Dec 20, 2011 - [2] Artzner P., Delbaen F., Eber J.-M., Heath D. (1997), Thinking Coher- ... [11] George M. Chryssochoidis (2004), Rolling out new products ...
OPTIMAL STRATEGY FOR STOCHASTIC PRODUCT ROLLOVER UNDER RISK USING CVAR ANALYSIS
Christian VAN DELFT HEC School of Management (Greghec), Paris, France Laoucine KERBACHE HEC School of Management (Greghec), Paris, France Hiba EL KHOURY
CR 958-2012 ISBN : 2-85418-958-2
© Groupe HEC, 78351 JOUY-EN-JOSAS CEDEX, France, 2012 ISBN : 2-85418-958-2
Optimal Strategy for Stochastic Product Rollover under risk using CVAR analysis Ch. van Delft, H. El Khoury and L. Kerbache
D´epartement Management des Op´erations et des Syst`emes d’Information HEC - Paris 1 Rue de La Lib´eration, Jouy-en-Josas 78351, France December 20, 2011
Abstract Motivated by many applications such as typical blockbuster product launches, we address in this paper, an inventory/production rollover process between an old and a new product, with a random availability/admissibility date for the new product. The optimization problem consists in finding the phase-in and phase-out dates which minimize a cost minimization objective function. We capture, via a CVar formulation, the risk phenomenon in the rollover decision making. Then, we provide explicit closed-form expressions for the optimal policies, which can be of three types: Planned Stock-out Rollover, Single-Product Rollover, and Dual-Product Rollover. The analysis led to several managerial insights which are provided in the paper. For instance, we exhibit, first, the impact of risk-aversion on the optimal strategy structure. Then, we show that increasing randomness of the availability date (in the stochastic dominance sense) reinforces the structure of the optimal strategy. We show that the stock-out period is increased in case of optimal Planned Stock-out Rollover and the overlap period is increased for optimal Dual-Product Rollover.
1
KEYWORDS: Product rollover; Uncertain approval date; Planned stockout rollover (PSR); Single product rollover (SPR); Dual product rollover(DPR); Risk sensitive optimization criterion; Conditional value at risk (CVaR); Stochastic dominance; Stochastic Comparisons
1
INTRODUCTION
Due to rapid technological development and increased variety demanded by consumers, product life cycles have shortened. Therefore, new products have to be introduced and old products phased out more and more frequently. As new product introduction is a source of growth, renewal and competitive advantage, decision makers are facing the issue of how to successfully manage product replacement and optimize the associated supply chain cost trade-offs. In an ideal setting, the optimal rollover strategy is clear : the old product is phased out at the planned introduction date of the new product, and the new product is readily available. Unfortunately, in real-life it is rarely the case. A study of U.S. durable goods companies ([26]) showed that, for various reasons, more than 50 percent of new products failed after being introduced to the market. These poor product launch performances are due to numerous potential random disruption in the process (unexpected logistic or industrial delays, quality problems, inaccurate demand forecasts, unexpected market reactions to the new product announcements, etc...). How to phase in new products while phasing out old ones has become a challenging managerial problem in companies. Obviously, when a company is planning the phase-out of an existing product and the phase-in of a replacement product, classical stochastic production/inventory tradeoffs have to be considered. The reason is if production of the existing product is stopped too early, i.e. before the new product is available for the market, the firm will lose sales and customer goodwill. On the other hand, if production of the existing product is stopped too late, the firm will experience an obsolescence cost for the existing product, because demand and/or price would have decreased as this product will be considered ”old generation” by the customers. Furthermore, if the production of the new product is launched too early, the firm will experience an inventory carrying cost until demand picks up. The process of launching a new product in the market 2
place and the removal of an old one is known as product rollover. In this paper, we focus on three fundamental strategies: planned stockout rollover, single-product rollover and dual-product rollover. An important issue in new product launch management is whether two product generations coexist in the market for a given some time period, i.e. whether the exists an overlapping between successive product generations. In the planned stockout rollover (PSR) strategy, the introduction of the new product is planned in such a way that a stockout phenomenon occurs during the product transition. During this stockout period, no product of any type is available for the market. In the single-product rollover (SPR) strategy, there is a simultaneous introduction of the new product and elimination of the old product, in such a way that at any time there is a unique product generation available in the market. In the dual-product rollover (DPR) strategy, the new product is introduced first and then the old product is phased out. Thus, in this setting, two product generations coexist in the market, for some period of time. The advantage of the DPR strategy, compared to the SPR policy, is to allow for some protection against backorders due to potential random events (delays, quality or market demand level) affecting the planned phasing. The drawback of the DPR strategy is the cost corresponding to the additional inventory in the supply chain. The purpose of this paper is to analyze and characterize the optimality of each type of strategy (PSR, SPR and DPR) for a setting with a stochastic availability date for the new product and taking risk into consideration. Efficient risk measure and optimization is a complex issue, extensively considered in finance research literature. A somewhat recent risk criterion, called conditional value at risk and usually denoted as CVaR, has emerged as exhibiting interesting tractable theoretical properties (see Rockafellar and Uryasev ([30, 31])). Our CVar model captures the risk issue in the rollover decision making and provides explicit closed-form expressions for the optimal policies. As we consider a quantitative approach, such an analysis requires a performance evaluation model for the supply chain rollover process between two successive products. We provide a newsboy type inventory planning model for the rollover process between two successive products inspired by Hill and Sawaya ([19]) and extended to a risk setting. By solving the associated optimization problem, we obtain the optimality conditions for PSR, SPR or DPR. Furthermore, we show how these different rollover strategies exhibit different 3
properties w.r.t. the risk. Furthermore, we characterize the influence on the optimal strategy structure of the parameters of the setting, of the size of the randomness and of the manager position with respect to the risk. Our theoretical analysis complements the work of Billington et al. [4] and rigorously show how each strategy (PSO, SPR, DPR) can be optimally associated to the risk aversion and uncertainty level. We formally prove several conjectures concerning optimal structures reported in other papers, that were obtained by empirical research. Also, we investigate the behavior of the optimal rollover policy in response to stochastically larger approval processes.
2
Literature Review
Several papers have addressed the question of efficient management of new product launch, old product termination, or combination of these two processes. A first trend of papers about new product development and launch is mainly of qualitative and descriptive nature (see [23] for a review, encompassing work in marketing, operations management, and engineering design). For instance, Chryssochoidis [10, 11] has studied empirically the whole process in many companies. This research describes the critical causes of delay in international product rollover implementation. Saunders and Jobber [32] identify the different types of strategies when implementing a phasein and phase-out process. Several papers have addressed the analysis of new product introduction and product rollover processes, under different assumptions. Erhun et al ([12]) conduct a qualitative study on different drivers affecting product rollovers at Intel Corp., and they develop a framework that guides managers to design and implement appropriate policies taking into consideration rollover risks related to the product, the manufacturing process, the supply chain features, and the managerial policies in a competitive environment. The authors suggest that companies should develop clear strategies for product rollover, in addition to contingency plans in case of failure. They compare and contrast single and dual product rollover strategies. They argue that a single product rollover strategy can be viewed as a high-risk with high return strategy and sensitive to potential random events. On the contrary, the dual product rollover strategy is less risky, but it induces higher inventory costs. Hendricks and Singhal ([18]) have shown by empirical research that delays in new product 4
introduction decrease the market value of the firm. The second trend of papers addresses quantitative modeling and optimization of rollover processes. Lim and Tang ([25]) developed a deterministic model that allows the determination of prices of old and new products and the times of phase-in and phase-out of the products. Moreover, they developed marginal cost based conditions to determine when a dual product rollover strategy is more favorable than a single product rollover strategy. Hill and Sawaya ([19]) examine the problem of simultaneously planning the phase-out of the old product and the phase-in of a new product, under an uncertain regulatory approval date for the new product. Under a usual expected profit criterion, they determine the structure of the optimal strategy, which can be linked to the well known newsvendor problem solution. As our paper deals with a risk-sensitive model, let us refer to the work of Tang ([37]) who provides a concise review of various quantitative models for managing supply chain risks. Most inventory-related papers maximize a predetermined target profit, despite the fact that this may lead to an increased risk. A way to take into account the risk consists of focusing on shortfall, through an absolute bound on the tolerable loss or by setting a bound on the conditional value at risk. Theoretical properties of the CVaR measure of risk has been extensively studied especially in finance (see for example [30, 31]). In inventory theory, some papers have adapted standard results to such risk criterion. For instance, Ozler et al ([29]) utilize Value at Risk (VaR) as a risk measure in a newsboy framework and investigate the multiproduct newsboy problem under a VaR constraint. Some papers ([17, 8]) developed closed form solutions due for a CVar newsboy problem. The structure of this paper is as follows: Section 2 is devoted to the literature while the introduction of the stochastic product rollover problem under consideration is done in section 3. In this same section, we explicit the notations, the optimization criterion, as well as the main assumptions. In section 4, we propose the formulation for the stochastic product rollover problem under risk and we develop the optimality conditions for the three different rollover strategies. Further, we present closed forms solutions and some numerical applications. Finally, in section 5, we propose various managerial insights, conclusions and future research perspectives.
5
3
The product rollover model
In this section, we will define the product rollover problem and introduce the different notations and assumptions.
3.1
Stochastic rollover process and profit/cost rates
The problem context requires a production plan for the phase-out of an existing product (hereafter called old product, or product 1) and the phase-in of a replacement product (called new product or product 2) under an uncertain admissibility date, denoted T , for the new product launch. Typical examples for such admissibility decisions are those of medical devices and pharmaceutical products which cannot be sold until an approval body grants permission. Two decision variables have to be fixed in such a rollover process: t1 , the date the firm plans to phase-out product 1 and t2 , the date product 2 is planned to be ready and available for the market. Product 1 continues to be sold until inventory is depleted or until it is replaced by the new approved product. The manufacturing and procurement lead times are assumed to be large, thus making it necessary to commit to the planning dates before the random approval date is revealed. Therefore, the decision process relies exclusively on the probability distribution of this date T . Such large procurement/manufacturing/distribution leadtimes are frequent in practice. During its regular commercial life span, each product is assumed to have a specific deterministic constant demand rate, namely d1 for product 1 and d2 for product 2. A channel inventory is needed to support each product in the market, which induces carrying inventory cost rates ch,1 and ch,2 . During the commercial life span, the contribution-to-profit rate for product i, is defined as mi = di (pi − ci ) − ch,i ,
(i = 1, 2),
(1)
with pi the selling price and ci the production cost. 3.1.1
Cost rates model for the PSO strategy
Furthermore, in our considered random setting, the profit and cost structure depends on the relative values of t1 , t2 and T . Indeed, if the strategy t1 ≤ t2 is chosen, the
6
structure of the profit and cost rates, defined over an infinite time horizon, is given in Figure 1,
Figure 1: the profit rates when t1 ≤ t2
As shown in Figure 1, there are three main cases to be considered. First if T ≤ t1 , the profit rate is m1 over the time interval [0, T [. Then, over [T, t1 [, product 2 is admissible, but not yet physically available in the supply chain. As the market is assumed to be informed that product 2 will soon substitute product 1, profit rate of product 1 changes from m1 to m′1 as long as product 1 is available, i.e. over [T, t1 [. This contribution rate m′1 is formally given by m′1 = d′1 (p′1 − c1 ) − ch,1 .
(2)
Then, over the interval [t1 , t2 [, when product 1 is sold out, shortages occur until product 2 delivery date t2 , at a corresponding shortage cost rate g. Once product 2 is available, at t2 , the profit rate becomes m2 over the remaining time horizon [t2 , ∞[. Then, when t1 ≤ T ≤ t2 , the rates are similar to the first case, except over [0, t1 ], where the profit rate is m1 . Finally, if t2 ≤ T , the profit/cost rates are similar to the previous situation, except over the interval [t2 , T [, where product 2 is physically available in the supply chain, but still not admissible. A shortage cost rate g occurs until product 2 is admissible. In addition, an inventory cost rate ch,2 associated with the physical inventory of product 2 is incurred. 3.1.2
Cost rates model for the DPR strategy
If the strategy t2 ≤ t1 is chosen, the structure of the costs and profit rates is given in Figure 2. 7
Figure 2: the profit rates when t2 ≤ t1
First, let us consider the instance where T < t2 . The profit rate is m1 over the time interval [0, T [ and m′1 over [T, t2 [. Then, over the time interval [t2 , t1 [, as product 2 is admissible and physically available, it is sold with a profit rate m2 . However, in the current setting, it is assumed that the firm immediately scraps, at a cost rate s1 , all the remaining inventory of product 1 when an approved product 2 is available for sale, i.e. over the time interval [T, t1 ]. This is justified by the higher margins for product 2 and by the need to maintain brand equity as a leading-edge provider. Finally, over the remaining time horizon [t1 , ∞[, the profit rate resumes to m2 . When t2 ≤ T ≤ t1 , the profit rate is m1 over [0, t2 [. Then over the interval [t2 , T [, the profit rate is still m1 , but as product 2 is physically available in the supply chain, but not admissible for sale, an inventory cost rate ch,2 is incurred. Over the remaining horizon starting at T , product 2 is sold with a profit rate m2 . In the time interval [T, t1 [, product 1 is scrapped at a cost rate s1 . Finally, when t1 ≤ T , the profit rate is m1 over [0, t2 [. Then, over the interval [t2 , t1 [ the profit rate is still m1 , but an inventory cost rate ch,2 has to be incurred. Over [t1 , T [, product 1 is phased-out and product 2 is not yet admissible, thus creating a shortage cost rate g. Over the remaining time horizon [T, ∞[, the profit rate reverts to m2 .
3.2
Model Notations
The main notations used in the product rollover model are summarized here. For each product type i ∈ {1, 2}, we define ci : the unit cost of product i, 8
pi : the unit price of product i, di : the demand rate of product i, ch,i : the carrying cost rate of product i, mi : the contribution-to-profit rate of product i, defined as mi = di (pi − ci ) − ch,i . Furthermore, we define g : the lost of goodwill rate when the firm has none of the products for sale, m′1 : the new contribution-to-profit rate of product 1 after the admissibility of product 2 is granted, s1 : the scrap cost rate for product 1. Furthermore, we denote T : the random approval date for product 2. This random variable has a density probability function f (⋅) and an associated probability distribution function F (⋅), both defined over [0, ∞[. The decision variables are t1 : the planned run-out date for product 1, t2 : the planned availability date for product 2. Further, we note [Y ]+ ∶= max(Y, 0).
3.3
The Performance Criterion for the Rollover Problem
Here, we consider a performance criterion defined as the difference between the profit under complete information about admissibility date, and the profit when the admissibility date is random and known exclusively through its probability distribution. This performance criterion is defined as follows: when the admissibility date is known before the decisions t1 and t2 are made, as depicted in Figure 3,
Figure 3: Full information case
9
the optimal strategy is clearly given by : t1 = t2 = T , i.e., product 1 is phased out at the planned introduction date of product 2, corresponding to the admissibility date. We can see that over the time interval [0, T [, the profit rate is m1 , while on the remaining horizon [T, ∞], the profit rate is m2 . In order to characterize the impact of randomness on the rollover process, we consider an infinite horizon objective function defined as the difference between the perfect information cost rate function (Figure 3) and the cost rates functions with imperfect information (Figures 1 and 2). This difference can be interpreted as the loss caused by the randomness plaguing the availability date T . Formally, according to the description given above, these loss functions are piecewise linear and exhibit different structures depending on the relative values of the decision variables t1 and t2 . If a strategy with t1 ≤ t2 is chosen, the loss rate function is denoted as L1 (t1 , t2 , T ) and can be expressed as:
L1 (t1 , t2 , T ) =
⎧ ⎪ ⎪ (m′1 − m2 )(t1 − T ) − (m2 + g)(t2 − t1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −(g + m1 )(T − t1 ) − (g + m2 )(t2 − T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −(g + m1 )(t2 − t1 ) − (g + m1 + ch,2 )(T − t2 )
if 0 ≤ T ≤ t1 , if t1 ≤ T ≤ t2 , if t2 ≤ T.
It is rewritten as L1 (t1 , t2 , T ) = (m1 + g)[T − t1 ]+ − (g + m′1 )[t1 − T ]+ + ch,2 [T − t2 ]+ + (m2 + g)[t2 − T ]+ .
(3)
If a strategy t2 ≤ t1 is chosen, the loss function is denoted as L2 (t1 , t2 , T ) and is given by
L2 (t1 , t2 , T ) =
⎧ ⎪ ⎪ (m′1 − m2 )(t2 − T ) − s1 (t2 − t1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −ch,2 (T − t2 ) − s1 (t1 − T ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −ch,2 (t2 − t1 ) − (g + m1 )(T − t1 )
if 0 ≤ T ≤ t1 , if t1 ≤ T ≤ t2 , if t2 ≤ T.
It is rewritten as L2 (t1 , t2 , T ) = (m2 − m′1 − s1 )[t2 − T ]+ + ch,2 [T − t2 ]+ + (m1 + g)[T − t1 ]+ + s1 [t1 − T ]+ . 10
(4)
If we formally introduce the two regions, R1 = {(t1 , t2 ) ∈ IR2+ ∶ t1 ≤ t2 } and R2 = {(t1 , t2 ) ∈ IR2+ ∶ t1 ≥ t2 }, the piecewise loss rate function can be rewritten as L(t1 , t2 , T ) = Li (t1 , t2 , T )
if (t1 , t2 ) ∈ Ri
(i = 1, 2).
(5)
On Rb , defined as the boundary between regions R1 and R2 , i.e. for Rb = {(t1 , t2 ) ∈ IR2+ ∶ t1 = t2 }, the expression of the objective function is obtained from (3) and (4) as Lb (t, T ) = (m2 − m′1 )[t − T ]+ + (m1 + g + ch,2 )[T − t]+ .
3.4
(6)
Assumptions for Problem Parameters
In order to guarantee the significance of the model, it is necessary to introduce some assumptions for the different problem parameters. First the contribution-to-profit rate for the products during regular sale period is positive, i.e. m1 , m2 > 0.
(7)
Furthermore, for product 1, the contribution-to-profit rate during regular sale period is strictly greater than the (possibly negative) contribution to profit rate after product 2 is available, i.e. m1 > m′1 .
(8)
In order to avoid cases for which it would be optimal to delay infinitely the new product launch, we assume m2 > m′1 .
(9)
Finally, as for any classical inventory problem, we assume, g, ch,2 , s1 > 0.
4
(10)
CVar Formulation of the Rollover Problem
In order to introduce the impact of risk aversion in the decision process, we consider our problem in a CVaR-minimization context (see [30, 31]). First, we introduce CVaR formulation of the rollover problem. Then, we establish the corresponding expression of the analytical optimal solutions. Finally, we analyze the impact of the risk-aversion on the selected rollover policies. 11
4.1
Conditional Value at Risk formulation
For a given probability distribution F (⋅) associated with the random date T , let us denote the probability distribution function of the loss function L(t1 , t2 , T ) by LF (η∣t1 , t2 ) = P r{L(t1 , t2 , T ) ≤ η}.
(11)
For any β ∈ [0, 1), we define the β-VaR of this distribution by αβ (t1 , t2 ) = min{α∣LF (α∣t1 , t2 ) ≥ β}.
(12)
It is now possible to introduce the β-tail distribution function to focus on the upper tail part of the loss distribution as LF,β (η∣t1 , t2 ) = {
for η < αβ (t1 , t2 ),
0 Lβ (η∣t1 ,t2 )−β 1−β
for η ≥ αβ (t1 , t2 ).
(13)
Using the expectation operator Eβ [⋅] under the β-tail distribution LF,β (⋅∣⋅, ⋅), we define the β-conditional value-at-risk of the loss L(t1 , t2 , T ) by Eβ [L(t1 , t2 , T )].
(14)
Finding the optimal rollover structure and the optimal values of the phase-in and phase-out dates which minimize the a CVar cost criterion requires solving the following optimization problem min
2
(t1 ,t2 )∈IR+
Eβ [L(t1 , t2 , T )].
(15)
According to [30, 31], the minimization of Eβ [L(t1 , t2 , T )] with respect to the decision variables t1 and t2 amounts to the minimization of the following auxiliary function lβ (t1 , t2 , α) ∶= α +
1 EF [[L(t1 , t2 , T ) − α]+ ]. 1−β
(16)
lβ (t1 , t2 , α) is known to be convex with respect to α (see [30, 31]). According to the specific structure of the loss function (3)-(4), it is natural to associate to (16) a pair of auxiliary functions for i = 1, 2, defined as follows lβ,i (t1 , t2 , α) =
{α +
1 EF [Li (t1 , t2 , T ) − α]+ }, 1−β
(17)
and an auxiliary function on the boundary, lβ,b (t, α) = {α +
1 EF [Lb (t, T ) − α]+ }. 1−β 12
(18)
4.2
Convexity and structure of the optimal solutions
The optimal solution structure is essentially determined by concavity and convexity characteristics of the functions (17)-(18) over the regions R1 and R2 . Property 1: Under assumption (8), the CVaR loss function lβ,1 (⋅, ⋅, ⋅) is strictly jointly convex on R3+ . Proof. This property stems from convexity of the loss function L1 (⋅, ⋅), combined with a specific property of the CVar formulation (see Appendix A). Corollary: The boundary loss function lβ,b (⋅, ⋅) is strictly jointly convex on R2+ . Proof. The proof is direct as lβ,b (⋅) can be viewed as the intersection of lβ,1 (α, t1 , t2 ) by an hyperplane defined by t1 = t2 (with t1 , t2 ≥ 0). ∎ Property 2: Under the assumption (m2 − m′1 − s1 + ch,2 ) > 0, the CVaR-loss function lβ,2 (⋅, ⋅, ⋅) is strictly jointly convex over R3+ . Otherwise, lβ,2 (t1 , ⋅, T ) is a strictly decreasing function. Proof. This property stems from convexity of the loss function L2 (⋅, ⋅), combined with a specific property of the CVar formulation (see Appendix A). The structure of the optimal strategy depends on the combination of the convexity properties of the functions lβ,1 (⋅, ⋅, ⋅) and lβ,2 (⋅, ⋅, ⋅) and on the location of their minimum. As defined in the introduction, we observe three types of strategies: planned stock-out rollover , single product rollover, and dual product rollover. If one denotes ∗ the respective minimum of the lβ,i (⋅, ⋅, ⋅) functions over R3+ as (αβ,i , t∗β,1,i , t∗β,2,i ), the
optimal strategy structure is displayed in Table 1,
13
lβ,1 (t1 , t2 ) properties: lβ,2 (t1 , t2 ) properties
Strictly decreasing w.r.t. t2 or convex (t∗β,1,2 , t∗β,2,2 )
Global Optimal Solution
∈ R1
∈ R2
(t∗β,1,1 , t∗β,1,2 )
undetermined
↓
↓ ?
lβ,1 (t1 , t2 ) properties:
(t∗β,1,1 , t∗β,1,2 )
Strictly decreasing w.r.t. t2 or convex (t∗β,1,2 , t∗β,2,2 )
Global Optimal Solution
Convex (t∗β,1,2 , t∗β,2,2 )
Planned Stockout
Optimal Strategy Structure
lβ,2 (t1 , t2 ) properties
(t∗β,1,1 , t∗β,1,2 ) ∈/ R2
∈/ R2
∈/ R1
Convex (t∗β,1,2 , t∗β,2,2 )
∈ R2
On the boundary R1 ⋂ R2
(t∗β,1,2 , t∗β,2,2 )
↓
↓
Single Product Rollover
Dual Product Rollover
Optimal Strategy Structure
Table 1: Convexity properties and structure of the optimal rollover strategy In the next sections, we show how the cost/profit parameters fix the optimal strategy structure.
4.3
First-order Optimality Conditions
The minimum of the auxiliary loss functions lβ,1 (⋅, ⋅, ⋅), lβ,2 (⋅, ⋅, ⋅) and lβ,b (⋅, ⋅) are characterized in the following properties by classical first-order conditions. Property 3.a : Consider the setting m2 ≥ m1 > m′1 . Under the assumption ch,2 m1 + g < , m1 − m′1 m2 + ch,2 + g
(19)
the first-order conditions solutions OC t∗,F β,1,1
=
F −1 (
OC t∗,F β,1,2
=
(
(m1 + g)(1 − β) ), m1 − m′1
m1 + ch,2 + g m2 + ch,2 + g
)F −1 (
(20)
ch,2 + β(m2 + g) m2 + ch,2 + g
)+(
ch,2 (1 − β) m2 − m1 )F −1 ( ) m2 + ch,2 + g ch,2 + m2 + g
(21)
correspond to the unique finite minimum of the CVaR loss function lβ,1 (⋅, ⋅, ⋅) in R1 . If condition (19) is not satisfied, there is no finite minimum in R1 for lβ,1 (⋅, ⋅, ⋅). Proof. See Appendix B-1. Property 3.b: Consider the setting m1 > m2 ≥ m′1 . Under the assumptions m′1
m1 − ch,2 ,
(23)
14
and for ∈
β
[β1,inf , 1 +
m2 − m1 ], ch,2
(24)
with ⎧ if m1 − m′1 > m2 + g + ch,2 , ⎪ ⎪ 0 β1,inf = ⎨ (m1 +g)(m2 +g+ch,2 −m1 +m′ ) 1 ′ ⎪ ⎪ ⎩ (m′1 +g)(m2 +g)+ch,2 (m1 +g) if m1 − m1 ≤ m2 + g + ch,2 ,
(25)
the first-order conditions solutions m2 − m′1 m1 − βm′1 + g(1 − β) (m1 + g)(1 − β) m1 − m2 )F −1 ( )+( )F −1 ( ), ′ ′ ′ m1 − m1 m1 − m1 m1 − m1 m1 − m′1
OC t∗,F β,1,1
=
(
OC t∗,F β,1,2
=
F −1 (
m1 + g + ch,2 β m2 + g + ch,2
).
(26) (27)
correspond to the unique finite minimum of the CVaR loss function lβ,1 (⋅, ⋅, ⋅) in R1 . If condition (22), (23), or (24) is not satisfied, there is no finite minimum in R1 . Proof. See Appendix B-2. Property 4.a : Consider the setting ch,2 > s1 . Under the assumptions m2 − m′1 − s1 m1 + g m1 + g + s1
≥ >
0,
(28) ch,2
m2 − m′1 − s1 + ch,2
,
(29)
the first-order conditions solutions OC t∗,F β,2,1
=
F −1 (
OC t∗,F β,2,2
=
(
+
(
m1 + g + s1 β ), m1 + g + s1
(30)
ch,2 (1 − β) m2 − m′1 )F −1 ( ) m2 − m′1 − s1 + ch,2 m2 − m′1 − s1 + ch,2 ch,2 − s1 m2 − m′1
− s1 + ch,2
)F −1 (
ch,2 + β(m2 − m′1 − s1 ) m2 − m′1 − s1 + ch,2
)
(31)
correspond to the unique finite minimum of the CVaR loss function lβ,2 (⋅, ⋅, ⋅) in R2 . If condition (28) or (29) is not satisfied, there is no finite minimum in R2 . Proof. See Appendix B-3. Property 4.b: Consider the setting ch,2 ≤ s1 . Under the assumptions m2 − m′1 − s1 + ch,2
>
0,
m2 − m′1 − s1 m1 + g m1 + g + s1
(32) (33) ch,2
m2 − m′1 − s1 + ch,2
,
(34)
and for β values satisfying β
>
m2 − m′1 − s1 ch,2
15
,
(35)
the first-order conditions solutions OC t∗,F β,2,1
OC t∗,F β,2,2
m1 + g + ch,2
=
(
+
(
=
F −1 (
m1 + g + s1 s1 − ch,2 m1 + g + s1
)F −1 (
)F −1 (
m1 + g + βs1 ) m1 + g + s1
(m1 + g)(1 − β) ), m1 + g + s1
ch,2 (1 − β) m2 − m′1 − s1 + ch,2
(36)
)
(37)
correspond to the unique finite minimum of the CVaR loss function lβ,2 (⋅, ⋅, ⋅) in R2 . If condition (32), (33), (34) or (35) is not satisfied, there is no finite minimum in R2 for lβ,2 (⋅, ⋅, ⋅). Proof. See Appendix B-4. Property 5: Under assumption (9), the boundary loss function lβ,b (⋅, ⋅) has a unique finite minimum over R+ corresponding to OC t∗,F b
=
(
m2 − m′1
m2 − m′1
+(
+ m1 + ch,2 + g
)F −1 (
m1 + ch,2 + g m2 − m′1 + m1 + ch,2 + g
(m1 + ch,2 + g)(1 − β) m2 − m′1 + m1 + ch,2 + g
)F −1 (
)
m1 + ch,2 + g + β(m2 − m′1 ) m2 − m′1 + m1 + ch,2 + g
).
(38)
Proof. See Appendix B-5.
4.4
Optimal Product Rollover Strategies
The above properties are used in Table 2, which exhibits the optimal product rollover strategy structure depending on the critical constraints on the parameters. 4.4.1
Parameters and associated optimal strategy structure.
First, it can be observed in Table 2 that the different strategy structures (PSO, DPR and SPR) can apply to the four main cases, namely (m2 ≥ m1 > m′1 ;ch,2 ≥ s1 ), (m2 ≥ m1 > m′1 ; ch,2 < s1 ), (m1 > m2 ≥ m′1 ; ch,2 ≥ s1 ) and finally (m1 > m2 ≥ m′1 ;ch,2 < s1 ). Indeed, this shows that the structure of the optimal rollover strategy depends simultaneously, and in a complex manner, on all the parameters. Furthermore, for most cases (highlighted in grey) the optimal strategy structure is independent from the probability distribution F (⋅). Second, the PSO strategy can be optimal exclusively for products with significantly negative margins m′1 (i.e. when the market or the price of product 1 collapses once 16
17
ch,2 < s1
ch,2 ≥ s1
ch,2 m2 −m′ −s1 +ch,2 1
Otherwise
≥
Otherwise
β ≥ (m2 − m′1 − s1 )/ch,2
m2 − m′1 − s1 − ch,2 < 0 ch,2 m1 +g ≥ m1 +g+s1 m2 −m′ −s1 +ch,2 1
m2 − m′1 − s1 + ch,2 > 0
m1 +g m1 +g+s1
m2 − m′1 − s1 ≥ 0
→ SPR
→ PSO
lβ,1 : PSO → PSO
lβ,2 : not DPR
]
m1 > m2 ≥ m′1
β∈
→ PSO
lβ,2 : not DPR
lβ,1 : PSO
→ PSO or DPR
lβ,2 : DPR
lβ,1 : PSO
→ PSO
lβ,2 : not DPR
lβ,1 : PSO
→ PSO or DPR
lβ,2 : DPR
lβ,1 : PSO
(m1 +g)(m2 +g+ch,2 −m1 +m′1 ) m −m1 , 1 + c2 [ (m′ +g)(m2 +g)+ch,2 (m1 +g) h,2 1
m1 − m′1 ≤ m2 + g + ch,2
m2 > m1 − ch,2
m′1 < −g
Table 2: Optimal Rollover Strategy Structure
lβ,1 : not PSO lβ,2 : not DPR
lβ,1 : PSO
→ PSO or DPR
→ DPR
→ PSO or DPR
lβ,2 : not DPR
lβ,2 : DPR
lβ,2 : DPR
lβ,1 : PSO
→ PSO
lβ,2 : not DPR
lβ,1 : not PSO
→ SPR
→ PSO
lβ,1 : PSO
lβ,2 : not DPR
lβ,2 : not DPR
lβ,2 : DPR
lβ,1 : not PSO
lβ,1 : PSO
lβ,1 : PSO
→ PSO or DPR
→ DPR
→ PSO or DPR
lβ,2 : DPR
lβ,1 : PSO
β ∈ [0, 1 +
m2 −m1 ch,2
lβ,2 : DPR
>
ch,2 m2 +g+ch,2
lβ,1 : not PSO
m1 +g m1 −m′ 1
lβ,1 : PSO
≤
ch,2 m2 +g+ch,2
m1 − m′1 ≥ m2 + g + ch,2
lβ,2 : DPR
m1 +g m1 −m′ 1
m2 ≥ m1 > m′1
]
→ SPR
lβ,2 : not DPR
lβ,1 : not PSO
→ DPR
lβ,2 : DPR
lβ,1 : not PSO
→ SPR
lβ,2 : not DPR
lβ,1 : not PSO
→ DPR
lβ,2 : DPR
lβ,1 : not PSO
Otherwise
product 2 is approved). If the margin m′1 remains positive, then the optimal strategy should be SPR or DPR. Third, high scrap cost rates s1 exclude DPR, while for high holding cost rates ch,2 , PSO appears to be attractive. Fourth, if the margins of products 1 and 2 are similar, i.e. m1 ≈ m′1 ≈ m2 , then the optimal product rollover strategy is SPR. 4.4.2
Risk Aversion and Optimal Strategy Structure
As shown in above subsection, the optimal strategy structure depends simultaneously on the different parameters of the problem, on the probability distribution F (⋅), as well as on the risk aversion defined through β. While, it is tedious to find explicit necessary and sufficient optimality conditions for each type of rollover strategy based on these different factors, the specific impact of risk aversion over the optimal strategy structure can be analyzed. As mentionned previously, β reflects the degree of risk aversion for the planner: the larger β is, the more risk averse the planner is. By using the above properties of the CVar-loss functions, Tables 3 and 4 are developed based on two significantly different situations : low risk aversion (β value close to zero) and high risk aversion (β value close to 1). 4.4.3
The Low Risk Aversion Case
For this case (depicted in Table 3), the analysis is done with β close to zero.
18
19
ch,2 < s1
ch,2 ≥ s1
ch,2 m2 −m′ −s1 +ch,2 1
Otherwise
≥
DPR→SPR
SPR
PSO or DPR→PSO
PSO
SPR
PSO
ch,2
> m +g+c 2 h,2
DPR
m1 +g m1 −m′ 1
PSO or DPR
ch,2
≤ m +g+c 2 h,2
PSO or DPR
PSO
PSO or DPR→PSO
PSO→SPR
PSO or DPR→DPR
PSO→SPR
PSO or DPR →DPR
m1 − m′1 ≤ m2 + g + ch,2
m2 > m1 − ch,2 m1 − m′1 ≥ m2 + g + ch,2
PSO
m1 > m2 ≥ m′1 m′1 < −g
Table 3: Optimal Rollover Policy Structure under low risk aversion
Otherwise
m2 − m′1 − s1 − ch,2 < 0 ch,2 m1 +g ≥ m1 +g+s1 m2 −m′ −s1 +ch,2 1
m2 − m′1 − s1 > 0
m1 +g m1 +g+s1
m2 − m′1 − s1 ≥ 0
m1 +g m1 −m′ 1
m2 ≥ m1 > m′1
SPR
DPR→SPR
SPR
DPR
Otherwise
It is worth noting that the optimal strategy structure is dependent on the decision maker risk aversion. As a matter of fact, by changing β value, it can be seen in Table 3 and Table 4 that for some parameters combinations the structure of the optimal strategy change for low or high β values. Even if the exact analysis of the impact of the risk aversion factor β requires several cases (as depicted in Table 3 and Table 4, it can be noted that lowering the β value induces optimal strategy structure change toward PSO (two cases), SPR (four cases) and DPR (two cases). Increasing the β value induces optimal strategy structure change toward SPR (four cases) and DPR (four cases). All these changes are independent from the probability distribution F (⋅). This short analysis shows that a conjecture, empirically given in ([4])), arguing single rollover to be a high-risk, high-return strategy while dual rollover to be less risky, has to be taken with care in practice. Clearly, the structure of the optimal strategy simultaneously depends on the costs structure and on the risk aversion.
20
21
ch,2 < s1
ch,2 ≥ s1
ch,2 m2 −m′ −s1 +ch,2 1
Otherwise
≥
− m′1 − s1 ≥ 0
DPR
SPR
PSO or DPR
PSO
SPR
PSO
ch,2
> m +g+c 2 h,2
DPR
m1 +g m1 −m′ 1
PSO or DPR
ch,2
≤ m +g+c 2 h,2
m1 > m2 ≥ m′1
PSO→SPR
PSO or DPR→DPR
PSO→SPR
PSO or DPR →DPR
m2 > m1 − ch,2
m′1 < −g
Table 4: Optimal Rollover Policy Structure under high risk aversion
Otherwise
m2 − m′1 − s1 + ch,2 > 0 m2 − m′1 − s1 − ch,2 < 0 ch,2 m1 +g ≥ m1 +g+s1 m2 −m′ −s1 +ch,2 1
m2 m1 +g m1 +g+s1
m1 +g m1 −m′ 1
m2 ≥ m1 > m′1
SPR
DPR
SPR
DPR
Otherwise
5
Impact of Uncertainty
In this section, we study the variation of the optimal product rollover policy structure and of the associated optimal cost, when increasing stochasticity of the random admissibility date T . We have shown in the preceding section that risk aversion level can change the structure of the optimal product rollover strategy. Billington et al., in their paper about efficient rollover strategies [4], present SPR as a high risk strategy, suited to situations with low uncertainty and DPR as a low risk strategy, suited to situations with a higher uncertainty. Increasing uncertainty level reinforces the rollover policy type (i.e. increase the overlap between t∗1 and t∗2 (positive or negative)), while the decision maker risk aversion can change the optimal strategy structure (i.e. change the t∗1 , t∗2 ordering). The main motivation of this section consists in theoretically analyzing this conjecture by Billington et al. (see [4]) claiming that when the variability of the new product admissibility date increases, then basically the overlap, i.e. the positive or negative gap between t1 and t2 , in the optimal solution has to increase too. This property is theoretically known as a dispersive ordering property. Here, we formally give the conditions guaranteeing this conjecture. It can be seen that the impact of the variability of the admissibility date is threefold : impact on the optimal global cost, impact on the optimal value of each of the two decision variables and impact on the structure of the optimal strategy. In such an analysis, the key element is the formal definition of variability or stochasticity increase between a pair of probability distribution functions. In order to assess the variability effects on the considered model, we conduct a stochastic comparison between two rollover processes. We consider two rollover processes i = 1, 2, with approval dates Ti , known through their cumulative probability distribution functions Fi . Here, we focus on the variability effects of Ti and thus we assume that the admissibility dates have equal means, E[T1 ] = E[T2 ]. In order to compare the variabilities of the pair of random variables T1 and T2 , we will have to define as stochastic ordering criterion. First, we focus on the changes of the optimal solution values and on the change of the optimal cost when the problem variability increases. These changes can be theoretically characterized along the lines of ([36, 39]). In order to define the concept 22
of variability increase, we consider a stochastic ordering based on a comparison of the spread of the probability density functions. Second, we focus on the change of the optimal product rollover strategy structure, namely the change of the overlap size associated with the optimal policies. We recall that in case of a positive overlap (corresponding to DPR), the pair of products are simultaneously available on the market during some time period, while in case of a negative overlap (corresponding to PSO), no product is available on the market. To do so, we need to use a more restrictive stochastic ordering assumption, known as dispersive ordering condition [20, 21, 35, 39].
5.1
Impact of Uncertainty on the average loss and on the optimal decisions
5.1.1
The Considered Stochastic Ordering
We consider here a usual stochastic ordering, based on the shapes of the distribution functions and defined as follows. Let u(t) be a real function defined on an ordered set U of the real line and let S(u) be the number of sign changes of u(t) when t ranges over the entire set U . Definition. Consider two random variables T1 and T2 with the same mean, i.e. E[T1 ] = E[T2 ], having probability distributions F1 (⋅) and F2 (⋅) with densities f1 (⋅) and f2 (⋅). We say that T1 is more variable than T2 , denoted T1 ≥var T2 , if S(f1 − f2 ) = 2 with sign sequence +, -, +.
(39)
That is, f1 (⋅) crosses f2 (⋅) exactly twice, first from above and then from below. It is known (see [39]), that when E[T1 ] = E[T2 ], condition (39) implies that F1 (x) ≤ F2 (x)
for all x
and
E[h(T1 )] ≥ E[h(T2 )]
(40)
for all nondecreasing functions h(⋅). Observe that condition (39) also implies that S(F1 − F2 ) = 1
(41)
with sign sequence +,−, +, in other words, F1 (⋅) crosses F2 (⋅) exactly once, and the cross is from above. Furthermore, it is also known (see [39]) that equation (41) implies t
∫−∞ (F1 (x) − F2 (x))dx ≤ 0. 23
(42)
Examples of pairs of distributions satisfying condition (41) are given in reference [36] and include a large number of important standard unimodal densities arising in statistical applications, as seen from the following pairs (i = 1, 2): - fi (⋅) are Gamma (Weibull) with shape parameter η1 , η2 , with η2 < η1 ; - fi (⋅) are Uniform (ai , bi ), with a1 < a2 , b1 > b2 , but a1 + b1 = a2 + b2 ; - Fi (⋅) are Gaussian with parameters µi and σi , with µ1 = µ2 and σ2 < σ1 ; - fi (⋅) are truncated Gaussian with parameters µi and σi , with µ1 = µ2 ≫ 0 and σ2 < σ1 ; - f1 (⋅) is decreasing (e.g., exponential) and f2 (⋅) is Uniform.
5.1.2
Impact of variability on the decision variables
We now present our results regarding the effect of approval date variability on the optimal times. Property 6. If T1 ≥var T2 , then there exists a critical number θF1 ,F2 such that ⎧ −1 −1 ⎪ ⎪ ⎪ F1 (r) ≤ F2 (r) ⎨ ⎪ −1 −1 ⎪ ⎪ ⎩ F1 (r) ≥ F2 (r)
if
0 ≤ r ≤ θF1 ,F2 ,
if
θF1 ,F2 ≤ r ≤ 1.
Proof : the proof follows reference [36]. Condition T1 ≥var T2 implies that F1 (⋅) crosses F2 (⋅) exactly once for x = x∗ (i.e. one has F1 (x∗ ) = F2 (x∗ )), and the cross is from above. That means that there exists x∗ such that, for 0 < x < x∗ , F1 (x) is at least as large as F2 (x) and for x < x∗ , F1 (x) is at most as large as F2 (x). Setting θF1 ,F2 = F1 (x∗ ) = F2 (x∗ ), the results regarding the order of Fi−1 (r) are immediate. ∎ A direct application of the above proposition is the following corollary. Corollary 6.1 If T1 ≥var T2 , the value of the different optimal solutions corresponding to the first order conditions (20)-(21), (26)-(27), (30)-(31), (36)-(37) and (38) can increase or decrease, depending on the parameter values and on the probability distributions F1 (⋅) and F2 (⋅). Proof : Let us denote the distribution dependence of the different first order solutions respectively corresponding to equations (20)-(21), (26)-(27), (30)-(31), (36)-(37) and (38), as t∗,F OC (F ), which can be formally expressed as a linear combination of
24
quantiles as follows, t∗,F OC (F ) = γF −1 (r1 ) + (1 − γ)F −1 (r2 ),
with
0 ≤ γ ≤ 1,
(43)
with the parameters γ, r1 and r2 extensively defined in equations (20)-(21), (26)-(27), (30)-(31), (36)-(37) and (38). Now, as the optimal solutions t∗,F OC (F ) are convex combinations of quantiles, one finds the following orderings ⎧ ⎪ ⎪ ⎪ t∗,F OC (F1 ) ≤ t∗,F OC (F2 ) ⎨ ⎪ ∗,F OC (F ) ≥ t∗,F OC (F ) ⎪ ⎪ 1 2 ⎩ t
if
0 ≤ r1 , r2 ≤ θF1 ,F2 ,
if
θF1 ,F2 ≤ r1 , r2 ≤ 1,
while the ordering is undefined when 0 ≤ r1 ≤ θF1 ,F2 ≤ r2 or 0 ≤ r2 ≤ θF1 ,F2 ≤ r1 . ∎ This shows that, for increasingly variable distributions, the sign of the change of the optimal solutions is not straightforward and depends on the order relationship between the threshold θF1 ,F2 and the different ratios defining the optimal solution values. 5.1.3
Impact of variability on the average loss
The following proposition establishes the intuitive result that increasing variability increases the expected loss. Property 7. If T1 ≥var T2 , then min+
+
(t1 ,t2 ) ∈ IR ×IR
EF1 [L(t1 , t2 , T )] ≤
min+
+
(t1 ,t2 ) ∈ IR ×IR
EF2 [L(t1 , t2 , T )].
(44)
Proof. This result is obtain by applying the stochastic ordering assumption to the expected loss expression (See Appendix E).
5.2
Impact of Uncertainty on structure of the optimal product rollover strategy
5.2.1
Stochastic Ordering Definitions
This subsection analyzes the impact of uncertainty on the structure of the product rollover optimal strategy, e.g. on the size of the overlap between the planning. The analysis, focused on the difference between the optimal decisions for t1 and t2 , and not their individual values, relies on another class of stochastic ordering, called dispersive ordering, as defined below. 25
Definition. Consider two random variables T1 and T2 with same mean E[T1 ] = E[T2 ], having distributions F1 (⋅) and F2 (⋅) with densities f1 (⋅) and f2 (⋅). T1 is said to be less dispersed than T2 , denoted by T1 t∗2 (F ), then the first order necessary conditions (see proof of properties 4.a and 4.b) require that r1,1 , r1,2 > r2,1 , r2,2 .
(50)
Furthermore, we have for any optimal solution, t∗1 (F ) − t∗2 (F ) = γ(F −1 (r1,1 ) − F −1 (r2,1 )) + (1 − γ)(F −1 (r1,2 ) − F −1 (r2,2 ))). (51) As T1 ≥disp T2 , for any PSO optimal solution, one has F1−1 (r1,j ) − F1−1 (r2,j ) ≥ F2−1 (r1,j ) − F2−1 (r2,j ),
(52)
for j = 1, 2,
(53)
which implies that t∗1 (F1 ) − t∗2 (F1 ) ≥ t∗1 (F2 ) − t∗2 (F2 ). As a similar proof applies for the case of a DPR optimal solution, this amounts to t∗2 (F1 ) − t∗1 (F1 ) ≥ t∗2 (F2 ) − t∗1 (F2 ). ∎ This proposition establishes general conditions guaranteeing that when the regulatory date process is more random, then the optimal policies are reinforced. In the case of PSO, the stockout period is increased, and in case of DPR, the dual product pipe-line inventory period is increased. This formally analyzes a conjecture, empirically given in ([4])). These authors argue that SPR is a high-risk, high-return strategy while DPR to be less risky.
6
Conclusions, managerial insights, and future research
In this paper, we apply the CVaR minimization to a product rollover problem with uncertain admissibility date. Results show that the optimal strategy depends on the cost and price parameters, on the probability distribution and the risk. We derive optimality conditions and unique closed-form solutions for SPR and DPR. Furthermore, we analyze the variation of optimal costs and solutions under different probability distribution families. Many potential extensions and directions for this research are under consideration. For instance, we are looking into an optimization with respect to a distribution free admissibility date, different products and lifecycles, and product rollover strategies for time-dependent demand. Further, we are also working on the expected value criterion under a Bass diffusion rate demand. 27
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31
A
A: Convexity Properties
Lemma 1: The loss function L1 (⋅, ⋅, T ) is strictly jointly convex over R2+ . Proof. We can rewrite L1 (⋅, ⋅, T ) as: L1 (t1 , t2 , T ) = La (t1 , t2 , T ) + Lb (t1 , t2 , T ),
(54)
with La (t1 , t2 , T ) = ch [T − t2 ]+ + (m2 + g)[t2 − T ]+
(55)
Lb (t1 , t2 , T ) = −(g + m′1 )[t1 − T ]+ + (m1 + g)[T − t1 ]+ .
(56)
Note that L1 (⋅, ⋅, T ) is the non negative sum of La (⋅, ⋅, T ) and Lb (⋅, ⋅, T ), where La (⋅, ⋅, T ) is convex and Lb (⋅, ⋅, T ) is convex by assumption (8), therefore L1 (⋅, ⋅, T ) is jointly convex. Property 1: The CVaR loss function lβ,1 (t1 , t2 , α) is strictly jointly convex. Proof. It is a known result that if L1 (⋅, ⋅, T ) is convex for any fixed value T , then the CvaR minimization leads to a convex problem (see Rockafellar and Uryasev, 2000,2002). Convexity of L1 (⋅, ⋅, T ) in R1 was previously proved in Lemma 1. Lemma 2: The loss function L2 (⋅, ⋅, T ) is strictly jointly convex over R+ × R+ under the assumption m2 − m′1 − s1 + ch > 0. Proof. We can rewrite L2 (⋅, ⋅, T ) as: L(t1 , t2 , T ) = Lc (t1 , t2 , T ) + Ld (t1 , t2 , T ),
(57)
Lc (t1 , t2 , T ) = (m2 − m′1 − s1 )[t2 − T ]+ + ch [T − t2 ]+
(58)
Ld (t1 , t2 , T ) = (m1 + g)[T − t1 ]+ + s1 [t1 − T ]+ .
(59)
with
Note that L2 (⋅, ⋅, T ) is the non negative sum of Lc (⋅, ⋅, T ) and Ld (⋅, ⋅, T ), with Ld (⋅, ⋅, T ) convex and Lc (⋅, ⋅, T ) convex if m2 −m′1 −s1 +ch > 0, therefore L2 (⋅, ⋅, T ) is jointly convex. Property 2: The CVaR loss function lβ,2 (t1 , t2 , α) is strictly jointly convex. Proof. It is a known result that if L2 (⋅, ⋅, T ) is convex for any fixed value T , then the CvaR minimization leads to a convex problem (see Rockafellar and Uryasev, 2000,2002). Convexity of L2 (⋅, ⋅, T ) in R2 was previously proved in Lemma 2. 32
B
B-1: First Order Conditions for lβ,1(t1, t2, α)
Lemma: The CVaR loss function lβ,1 (t1 , t2 , α) is differentiable inside R2+ . Proof. This property is direct from the expression of the derivatives of the loss function (See Appendix C). Property 3.a : Consider the setting m2 ≥ m1 > m′1 . Under the assumption ch,2 m1 + g < , m1 − m′1 m2 + ch,2 + g
(60)
the first-order conditions solutions OC t∗,F β,1,1
=
F −1 (
OC t∗,F β,1,2
=
(
(m1 + g)(1 − β) ), m1 − m′1
m1 + ch,2 + g m2 + ch,2 + g
)F −1 (
(61)
ch,2 + β(m2 + g) m2 + ch,2 + g
)+(
ch,2 (1 − β) m2 − m1 )F −1 ( ). m2 + ch,2 + g ch,2 + m2 + g
(62)
corresponds to the unique finite minimum of the CVaR loss function lβ,1 (⋅, ⋅, ⋅) in R1 . Otherwise, if condition (60) is not satisfied, there is no finite minimum in R1 for lβ,1 (⋅, ⋅, ⋅). Proof. The piecewise linear function L1 (t1 , t2 , T ), with t1 , t2 assumed to be given, is depicted in figure (4). On the same figure are displayed the critical values for the α parameters corresponding to the slope discontinuities of the L1 (t1 , t2 , ⋅) function. When m2 ≥ m1 > m′1 , these critical values are given, as functions of t1 and t2 , by α ˜ 1,1 (t1 , t2 ) = m1 (t2 − t1 ) + g(t2 − t1 ),
(63)
α ˜ 1,2 (t1 , t2 ) = m2 (t2 − t1 ) + g(t2 − t1 ),
(64)
α ˜ 1,3 (t1 , t2 ) = m2 t2 − m′1 t1 + g(t2 − t1 ),
(65)
with α ˜ 1,1 (t1 , t2 ) ≤ α ˜ 1,2 (t1 , t2 ) ≤ α ˜ 1,3 (t1 , t2 ) (see Figure (4)),
In order to characterize the first order conditions, we define the following regions: C1,1 = {(t1 , t2 , α) with (t1 , t2 ) ∈ R1
and α ∈]∞, α ˜ 1,1 (t1 , t2 )[,
C1,2 = {(t1 , t2 , α) with (t1 , t2 ) ∈ R1
and α ∈]˜ α1,1 (t1 , t2 ), α ˜ 1,2 (t1 , t2 )[,
C1,3 = {(t1 , t2 , α) with (t1 , t2 ) ∈ R1
and α ∈]˜ α1,2 (t1 , t2 ), α ˜ 1,3 (t1 , t2 )[,
C1,4 = {(t1 , t2 , α) with (t1 , t2 ) ∈ R1
and α ∈]˜ α1,3 (t1 , t2 ), ∞[.
33
Figure 4: Function L1 (t1 , t2 , T ): the m2 ≥ m1 > m′1 setting.
First order conditions by regions. Region C1,1 . In this region, the objective function becomes lβ,1 (t1 , t2 , α) = α
+
1 [(m2 t2 − m′1 t1 )F (t1 ) + (m′1 − m2 )G(t1 ) 1−β m1 (µ − G(t1 ) − t1 (1 − F (t1 ))) + m2 t2 (F (t2 ) − F (t1 ))
+
(ch,2 + g)(µ − G(t2 ) − t2 (1 − F (t2 ))) + g(t2 − t1 ) − α]
+
(66)
and the first order derivatives are given by dlβ,1 (t1 , t2 , α) dα dlβ,1 (t1 , t2 , α) dt1 dlβ,1 (t1 , t2 , α) dt2
= = =
−β < 0, 1−β (m1 − m′1 ) (m1 + g) F (t1 ) − , 1−β 1−β (m2 + ch,2 + g) ch,2 F (t2 ) − . 1−β 1−β
(67) (68) (69)
Region C1,2 . According to Figure 4, let’s define T2 (α, t1 , t2 ) as the T value solving α = m1 (T − t1 ) + g(t2 − t1 ) + m2 (t2 − T )
(70)
and T3 (α, t1 , t2 ) as the T value solving α = m1 (T − t1 ) + g(t2 − t1 ) + (ch,2 + g)(T − t2 ).
34
(71)
In this region, the objective function becomes lβ,1 (t1 , t2 , α) = α
+
1 ((m1 − m′1 )t1 F (t1 ) + (m′1 − m1 )G(t1 ) 1−β (m1 − m2 )G(T2 (t1 , t2 , α))
+
(−m1 t1 + g(t2 − t1 ) + m2 t2 − α)F (T2 (t1 , t2 , α))
+
(m1 + ch,2 + g)(µ − G(T3 (t1 , t2 , α)))
+
(−m1 t1 − gt1 − ch t2 − α)(1 − F (T3 (t1 , t2 , α))))
+
(72)
and the first order derivatives are given by dlβ,1 (t1 , t2 , α)
=
dt1
− dlβ,1 (t1 , t2 , α)
=
dt2 dlβ,1 (t1 , t2 , α)
=
dα
(m1 − m′1 ) (m1 + g) F (t1 ) − F (T2 (α, t1 , t2 )) 1−β 1−β (m1 + g) (1 − F (T3 (α, t1 , t2 ))), 1−β ch,2 (m2 + g) F (T2 (α, t1 , t2 )) − (1 − F (T3 (α, t1 , t2 ))), 1−β 1−β
(73)
F (T3 (t1 , t2 , α)) − F (T2 (t1 , t2 , α)) − β . 1−β
(75)
(74)
Region C1,3 . According to Figure 4, let’s define T1 (α, t1 , t2 ) as the T value solving α = −m′1 (t1 − T ) + g(t2 − t1 ) + m2 (t2 − T ).
(76)
In this region, the objective function becomes lβ,1 (t1 , t2 , α)
1 [(m1 − m2 )G(T1 (t1 , t2 , α)) + (m1 + ch,2 + g)(µ − G(T3 (t1 , t2 , α))) 1−β
=
α+
+
(−m′1 t1 + g(t2 − t1 ) + m2 t2 − α)F (T1 (t1 , t2 , α))
+
(−m1 t1 − gt1 − ch t2 − α)(1 − F (T3 (t1 , t2 , α)))]
(77)
and the first order derivatives are given by dlβ,1 (t1 , t2 , α) dt1 dlβ,1 (t1 , t2 , α) dt2 dlβ,1 (t1 , t2 , α) dα
(m′1 + g)
(m1 + g) (1 − F (T3 (α, t1 , t2 ))), 1−β ch,2 (m2 + g) F (T1 (α, t1 , t2 )) − (1 − F (T3 (α, t1 , t2 ))), 1−β 1−β
=
−
=
1−β
F (T1 (α, t1 , t2 )) −
F (T3 (t1 , t2 , α)) − F (T1 (t1 , t2 , α)) − β . 1−β
=
(78) (79) (80)
Region C1,4 . In this region, the objective function becomes lβ,1 (t1 , t2 , α)
1 [(m1 + ch,2 + g)(µ − G(T3 (t1 , t2 , α))) 1−β
=
α+
+
(−m1 t1 − gt1 − ch t2 − α)(1 − F (T3 (t1 , t2 , α)))].
(81)
The first order derivatives are given by: dlβ,1 (t1 , t2 , α) dt1 dlβ,1 (t1 , t2 , α) dt2 dlβ,1 (t1 , t2 , α) dα
m1 + g )(1 − F (T3 (t1 , t2 , α))), 1−β
=
−(
=
−(
=
F (T3 (t1 , t2 , α) − β . 1−β
ch,2 1−β
35
)(1 − F (T3 (t1 , t2 , α))),
(82) (83) (84)
Analysis of the first order conditions. Direct computations show that the only case where the first order conditions have a solution is the region C1,2 . Under adequate parameters assumptions, the first order conditions associated to (73)-(75) have the solution (m1 + g)(1 − β) ), m1 − m′1
OC t∗,F β,1,1
=
F −1 (
OC t∗,F β,1,2
=
(
+
(
=
(m1 + ch,2 + g)F −1 (
OC α∗,F β,1
m1 + ch,2 + g m2 + ch,2 + g
)F −1 (
(85)
ch,2 + β(m2 + g) m2 + ch,2 + g
)
ch (1 − β) m2 − m1 )F −1 ( ), m2 + ch,2 + g m2 + ch,2 + g ch,2 + β(m2 + g) m2 + ch,2 + g
(86) OC OC ) − (m1 + g)t∗,F − ch t∗,F , β,1,1 β,1,2
(87)
and we also have the following parameter values OC ∗,F OC OC T3 (t∗,F , tβ,1,2 , α∗,F ) β,1,1 β,1
=
F −1 (
OC ∗,F OC OC T2 (t∗,F , tβ,1,2 , α∗,F ) β,1,1 β,1
=
F −1 (
ch,2 + β(m2 + g) m2 + ch,2 + g
),
ch (1 − β) ). m2 + ch,2 + g
(88) (89)
In order to guarantee that solution (85)-(87) exists and belongs to the interior of C1,2 , the following assumptions are required (m1 + g)(1 − β) m1 − m′1
