z (F 0

I )

c

√e−(γ∗+γ′)τ(f)

γ∗ · γ′

+

e−γ∗τ(f) [z (F 0

I ∪f)− z (F 0I )]

z (F 0I )

+ 1 by defn. of x

=e−γ

∗τ(f) [z (F 0I ∪f)− z (F 0

I )]

z (F 0I )

1−

√γ∗

γ′e−

12(γ′−γ∗)τ(f)

+ 1

≥ J ⇐⇒[z(F 0I ∪f

)− z

(F 0I

)]≥H :=

z (F 0I ) (J − 1)

e−γ∗τ(f)

[1−

√γ∗

γ′ e− 1

2 (γ′−γ∗)τ(f)] .

Note that H ≥ 0 and H <∞ since γ∗ <γ′ by Proposition 16. In summary, the certainty-equivalent heuristic performs poorly in this case because the associated

discount factor γ′ induces an overly conservative termination threshold xCE. The penalty for thisconservatism increases without bound as the incremental value of the feature f increases.

6.4. Numerical Performance We evaluated the performance of the approximation-basedpolicy d∞Θ∗ as follows: (i) solve ALP (x0,F0

A) to obtain an upper bound on V ∗ (x0,F0A), (ii) simulate

d∞Θ∗ along 1,000 randomly generated paths, re-solving ALP at each decision epoch and identifyingthe maximizing action (39) via enumeration, and (iii) compare the average performance of thepolicy to the upper bound calculated in (i) to obtain an upper bound on the optimality gap ofd∞Θ∗ . In addition, we also simulated the CE-DP heuristic along the same set of paths and comparedits average performance to that of the approximation-based policy. Unless otherwise noted, we

assumed: (i) g (x,FI) = z(FI )

xαas in (9) with α = 1, (ii)

∣∣∣F 0A

∣∣∣ = 10, (iii) c = 10, (iv) x0 = 10, (v)

z (F 0I ) = 10, (vi) ξ ∼Unif [1,1.1], (vii) τ is linear, (viii) z is linear, (ix) λ corresponds to an expected

12 releases per year, and (x) β corresponds to a 10% annual interest rate. All linear programs werecoded in AMPL and solved with Gurobi 5.0 (6 core Intel Xeon at 3.33 GHz, 4 GB of RAM). The

average solution time for ALP (x0,F0A) was 1.69 seconds, with a standard deviation of 0.15 seconds.

The statistics for solving CE-DP via linear programming were nearly identical.By construction, the policy d∞Θ∗ is exact when c = 0. The first plot in Figure 2 indicates that

the approximation-based policy also performs well for strictly positive fixed costs. Even thoughthe policy’s performance degrades slightly for higher values of c, it always performs within 3.5% ofoptimality and consistently beats the CE-DP heuristic. The second plot shows that the performancegap between the two policies tends to increase in c, with a maximum observed gap of 5.5%.

Our second set of trials evaluates the approximation-based policy against markets with varyingshock frequency and magnitude. Fixed costs were held constant at the default value of 10. Figure3 displays the results for the case of uniformly distributed shocks. Lognormal shocks were alsoconsidered, with nearly identical results. As indicated by the first plot, the approximation-basedpolicy performed within 2% of optimality in all scenarios regardless of the market shock frequency.In some cases, the expected value of our policy slightly exceeds the upper bound obtained fromALP (x0,FA). This behavior typically indicates that the policy is high-performing, and a much largernumber of paths is required to obtain a more accurate estimate of the policy’s value. As in the firstset of trials, the second plot shows that our policy consistently outperforms the certainty-equivalentheuristic. Furthermore, the performance gap increases as the market worses, i.e., exhibits higheraverage shock frequency and expected shock magnitude.

Finally, we discuss an additional advantage of our approximation-based policy relative to the CE-DP heuristic. Even though the CE-DP has an uncountable state space, the fact that its transitions


0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 10 20 30 40 50 60

LB

/U

B

Fixed Cost (c)

d∞Θ∗ Optimality Gap vs. Fixed Cost

Approx. PolicyCE-DP Heuristic

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

1.05

1.055

0 10 20 30 40 50 60

VΘ∗

(x0,F

0 A)/VCE

(x0,F

0 A)

Fixed Cost (c)

Policy Comparison by Fixed Cost

Figure 2. Sensitivity of Policy Performance to Fixed Cost

0.984

0.986

0.988

0.99

0.992

0.994

0.996

0.998

1

1.002

1.004

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

LB

/UB

100 · (E [ξ]− 1)

d∞Θ∗ Optimality Gap vs. Rival Strength

AnnuallyQuarterlyMonthly

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

VΘ∗

(x0,F

0 A)/V

CE

(x0,F

0 A)

100 · (E [ξ]− 1)

Policy Comparison by Rival Strength

AnnuallyQuarterlyMonthly

Figure 3. Optimality Gap of Approximate Policy vs. Expected Shock Strength by Average Shock Frequency

are deterministic means we can solve it over a finite state space by tracking the set of availablefeatures as well as the expected intensity level given the amount of time that has elapsed since thebeginning of the horizon. Suppose the release manager has the set FA ⊆ F 0

A of features availablefor development. If τ is additive, then, regardless of the order in which previous releases wereimplemented, the total time required for all earlier releases must be τ (F 0

A\FA). However, if τ is


not additive, then we must account for all possible order-dependent release sequences that resultin the current set of available features FA. The memory requirements for storing the expandedstate space quickly become prohibitive. For instance, with only ten features the number of statesin the expanded CE-DP is roughly 660 times greater than the number of states required when τis additive. Our approximation-based policy does not suffer from this issue. Furthermore, whenwe performed trials with non-additive specifications for τ , the results and solution times for ourapproximation-based policy were very similar to those from the additive case discussed earlier.

7. Conclusion By adapting the assumptions from Adelman and Mancini (2014) to the releasemanagement problem, we obtained two of the key ingredients for our analysis: (i) the existence ofa quasi-open-loop optimal policy, and (ii) the ability to study this policy by analyzing a reduceddynamic program that utilizes a modified discount factor. While we can relax some of the assump-tions presented here without invalidating our approach, doing so would complicate our analysiswithout leading to additional insights. For example, we retain the existence of a quasi-open-loopoptimal policy even if the shock-time process N is a generic counting process, so long as it stillsatisfies the exogeneity assumptions in §4. However, there no longer exists a modified discountfactor that encodes the deflationary impact of market intensity on revenues. Similarly, Assumption6, part (iii) of Asssumption 9, and Assumption 11 together suffice for the existence of a modifieddiscount factor akin to the one identified in Proposition 1. However, parts (i) and (ii) of Assumption9, as well as Assumption 10, ease our analysis by ensuring that this modified discount factor doesnot depend on the intensity level, the release manager’s actions, or the release manager’s product.Hence, the same modified discount factor applies at every epoch.

The third crucial component of our analysis is the assumption that development times aredeterministic and depend only on the features being developed. The determinism of τ makes thereduced dynamic program used to analyze the quasi-open-loop optimal policy deterministic and,therefore, more tractable. However, if the development time mapping τ is a strictly positive randomvariable parameterized by the release manager’s action φ and independent of the release manager’sexisting product FI and market intensity, then we essentially retain all of our analytical results bytaking expectations. For instance, the single-feature index from Corollary 1 becomes

E[e−γ

∗τ(f)]g (x0,f)

1−E [e−γ∗τ(f)],

which corresponds to the “weighted discounted shortest expected processing time first” rule fromthe stochastic scheduling literature (Pinedo (2012)). A second extension is to allow developmenttimes to depend on the set of features in the release manager’s product. Theorems 2 and 3 remainvalid in this case with only minor adjustments, as does the ADP implementation discussed in §6.However, this extension requires modifications to the exchange condition in Proposition 8 thatobfuscate any underlying insights (Appendix D). Finally, we note that a third consequence of ourassumption on development times is that the release manager can not react to a market shockuntil the current iteration of her product is complete. In fact, we can model τ as a randomizedstopping time of the shock-time process N without sacrificing the existence of a quasi-open-loopoptimal policy (Adelman and Mancini (2014)). However, we lose the tractability afforded by themodified discount factor from Proposition 1.

Our results suggest several avenues for further research. First, one could consider embedding ourtactical model in a framework in which a strategic analysis of rival products directs the research anddevelopment efforts that ultimately furnish the features available to the release manager. Similarly,strategic investment decisions would influence the release manager’s development capabilities, andtherefore define the τ mapping. Second, our SMDP does not allow for new features to arrive overtime. While a release manager could accommodate random arrivals by updating the set of available


features at each decision epoch when implementing the approximate policy discussed in §6, ourresults would no longer provide an upper bound on the value function with which to evaluate theperformance of the approximate policy. Recovering an upper bound entails the difficult task ofextending the SMDP state space to account for the entire universe of possible features, and solvingthe associated infinite dimensional linear program. Third, we believe that exploring the relationshipbetween multiplicative separability and primitive models of consumer choice would further informmanagers of the applicability of our results. Fourth, our model assumes that new releases have aninstantaneous impact on revenues. Allowing consumer decisions to take place over time necessitatesa generalization of the crucial modified discount factor result (Proposition 1). Similarly, we do notconsider the switching costs consumers incur when transitioning to a new release, or the impact oflimited market size on the release manager’s decisions.

From a computational perspective, the numerical performance of the approximate policy in §6justifies research into the scalability of the underlying linear program ALP . Since the size of thislinear program grows exponentially in the number of available features, we likely can not capturelarge-scale instances of the release management problem. However, discussions with managers at amajor software firm indicate that it is typical for a small number of high priority features to dom-inate the feature selection process, and so our existing approach still provides value. Nevertheless,a scalable algorithm for our approximate linear program would make the model more applicable.

To the best of our knowledge, our model is the first to consider release management from anoperational perspective in a dynamic, combinatorial setting. We provided significant managerialinsights in the case with no fixed costs, and built on this analysis to construct a high-performingpolicy for release managers facing positive fixed costs. Our hope is that this work serves as thefoundation for a fruitful discussion of the combinatorial issues involved in dynamic release man-agement.

Appendix A: Proof of Proposition 7 Combining the relation (17) with a recursive appli-cation of (11) yields:

Λ∗(F 0A

)=(

1− e−γ∗τ(φ∗1)

)+ e−γ

∗τ(φ∗1)b(F 0I ∪φ∗1,F 0

I

)Λ∗(F 0A\φ∗1

)=(

1− e−γ∗τ(φ∗1)

)+ e−γ

∗τ(φ∗1)b(F 0I ∪φ∗1,F 0

I

)·[(

1− e−γ∗τ(φ∗2)

)+ e−γ

∗τ(φ∗2)b(F 0I ∪φ∗1 ∪φ∗2,F 0

I ∪φ∗1)

Λ∗(F 0A\ (φ∗1 ∪φ∗2)

)]by (11)

=(

1− e−γ∗τ(φ∗1)

)+(

1− e−γ∗τ(φ∗2)

)e−γ

∗τ(φ∗1)b(F 0I ∪φ∗1,F 0

I

)+

e−γ∗(τ(φ∗1)+τ(φ∗2))b

(F 0I ∪φ∗1 ∪φ∗2,F 0

I

)Λ∗(F 0A\ (φ∗1 ∪φ∗2)

)by defn. (8) of b

...

=

[j∗∑i=1

(1− e−γ

∗τ(φ∗i ))e−γ

∗[∑i−1n=1 τ(φ

∗n)]b

(F 0I ∪(∪i−1n=1φ

∗n

),F 0

I

)]+

e−γ∗

[∑j∗i=1 τ(φ

∗i )]b(F 0I ∪F 0

A,F0I

). (46)

Substituting the definition (12) into (46) and applying the definition (8) of b, we obtain:

V ∗(x0,F

0A

)=

[j∗∑i=1

(1− e−γ

∗τ(φ∗i )

γ∗

)e−γ

∗[∑i−1n=1 τ(φ

∗n)]g

(x0,F

0I ∪(∪i−1n=1φ

∗n

))]+

e−γ∗

[∑j∗i=1 τ(φ

∗i )]

γ∗g(x0,F

0I ∪F 0

A

)


=

j∗∑i=1

e−γ∗[∑i−1n=1 τ(φ

∗n)] g

(x0,F

0I ∪(∪i−1n=1φ

∗n

))γ∗

−

j∗∑i=1

e−γ∗[∑in=1 τ(φ∗n)] g

(x0,F

0I ∪(∪i−1n=1φ

∗n

))γ∗

+ e−γ∗

[∑j∗i=1 τ(φ

∗i )]g (x0,F

0I ∪F 0

A)

γ∗

=g (x0,F

0I )

γ∗+

j∗−1∑i=1

e−γ∗[∑in=1 τ(φ∗n)] g (x0,F

0I ∪ (∪in=1φ

∗n))

γ∗−

j∗∑i=1

e−γ∗[∑in=1 τ(φ∗n)] g

(x0,F

0I ∪(∪i−1n=1φ

∗n

))γ∗

+ e−γ∗

[∑j∗i=1 τ(φ

∗i )]g (x0,F

0I ∪F 0

A)

γ∗

=g (x0,F

0I )

γ∗+

j∗∑i=1

e−γ∗[∑in=1 τ(φ∗n)]

[g (x0,F

0I ∪ (∪in=1φ

∗n))− g

(x0,F

0I ∪(∪i−1n=1φ

∗n

))γ∗

],

which completes the proof.

Appendix B: Proof of Proposition 13 By construction, LP ∗(x0,F0

A)≥ LP

∗

(x0,F0A). Suppose

now that v is an arbitrary feasible solution to LP (x0,F0A), and for all FA ⊆ F 0

A let

v (x,FA) :=

v (x,FA) , ∀x∈Xϑ∗ ∩Rx0

C, ∀x∈x∈ [x0,∞)

∣∣∣x /∈Xϑ∗ ∩Rx0

for some C ∈ [0,∞). By construction, v ∈ Vb. We proceed by showing that for sufficiently large

values of C, the function v is a feasible solution to LP(x0,F0A). Hence, LP ∗

(x0,F0A)≤ LP

∗

(x0,F0A) and

so the result follows.By Lemma 2 in Adelman and Mancini (2014), if x∈Xϑ∗ ∩Rx0 , then

P(X (u)∈ (Xϑ∗ ∩Rx0)

∣∣∣X (0) = x)

= 1 for all u ∈ [0,∞). The constraints of the linear program

LP(x0,F0A) evaluated at v thus become:

v (x,FA) ≥ r ((x,FA) , φ) + e−βτ(φ)E[v (X (τ (φ)) ,FA\φ)

∣∣∣X (0) = x]

∀ ((x,FA) , φ)∈K s.t. x∈Xϑ∗ ∩Rx0

C ≥ r ((x,FA) , φ) +


∣∣∣X (τ (φ))∈ (Xϑ∗ ∩Rx0) ,X (0) = x]·

P(X (τ (φ))∈ (Xϑ∗ ∩Rx0)

∣∣∣X (0) = x)

+

C ·P(X (τ (φ)) /∈ (Xϑ∗ ∩Rx0)

∣∣∣X (0) = x)

∀ ((x,FA) , φ)∈K s.t. x /∈Xϑ∗ ∩Rx0 .

The first set of constraints are satisfied since v is a feasible solution of LP (x0,F0A). Furthermore, the

following inequality implies that, for sufficiently large values of C, the second set of inequalities isalso satisfied:

e−βτ(φ)P(X (τ (φ)) /∈ (Xϑ∗ ∩Rx0)

∣∣∣X (0) = x)≤ 1 ∀φ⊆ F 0

A ∀x≥ 0.

Appendix C: Supplement to Proof of Proposition 17 We start by showing that

v∗CE (x,f)≥ z (F 0I )

γ′x∀x∈ [x0,∞) , (47)


with the CE-DP value function v∗CE defined as in (41).Recall that τ (∅) =w> 0. The CE-DP optimality equations imply

v∗CE (x,f)≥ z (F 0I )

γ′x

(1− e−γ

′w)

+ e−βwv∗CE(xeλ(E[ξ]−1)w,f

)∀x∈ [x0,∞) . (48)

Fixing a value x∈ [x0,∞) and recursively applying (48), we obtain:

v∗CE (x,f) ≥ z (F 0I )

γ′x

(1− e−γ

′w)

+ e−βwv∗CE(xeλ(E[ξ]−1)w,f

)≥ z (F 0

I )

γ′x

(1− e−γ

′w)

+ e−βw[

z (F 0I )

γ′xeλ(E[ξ]−1)w

(1− e−γ

′w)

+ e−βwv∗CE(xeλ(E[ξ]−1)(2w),f

)]=z (F 0

I )

γ′x

(1− e−γ

′w)(

1 + e−γ′w)

+ e−β(2w)v∗CE(xeλ(E[ξ]−1)(2w),f

)≥ z (F 0

I )

γ′x

(1− e−γ

′w)(

1 + e−γ′w)

+ e−β(2w)

[z (F 0

I )

γ′xeλ(E[ξ]−1)(2w)

(1− e−γ

′w)

+ e−βwv∗CE(xeλ(E[ξ]−1)(3w),f

)]=z (F 0

I )

γ′x

(1− e−γ

′w)(

1 + e−γ′w + e−γ

′(2w))

+ e−β(3w)v∗CE(xeλ(E[ξ]−1)(3w),f

)...

≥ z (F 0I )

γ′x

(1− e−γ

′w)( n∑

i=0

e−(γ′w)i

)+ e−β((n+1)w)v∗CE

(xeλ(E[ξ]−1)(n+1)w,f

)∀n∈N.

Since β > 0 and v∗CE is bounded, letting n→∞ yields (47).Next, a similar recursion indicates that

v∗CE (x,∅) =z (F 0

I ∪f)γ′x

∀x∈ [x0,∞) (49)

because the release manager must terminate development once the only available feature f hasbeen implemented.

Finally, let x be an arbitrary intensity level that exceeds the certainty-equivalent threshold, i.e.,

x> xCE = sup

x∈R

∣∣∣− c+e−γ

′τ(f)

γ′

[z (F 0

I ∪f)− z (F 0I )

x

]≥ 0

. (50)

Applying (47), we obtain:

v∗CE (x,f) ≥ z (F 0I )

γ′x

>z (F 0

I )

γ′x+

−c+

e−γ′τ(f)

γ′

[z (F 0

I ∪f)− z (F 0I )

x

]by (50)

= −c+z (F 0

I )

γ′x

(1− e−γ

′τ(f))

+ e−γ′τ(f) z (F 0

I ∪f)γ′x

= −c+z (F 0

I )

γ′x

(1− e−γ

′τ(f))

+ e−βτ(f)v∗CE(xeλ(E[ξ]−1)τ(f),∅

)by (49).

The final inequality indicates that developing feature f in state (x,f) is suboptimal, and so thewaiting action is optimal. .


Appendix D: Product-Dependent Development Times Suppose that the developmenttime mapping τ is deterministic, but depends on the set of features currently in the release man-ager’s product. Letting τ (φ;φ′) denote the development time for the bundle φ when the existingproduct consists of the features F 0

I ∪φ′, the exchange argument applied in Proposition 8 yields: forall m∈ 1,2, . . . , j∗− 1,

e−γ∗τ(φ∗m;∪m−1

n=1 φ∗n) [g (x0,F

0I ∪ (∪mn=1φ

∗n))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]1− e−γ∗τ(φ∗m;∪m−1

n=1 φ∗n∪φ∗m+1)

+ (51)

e−γ∗[τ(φ∗m;∪m−1

n=1 φ∗n)+τ(φ∗m+1;∪mn=1φ

∗n)] [g (x0,F

0I ∪(∪m+1n=1 φ

∗n

))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))](1− e−γ∗τ(φ∗m;∪m−1

n=1 φ∗n∪φ∗m+1)

)(1− e−γ∗τ(φ∗m+1;∪mn=1φ

∗n)) ≥ (52)

e−γ∗τ(φ∗m+1;∪m−1

n=1 φ∗n) [g (x0,F

0I ∪(∪m−1n=1 φ

∗n

)∪φ∗m+1

)− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))]1− e−γ∗τ(φ∗m+1;∪mn=1φ

∗n)

+ (53)

e−γ∗[τ(φ∗m+1;∪m−1

n=1 φ∗n)+τ(φ∗m;∪m−1

n=1 φ∗n∪φ

∗m+1)]

[g(x0,F

0I ∪(∪m+1n=1 φ

∗n

))− g

(x0,F

0I ∪(∪m−1n=1 φ

∗n

))](1− e−γ∗τ(φ∗m;∪m−1

n=1 φ∗n∪φ∗m+1)

)(1− e−γ∗τ(φ∗m+1;∪mn=1φ

∗n)) . (54)

When development times do not depend on existing functionality, terms (51) and (53) reduce tothose from Proposition 8. Furthermore, the terms (52) and (54) cancel because it takes the sameamount of time to develop φ∗m followed by φ∗m+1 as it does to develop the bundles in reverse order.If development times are sensitive to the release manager’s product, however, (52) and (54) reflectthe possibility that swapping φ∗m and φ∗m+1 impacts the time at which both bundles are completed.

Acknowledgements. Both authors are grateful for the financial support provided by theUniversity of Chicago Booth School of Business.

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Yale School of Management · DO NOT DISTRIBUTE Dynamic Release Management of Product Features Daniel Adelman, Angelo J. Mancini University of Chicago Booth School of Business, Chicago,