Monotone trends in inventory-price control under time-consistent coherent risk measure

Accepted Manuscript Monotone trends in inventory-price control under time-consistent coherent risk measure Jian Yang PII: DOI: Reference:

S0167-6377(17)30238-9 http://dx.doi.org/10.1016/j.orl.2017.04.009 OPERES 6221

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Operations Research Letters

Received date: 20 July 2016 Revised date: 25 October 2016 Accepted date: 17 April 2017 Please cite this article as: J. Yang, Monotone trends in inventory-price control under time-consistent coherent risk measure, Operations Research Letters (2017), http://dx.doi.org/10.1016/j.orl.2017.04.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Monotone Trends in Inventory-price Control under Time-consistent Coherent Risk Measure Jian Yang Department of Management Science and Information Systems Business School, Rutgers University Newark, NJ 07102 Email: [email protected] July 2016; revised, October 2016, April 2017 Abstract We use the concept of time-consistent coherent risk measure to study a risk-averse firm’s inventory and price control activities. Structural characterization for an optimal inventory policy reminiscent of the risk-neutral counterpart is easy to achieve. More interestingly, monotone properties can be derived for the pricing policy when the risk possesses certain order-theoretic structures. We also introduce the concept of optimism. Two risk measures thus ranked produce inventory and pricing decisions that can be ranked themselves. The involved coherent risk measure can be a mixture of the ordinary expectation and the conditional value at risk. Keywords: Coherent Risk Measure; Dynamic programming; Inventory and Price Control; Lattice and Supermodularity; Strong Set Order

1

1

Introduction

Classical inventory and price control (IPC) is concerned with the maximization of the expected total profit over a certain time horizon. Tremendous advances have been made on the understanding of behaviors of profit-to-go functions and characters of optimal control policies; see, e.g., Federgruen and Heching [11], Chen and Simchi-Levi [6], and the survey conducted by Chen and Simchi-Levi [7]. However, as noted in Schweitzer and Cachon [19], a great number of firms are concerned with avoiding excess swings in their fortunes and intent on protecting themselves from grave downside risks. So it is important to incorporate risk aversion into IPC problems. Progresses have been made on the optimal control of inventory-pricing activities with risk considerations. For instance, Bouakiz and Sobel [4] applied exponential utility to the total payoff of an inventory control problem and obtained an optimal time-varying base-stock policy. Meanwhile, Chen et al. [5] used the maximization of the total expected utility from consumptions in all periods as a risk criterion for IPC models. When the utility function is further additive with respect to period-wise contributions and the utility in each period is an exponential function of that period’s consumption level, policy shapes from the risk-neutral case was shown to be preservable. Using the same utility function, Chen and Sun [8] coped with ambiguities on demand distributions. They arrived to min-max type formulations and showed the optimality of simple policy forms when the time horizon is infinite. The utility-on-consumption criterion tacitly assumed that the concerned firm has access to a financial market for borrowing and lending with a risk-free saving and borrowing interest rate. This is not necessarily true all the time. Moreover, it is desirable to have an approach that is flexible, that does not add operational requirements on the concerned firm, and that lends itself to a general framework beyond the IPC application. We find one based on Ruszczy´ nski’s [16] work concerning the time-consistent coherent risk measure. In this approach, the riskiness of a sequence of random cost terms is assessed directly without resorting to an attendant consumption sequence or a financial market. The assessment is done in a backward-going fashion conducive to dynamic programming (DP) formulations. The adaptation of Ruszczy´ nski’s [16] approach to our current need is quite involved (see Yang [24]). However, the end result is very clear. For our IPC setting, the resulting DP formulation is of the min-max type, which differs from a traditional risk-neutral one only by an additional optimization over a convex set Mt of potential demand densities. With this formulation, it is straightforward to derive the optimality of the base-stock form of the optimal inventory control policy. Our focus is on how to make of the pricing portion of the optimal policy when Mt ’s are well structured. 2

When the convex density set Mt possesses the lattice-like property, such as when it leads to a mixture of the ordinary expectation and the conditional value at risk, concrete links between higher inventory levels and lower prices can be established (Proposition 1). Moreover, the monotone trend tends to be mild (Proposition 2). Finally, we make comparative statics comparisons between two systems under different risk measures. When Mt1 ≤ Mt2 in

the sense of the strong set order as proposed by Veinott [23], one can interpret that system 2 has a more optimistic expectation on future demand levels than system 1. If so, we can show that monotone trends exist in movements of policy parameters with regard to changes in the underlying risk measure (Proposition 3). In a number of ways this paper can add to people’s understanding of IPC problems. Propositions 1 and 2 have shown that monotone pricing trends could emerge in risk-averse situations. Indeed known for the more special risk-neutral case, these patterns are now tied to certain structures of the involved convex density sets. Moreover, Proposition 3 can be used to predict changes in ordering and pricing decisions when the decision maker’s risk attitude changes. For instance, when it changes from risk-neutral to one involving more pessimistic demand outlooks, then the proposition will forecast not only less immediate ordering but also changes in pricing that will result with a lower average next-period inventory level. The introduction of coherent risk measures can be credited to Artzner et al. [2], who also presented duality results that associated such measures with supremums over convex density sets. Using tools in convex analysis and optimization theory, researchers like Delbaen [10], F¨ollmer and Schied [12], Cheridito, Delbaen, and Kupper [9], Rockafellar, Uryasev, and Zabarankin [15], and Ruszczy´ nski and Shapiro [17] made steady progresses on the dual representation theory. The flexibility and versatility of coherent risk measures have been widely accepted. These measures encompass conditional values at risk at various percentile points, as well as certain mean-deviation and mean-semideviation risk functions. Multi-period optimization problems involving risk measures were studied, for instance, by Artzner et al. [3] and Riedel [14]. That DP could be used in such problems was realized by Ruszczy´ nski and Shapiro [18], who introduced conditional convex risk mappings, derived their representations in terms of conditional expectations, and developed DP relationships. By reducing the dependence of each period-wise risk measure on the entire history to that on the state-control pair only, Ruszczy´ nski [16] made the resultant DP more tractable. In addition, Ahmed, Cakmak, and Shapiro [1] used coherent risk measures on pure inventory control problems and obtained policy forms identical to their risk-neutral brethren. The remainder of the paper is organized as follows. After formulating our version of riskaverse IPC in Section 2, we spend Section 3 on the definition of a partial order for densities and the derivation of some preliminary results. Section 4 then focuses on the link between 3

lattice-like density sets and a monotone pricing trend. The latter is shown to be mild in Section 5. The optimism concept is introduced in Section 6 to facilitate comparative statics comparisons between two systems under different risk measures. The paper is concluded in Section 7. For more coverage on the general formulation and analysis of other cases such as infinite-horizon scenarios, the reader may consult the unabridged version Yang [24].

2

Formulation for Risk-averse IPC

We consider a firm that periodically reviews its inventory in periods 1 to T . Ordering occurs instantaneously and involves no setup cost. Unsatisfied demands are backlogged. In every regular period t = 1, 2, ..., T − 1, the unit ordering cost is ct , the unit holding cost rate is

ht , and the unit backlogging cost rate is bt . In the terminal period T , the inventory’s worth is tallied by cT · x. Demand is controllable through pricing. Suppose the firm can choose sales price p from a given interval [pt , pt ]; also, there exist continuous and strictly decreasing function z˜t (·) and random variable Θt , so that the random demand in period t follows ∆t (p) = z˜t (p) + Θt .

(1)

It will be more convenient to treat the demand lever z = z˜t (p) as being under control. For that matter, we let p˜t (·) be the inverse function of z˜t (·), z t = z˜t (pt ), and z t = z˜t (pt ). Given discount factor α ∈ [0, 1), the traditional risk-neutral problem will have the firm maximizing its total discounted expected profit.

Here, however, we suppose that the firm intends to minimize its exposure to risk ρ1T resulting from the costs Z1 , Z2 , ..., ZT that it incurs over the periods. The risk measure is considered time-consistent when for some operators ρ1 , ρ2 , ..., ρT −1 , ρ1T (Z1 , Z2 , ..., ZT ) = Z1 + ρ1 (Z2 + ρ2 (Z3 + · · · + ρT −2 (ZT −1 + ρT −1 (ZT )) · · · )).

(2)

Because each Zt is determined given the information Ft available at the beginning of period t, each operator ρt converts an Ft+1 -measurable cost to something “less random”: an Ft measurable cost. Time consistency was introduced by Ruszczy´ nski [16], who also showed

that the notion satisfies a number of reasonable axioms. When each ρt assumes the role of simple expectation, we will get the traditional risk-neutral setup as a special case. We suppose the more general case where each ρt is state-independent and coherent in the sense of Artzner et al. [2], so that it satisfies the axioms of positive homogeneity, monotonicity, translation equivariance, and convexity. Due to the dual representation of coherent risk measures, the end result is that in each period t, it is as though the firm has 4

to brace for the worst density on Θt out of a convex set Mt , where the densities are with respect to a given distribution Rt . For some real constant p ∈ (1, +∞), we suppose that Z +∞ p |θ|p · Rt (dθ) < +∞. (3) E[|Θt | ] = −∞

We also require that Mt ⊆ Mqt , the space of densities that belong to the linear topological

space Lq (<, B(<), Rt ), where B(<) stands for the Borel σ-filed built on the real line < and

q ∈ (1, +∞) is the real constant satisfying 1/p + 1/q = 1. Define revenue function r˜tm (z) = p˜t (z) · Em [z + Θt ] = p˜t (z) · z + Em [Θt ] · p˜t (z), where we have adopted the notation Em [f (Θt )] =

R +∞ −∞ m

(4)

f (θ) · m(θ) · Rt (dθ). When m happens

to be the all-one unit-rate function, we shall write E [f (Θt )] simply as E[f (Θt )]. Note that this has been done at (3). We assume that r˜tm (·) is concave at every t = 1, 2, ..., T − 1 and m ∈ Mqt . For instance,

when Em [Θt ] ≥ 0 is guaranteed, this can happen when both functions p˜t (z) and p˜t (z) · z are

concave in z. In turn, the latter can be true when p˜t (z) is linearly decreasing in z and hence p˜t (z) · z is quadratic in z with a downward opening. We want to add that the concavity

assumption is already common for the risk-neutral case; see Federgruen and Heching [11] and Chen and Simchi-Levi [6]. For technical reasons, we further assume that ct + ht − αct+1 > 0,

∀t = 1, 2, ..., T − 1.

(5)

This assumption appears often in the inventory control literature; see, e.g., Veinott [22] and Gavirneni [13]. It says that there is no speculative motive for the firm to order at an earlier time than needed. We also assume that the densities m in Mt have uniformly small tails. That is, for any t = 1, 2, ..., T − 1 and  > 0, there exists δ ∈ <, such that Z +∞ m E [1(Θt ≥ δ)] = m(θ) · Rt (dθ) < , ∀m ∈ Mqt .

(6)

δ

This restriction on Rt just specifies that predictions on demand are not outlandish. With the above assumptions in place, the problem of minimizing the firm’s risk exposure in periods 1 through T has a DP formulation. Let vt (x) be the minimum such value from period t to T . We have the terminal value vT (x) = −cT · x. For t = T − 1, T − 2, ..., 1, vt (x) = −ct · x + min ut (y),

(7)

ut (y) = ct · y + min qt (y, z),

(8)

y≥x

where z t ≤z≤z t

5

qt (y, z) = sup wtm (y, z),

(9)

˜ t + α · vt+1 )(y − z − Θt )]. wtm (y, z) = −˜ rtm (z) + Em [(h

(10)

m∈Mt

and ˜ t (x) = hx+ + bx− for convenience. Our min-max formulation In the above, we have used h from (7) to (10) is of a similar nature to Chen and Sun’s [8] in their (2.3) to (2.5). Whereas ours is due mainly to the dual representation of the coherent risk measure, theirs come from utility-on-consumption and ambiguity-on-demand considerations. Based on the above formulation, we can understand Mt as containing all potential demand densities perceivable by the firm. The latter in turn makes decisions in anticipation of the worst outcome; see the extra optimization done at (9). For t = T, T − 1, ..., 1, we can still

show that vt (·) is convex. Consequently, for period t = T − 1, T − 2, ..., 1, there exists basestock level St so that the order-up-to level yt∗ (x) = St ∨ x. This is not different from what

Federgruen and Heching [11] has to say about the risk-neutral case. Indeed, DP formulations and inventory policies can be derived for more general cases, say those involving ordering setup costs and multiplicative demand factors; see details in Yang [24]. However, we just present what is directly relevant to what will follow.

3

Preliminary Definitions and Results

For convenience, we first carry out the transformation w˜tm (υ, z) = wtm (−υ, z). The argument υ stands for the negative of the order-up-to level. By (10), ˜ t + α · vt+1 )(−υ − z − Θt )]. w˜tm (υ, z) = −˜ rtm (z) + Em [(h

(11)

˜ t + α · vt+1 )(x), the alignment of υ and z will render Due to the convexity of f (x) = (h

w˜tm (υ, z) supermodular in (υ, z). Define q˜t (υ, z) = qt (−υ, z), so that following (9), q˜t (υ, z) = qt (−υ, z) = sup wtm (−υ, z) = sup w˜tm (υ, z). m∈Mt

(12)

m∈Mt

From (8) and (12), it is quite clear that monotone trends of zt∗ (·) would come out of modularity properties of q˜t (·, ·). However, even with the supermodularity of w˜tm (·, ·), that of q˜t (·, ·) is not guaranteed if either w ˜tm (υ, z) possesses no further properties in terms of (m, υ) and

(m, z) or Mt is not well structured. The needed structure, as it turns out, is associated with a natural partial order among densities in Mqt . For a density m0 defined for any arbitrary probability space (Ω0 , F 0 , P 0 ), let us define R distribution m0 · dP 0 on the measurable space (Ω0 , F 0 ) through Z  Z 0 0 0 m · dP (Ω ) = m0 (ω 0 ) · P 0 (dω 0 ), ∀Ω0 ∈ F 0 . (13) Ω0

6

According to Shaked and Shanthikumar [20] (Section 1.A), two probability measures Q1 and Q2 on (<, B(<)) are ranked as Q1 ≤ Q2 in the usual stochastic sense when Q1 ([θ, +∞)) ≤ Q2 ([θ, +∞)),

∀θ ∈ <.

(14)

We can give a partial order to Mqt , so that m1 ≤ m2 for m1 , m2 ∈ Mqt when Z Z 1 m · dRt ≤ m2 · dRt in the usual stochastic sense of (14). Now m1 ≤ m2 if and only if, for every θ ∈ <, Z Z +∞ 1 0 0 m (θ ) · Rt (dθ ) ≤

θ

θ

+∞

m2 (θ0 ) · Rt (dθ0 ).

(15)

For m1 , m2 ∈ Mqt , we can define m so that R +∞ R +∞ m(θ) = m1 (θ) · 1( θ m1 (θ0 ) · Rt (dθ0 ) > θ m2 (θ0 ) · Rt (dθ0 )) R +∞ R +∞ +m2 (θ) · 1( θ m1 (θ0 ) · Rt (dθ0 ) ≤ θ m2 (θ0 ) · Rt (dθ0 )). The thus defined m is a member of Mqt and for any θ ∈ <,  _ Z Z +∞ Z +∞ 0 0 1 0 0 m (θ ) · Rt (dθ ) m(θ ) · Rt (dθ ) = θ

+∞

 m (θ ) · Rt (dθ ) , 2

θ

θ

(16)

0

0

(17)

where a ∨ b means the least upper bound max{a, b} for two real values a and b. In view

of (15), this means that m = m1 ∨ m2 , the least upper bound for m1 and m2 . Symmetrically, we can verify that m1 ∧ m2 , the greatest lower bound for m1 and m2 , belongs to Mqt as well.

Therefore, Mqt under the partial order specified in (15) is a lattice. ˜m For any m ∈ Mqt , let us define Θ t as a Borel-measurable mapping from [0, 1] to <, so that for any λ ∈ [0, 1],   Z Z m ˜ (λ) = inf θ ∈ < | Θ m · dRt ((−∞, θ]) = t

θ

−∞

0

0



m(θ ) · Rt (dθ ) ≥ λ .

R The defined entity is basically the λ-quantile of m · dRt . For f ∈ Lp (<, B(<), Rt ), Z 1 m ˜m f (Θ E [f (Θt )] = t (λ)) · dλ.

(18)

(19)

0

In addition, m1 ≤ m2 is translatable to

1 ˜m ˜ m2 Θ t (λ) ≤ Θt (λ),

∀λ ∈ [0, 1].

(20)

Hence, for every λ ∈ [0, 1], 1 ∨m2 1 ˜m ˜m ˜ m2 Θ (λ) = Θ t t (λ) ∨ Θt (λ).

1 ∧m2 1 ˜ m2 ˜m ˜m Θ (λ) = Θ t t (λ) ∧ Θt (λ),

(21)

With the partial order defined for densities in Mqt , which is in turn rendered a lattice, we can derive the following useful result. 7

Lemma 1 r˜tm (z) is submodular in (m, z) ∈ Mqt × [z t , z t ]. ˜ tq , z ∈ [z t , z t ], and positive ∆z satisfying z + ∆z ≤ z t , we have Proof: For any m1 , m2 ∈ M from (4), (19), and (21) that 1 ∨m2

r˜tm

(z + ∆z) + r˜tm

= (Em

1 ∨m2

which is equal to

R1 0

1 ∧m2 2

1

2

(z) − r˜tm (z) − r˜tm (z + ∆z)

1 ∧m2

1

[Θt ] − Em [Θt ]) · p˜t (z + ∆z) − (Em [Θt ] − Em

[Θt ]) · p˜t (z),

(22)

a(λ) · dλ with

1 ˜ m2 ˜ m2 ˜m ˜t (z + ∆z) a(λ) = [Θ t (λ) ∨ Θt (λ) − Θt (λ)] · p 1 2 1 m m ˜m ˜ t (λ) ∧ Θ ˜ t (λ) − Θ ˜t (z). −[Θ t (λ)] · p

(23)

1 ˜ m2 ˜m At a λ ∈ [0, 1] with Θ t (λ) ≤ Θt (λ), we have a(λ) = 0. Otherwise,

˜ m1 (λ) − Θ ˜ m2 (λ)] · [˜ a(λ) = [Θ pt (z + ∆z) − p˜t (z)]. t t

(24)

Since p˜t (·) is decreasing, p˜t (z + ∆z) − p˜t (z) is negative. Hence, a(λ) ≤ 0 here. After integration, we will have (22)’s negativity and hence the submodularity of r˜tm (z) in (m, z).

4

Lattice Risk Sets and Monotone Trends

When the risk set Mt is a lattice itself, we show that monotone pricing trends would emerge. First let us present a result that can help with the forthcoming Lemmas 3 and 4. Lemma 2 Suppose i(·) is a convex function defined on some real interval [x, x] and jtm (x) = Em [i(x + Θt )] for any t, m ∈ Mqt , and x ∈ [x, x]. Then, jtm (x) is supermodular in (m, x). Proof: For any t, any m1 , m2 ∈ Mqt , x ∈ [x, x], and positive ∆x satisfying x + ∆x ≤ x, jtm

1 ∨m2

1 ∧m2

(x + ∆x) + jtm 1 ∨m2

= Em

1

2

(x) − jtm (x) − jtm (x + ∆x) 1 ∧m2

[i(x + ∆x + Θt )] + Em m1

[i(x + Θt )]

(25)

m2

[i(x + Θt )] − E [i(x + ∆x + Θt )]. R1 According to (19) and (21), it is equal to 0 a(λ) · dλ with −E

˜ m1 (λ) ∨ Θ ˜ m2 (λ)) + i(x + Θ ˜ m1 (λ) ∧ Θ ˜ m2 (λ)) a(λ) = i(x + ∆x + Θ t t t t 1 m2 ˜m ˜ −i(x + Θ (λ)) − i(x + ∆x + Θ (λ)). t t

(26)

˜ m1 (λ) ≤ Θ ˜ m2 (λ), we have a(λ) = 0. Otherwise, At a λ ∈ [0, 1] with Θ t t 1 ˜m ˜ m2 ˜ m1 ˜ m2 a(λ) = i(x + ∆x + Θ t (λ)) + i(x + Θt (λ)) − i(x + Θt (λ)) − i(x + ∆x + Θt (λ)), (27)

8

which is positive due to the convexity of i(·). Since now a(λ) ≥ 0 for every λ ∈ [0, 1], we will have (25)’s positivity after integration, and hence the supermodularity of jtm (x) in (m, x). Now, we can claim that w ˜tm (υ, z) is also supermodular in both (m, υ) and (m, z). Lemma 3 w˜tm (υ, z) is supermodular in (m, υ, z) ∈ Mqt × < × [z t , z t ]. ˜ t + α · vt+1 )(−x) is convex. Meanwhile, (11) could be Proof: Note the function f (x) = (h understood as

m ˜m rtm (z) + u˜m w˜tm (υ, z) = −˜ t (υ, z) = E [f (υ + z + Θt )]. t (υ, z), with u

(28)

m At a fixed z ∈ [z t , z t ], we see that u˜m t (υ, z) = E [i(υ + Θt )] with i(x) = f (x + z). The latter

is convex since f (·) is. So by Lemma 2, we know that u˜m t (υ, z) is supermodular in (m, υ).

m At a fixed υ ∈ <, we see that u˜m t (υ, z) = E [i(z + Θt )] with i(x) = f (υ + x). The latter is

convex since f (·) is. So by Lemma 2, we know that u˜m t (υ, z) is supermodular in (m, z).

Right after (11), we have already argued that w˜tm (υ, z) is supermodular in (υ, z). Now in view of Lemma 1 and the properties just proved, we see that w˜tm (υ, z) is supermodular in both (m, υ) and (m, z). Therefore, w˜tm (υ, z) is supermodular in each of its three arguments, and also has increasing differences in (m, υ, z). So following Theorem 2.6.2 of Topkis [21], we can obtain that w˜tm (υ, z) is supermodular in (m, υ, z). For details, note that w˜tm1 (v1 , z1 ) − w˜tm1 ∧m2 (v1 ∧ v2 , z1 ∧ z2 ) = w˜tm1 (v1 , z1 ) − w˜tm1 (v1 , z1 ∧ z2 )

+w˜tm1 (v1 , z1 ∧ z2 ) − w˜tm1 (v1 ∧ v2 , z1 ∧ z2 )

+w˜tm1 (v1 ∧ v2 , z1 ∧ z2 ) − w˜tm1 ∧m2 (v1 ∧ v2 , z1 ∧ z2 )

≤ w˜tm1 ∨m2 (v1 ∨ v2 , z1 ) − w˜tm1 ∨m2 (v1 ∨ v2 , z1 ∧ z2 ) +w˜tm1 ∨m2 (v1 , z2 ) − w˜tm1 ∨m2 (v1 ∧ v2 , z2 )

≤

=

(29)

+w˜tm1 (v2 , z2 ) − w˜tm1 ∧m2 (v2 , z2 ) w˜tm1 ∨m2 (v1 ∨ v2 , z1 ∨ z2 ) − w˜tm1 ∨m2 (v1 ∨ v2 , z2 ) +w˜tm1 ∨m2 (v1 ∨ v2 , z2 ) − w˜tm1 ∨m2 (v2 , z2 ) +w˜tm1 ∨m2 (v2 , z2 ) − w˜tm2 (v2 , z2 ) w˜tm1 ∨m2 (v1 ∨ v2 , z1 ∨ z2 ) − w˜tm2 (v2 , z2 ),

where the first inequality is due to the various increasing differences, and the second inequality is due to the various individual supermodularities.

9

To move forward, we make the following assumption on our risk measure. IPC Assumption: For t = 1, 2, ..., T − 1, each Mt is a sublattice within the lattice Mqt , and

hence a lattice in its own right.

This assumption means that m1 , m2 ∈ Mt will lead to both m1 ∨ m2 ∈ Mt and m1 ∧ m2 ∈

Mt . It is a mild requirement. For instance, given β ∈ [0, 1] and γ ∈ (0, 1], we might have   β q ∀θ ∈ < . (30) Mt = m ∈ Mt | 1 − β ≤ m(θ) ≤ 1 − β + γ

For m1 , m2 ∈ Mt , we can check that m1 ∨ m2 as defined through (16) is a member of Mt . Symmetrically, we can verify that m1 ∧ m2 ∈ Mt as well. Therefore, Mt is a sublattice of the

lattice Mqt , and hence induces a risk measure that satisfies the IPC Assumption. Note that

the lower bound (1 − β) · 1 + β · 0 and upper bound (1 − β) · 1 + β · (1/γ) in (30) are both convex combinations with weights 1 − β and β. The corresponding time-t risk measure ρt

amounts to a mixture of the ordinary expectation and the conditional value at risk. For any Q random cost Z ∈ Lp (
(31)

At each fixed θ[1.t−1] = (θ1 , ..., θt−1 ) ∈ q˜t (υ, zt0 (υ)),

∀z ∈ (zt0 (υ), z t ].

(32)

Suppose υ 0 ≥ υ. Then, the supermodularity of q˜t (υ, z) in (υ, z) will lead to q˜t (υ 0 , z) > q˜t (υ 0 , zt0 (υ)),

∀z ∈ (zt0 (υ), z t ].

(33)

This means that zt0 (υ 0 ) ≤ zt0 (υ). In view of (12), we can let one optimal solution zt∗ (y) to (8)

be zt0 (−y). The thus constructed zt∗ (y) is increasing in y.

When zt∗ (y) is increasing in y, the price level p∗t (y) = p˜t (zt∗ (y)) will decrease in y. Hence, Proposition 1 delivers the message that sales price should be lowered when there is more inventory left. This agrees with Theorem 2(a) of Federgruen and Heching [11]. 10

5

Mild Monotonicity in Pricing

Let us extend the definition of the revenue function r˜tm (z) through linear extrapolation, so that it is concave on not only [z t , z t ] but also the entire <. Now define wˆtm (y, ζ) = wtm (y, y+ζ)

for (y, ζ) ∈ <2 , so that following (10),

˜ t + α · vt+1 )(−ζ − Θt )]. wˆtm (y, ζ) = −˜ rtm (y + ζ) + Em [(h

(34)

Note that −ζ can serve as an indicator of the period-(t + 1) starting inventory level, as

−ζ − Θt will be the actual level.

Lemma 4 wˆtm (y, ζ) is supermodular in (m, y, ζ) ∈ Mqt × <2 . Proof: Due to (34) and the concavity of r˜tm (·), for any ∆y, ∆ζ ≥ 0, wˆtm (y + ∆y, ζ + ∆ζ) + wˆtm (y, ζ) − wˆtm (y + ∆y, ζ) − wˆtm (y, z + ∆ζ)

= r˜tm (y + ζ + ∆y) + r˜tm (y + ζ + ∆ζ) − r˜tm (y + ζ + ∆y + ∆ζ) − r˜tm (y + ζ) ≥ 0.

(35)

Therefore, w ˆtm (y, ζ) is supermodular in (y, ζ). ˜ t + α · vt+1 )(−x) is convex too. Now consider Since vt+1 (·) is convex, f (x) = (h ˜ t + α · vt+1 )(−ζ − Θt )]. gtm (ζ) = Em [f (ζ + Θt )] = Em [(h

(36)

According to Lemma 2, gtm (ζ) is supermodular in (m, ζ). But by (34) and (36), we have rtm (y + ζ) + g m (ζ). So Lemma 1 and the above would together lead to the wˆtm (y, ζ) = −˜

supermodularity of wˆtm (y, ζ) in (m, y) and (m, ζ).

Taken together, we know that w ˆtm (y, ζ) is supermodular in (m, y, ζ). Define qˆt (y, ζ) = qt (y, y + ζ) for (y, ζ) ∈ <2 , so that following (9), qˆt (y, ζ) = qt (y, y + ζ) = sup wtm (y, y + ζ) = sup wˆtm (y, ζ). m∈Mt

(37)

m∈Mt

For the same reason that q˜t (·, ·) as defined at (12) is supermodular, we know that qˆt (y, ζ) is

supermodular in (y, ζ). Just because the supermodularity of q˜t (·, ·) leads to Proposition 1, that of qˆt (·, ·) will lead to a monotone pricing result as well.

Proposition 2 For the DP from (7) to (10), one optimal solution zt∗ (y) to (8) will render y − zt∗ (y) increasing in y.

11

Proof: Consider any t = T − 1, T − 2, ..., 1. Define function f (y, w) and set W (y): f (y, w) = −ˆ qt (y, −w),

and

W (y) = [y − z t , y − z t ].

(38)

Since qˆt (·, ·) as defined through (37) is supermodular on <2 , we know that f (·, ·) is supermodular as well. Note that W (y), a subset of the lattice <, is increasing in y in the strong set order sense defined by Veinott [23], so that w1 ∈ W (y 1 ) and w2 ∈ W (y 2 ) for y 1 ≤ y 2 would result with w1 ∧ w2 ∈ W (y 1 ) and w1 ∨ w2 ∈ W (y 2 ).

Then, by Lemma 2.8.1 of Topkis [21], one solution w∗ (y) to the optimization problem

maxw∈W (y) f (y, w) will have w∗ (·) increasing in y. From (37) and (38), it follows that max f (y, w) = −

w∈W (y)

min

ζ∈[z t −y,z t −y]

qˆt (y, ζ) = − min qt (y, z), z∈[z t ,z t ]

(39)

with the last optimization problem being identical to (8). Hence, seeing that zt∗ (y) = y − w∗ (y) is one optimal solution to (8), we can let y − zt∗ (y) be increasing in y. Proposition 2 advocates mild monotonicity in pricing. When the post-ordering inventory level y rises, the firm should follow the suggestion of Proposition 1 to lower its price and hence boost the demand lever zt∗ (y). But the increase in zt∗ (y) should not be so much as to erode the gain in y. We note that this same message was conveyed by Lemma 2 of Chen and Simchi-Levi [6] in the risk-neutral setting.

6

Optimism and Higher Inventory Levels

We now set out to conduct comparative statics studies concerning varying risk measures. To be compared are two systems, labeled by superscripts 1 and 2, that are otherwise identical but with two different risk measures. For the latter, we say one is less hesitant than the other, when Mt1 ⊆ Mt2 ,

∀t = 1, 2, ..., T − 1.

The conclusion reachable on the ranking of risk-to-go function values is straightforward: vt1 (x) ≤ vt2 (x) for every t = T, T − 1, ..., 1 and x ∈ <. However, this crude ranking does not imply much about the ranking between control policies.

Thus, we next turn to a ranking through a different angle. Recall that through (15) we R R have adopted the partial order for Mqt so that m1 ≤ m2 if and only if m1 · dRt ≤ m2 · dRt

in the usual stochastic sense. We consider risk measures 1 and 2 when each Mti for i = 1, 2

and t = 1, 2, ..., T − 1 satisfies the IPC Assumption and hence forms a sublattice within the lattice Mqt . Now between risk measures 1 and 2, we say one is less optimistic than another 12

when Mt1 ≤ Mt2 for each t = 1, 2, ..., T − 1; the inequality is in the strong set order sense defined by Veinott [23], meaning that for any m1 ∈ Mt1 and m2 ∈ Mt2 , m1 ∧ m2 ∈ Mt1 ,

and

m1 ∨ m2 ∈ Mt2 .

(40)

The above means that risk measure 1 contains more densities that correspond to low additive factors, while risk measure 2 contains more densities that correspond to high additive factors. Regarding the demand level, the former paints a less optimistic picture than the latter. However, the former does not necessarily represent a less or more risk averse attitude than the latter. Whether a low or high additive factor will benefit the firm more can really differ from one situation to another. Actually, the previous inclusion-based order is more directly associated with the degree of risk averseness in the coherent-risk context. When adopting risk measure 1 that is less hesitant than risk measure 2, the firm can worry about fewer demand scenarios; see the max-cost formulation of (9). We can verify that more optimistic demand outlooks entail not only higher ordering quantities, but also lower demand levers. Proposition 3 Consider the DPs from (7) to (10), involving risk-to-go functions vt1 (x) and vt2 (x), that are defined for two systems with different risk measures. Suppose risk measure 1 is less optimistic than risk measure 2. Then, vti (x) is supermodular in (i, −x) for t =

T, T − 1, ..., 1; consequently, the order-up-to points yti∗ (x) satisfy yt1∗ (x) ≤ yt2∗ (x), and the

demand levers zti∗ (y) satisfy zt1∗ (y) ≥ zt2∗ (y).

Proof: We prove by induction. First, as vT1 (x) = vT2 (x) = −cT · x, we certainly have i vTi (x)’s supermodularity in (i, −x). Next, for some t = T − 1, T − 2, ..., 1, suppose vt+1 (x) is

supermodular in (i, −x).

Recall the definition of w˜tm (υ, z) in (11). We first show that, for m1 , m2 ∈ Mqt and

y, z 1 , z 2 ∈ < with z 1 ≤ z 2 , 1 ∧m2

w˜t1,m

(y, z 1 ) + w˜t2,m

1 ∨m2

1

2

(y, z 2 ) ≥ w˜t1m (y, z 2 ) + w˜t2m (y, z 1 ).

(41)

Now, let us keep the key observation (21) in mind. Since z 1 ≤ z 2 , we have 1 ∨m2 1 1 1 ˜m ˜m ˜ m2 ˜ m1 ∧m2 (λ). (42) −υ − z 2 − Θ (λ) ≤ −υ − z 2 − Θ t t (λ), −υ − z − Θt (λ) ≤ −υ − z − Θt

Meanwhile, the sum of the left- and right-hand sides of (42) equals that of the two middle ˜ t (·), terms. Hence, due to the convexity of h 1 ∧m2 1 ∨m2 ˜ t (−υ − z 1 − Θ ˜ t (−υ − z 2 − Θ ˜m ˜m (λ)) + h (λ)) h t t 2 2 m1 1 ˜ ˜ ˜ ˜ ≥ ht (−υ − z − Θ (λ)) + ht (−υ − z − Θm (λ)),

t

t

13

(43)

1 1 ˜ m2 ˜m where we really have equality for those λ’s with Θ t (λ) ≥ Θt (λ). Note that vt+1 (·) is

convex. Thus, similarly to the above,

1 ∧m2 1 1 ˜ m1 ∨m2 (λ)) ˜m (λ)) + vt+1 (−υ − z 2 − Θ (−υ − z 1 − Θ vt+1 t t 1 2 m1 1 ˜ ˜ m2 (λ)). ≥ v (−υ − z − Θ (λ)) + v (−υ − z 1 − Θ

t+1

t

t+1

(44)

t

On the other hand, note that 1 ∨m2

˜m −υ − z 2 − Θ t

2

˜ m (λ). (λ) ≤ −υ − z 1 − Θ t

(45)

i So from the induction hypothesis on vt+1 (x)’s supermodularity in (i, −x), we have 2 ˜ m1 ∨m2 (λ)) − v 1 (−υ − z 2 − Θ ˜ m1 ∨m2 (λ)) vt+1 (−υ − z 2 − Θ t t+1 t 2 1 1 m2 2 ˜m ˜ ≥ vt+1 (−υ − z − Θt (λ)) − vt+1 (−υ − z 1 − Θ t (λ)).

(46)

Combine (43), (44), and (46), and we obtain ˜ t + α · v 2 )(−υ − z 2 − Θ ˜ t + α · v 1 )(−υ − z 1 − Θ ˜ m1 ∨m2 (λ)) ˜ m1 ∧m2 (λ)) + (h (h t t+1 t+1 t 1 2 m1 2 ˜ ˜ ˜ ˜ m2 (λ)). ≥ (ht + α · v )(−υ − z − Θ (λ)) + (ht + α · v )(−υ − z 1 − Θ t+1

t

t+1

(47)

t

Since p˜t (·) is decreasing and z 1 ≤ z 2 , we know ˜ m1 ∧m2 (λ) · p˜t (z 1 ) + Θ ˜ m1 ∨m2 (λ) · p˜t (z 2 ) ≤ Θ ˜ m1 (λ) · p˜t (z 2 ) + Θ ˜ m2 (λ) · p˜t (z 1 ), Θ t t t t

(48)

˜ m1 (λ) ≥ Θ ˜ m2 (λ). In the presence of (4), (11), and (19), we which is really equality when Θ t t can get (41) by integrating the sum of (47) and (48) over λ ∈ [0, 1].

We now show that q˜ti (υ, z) defined in (12) is supermodular in (i, z). Let υ, z 1 , z 2 with

z 1 ≤ z 2 be given. For any  > 0, let m1 ∈ Mt1 and m2 ∈ Mt2 be such that 1

w˜t1m (υ, z 2 ) ≥ sup w˜t1m (υ, z 2 ) − ,

and

m∈Mt1

2

w˜t2m (υ, z 1 ) ≥ sup w˜t2m (υ, z 1 ) − .

(49)

m∈Mt2

Because Mt1 ≤ Mt2 in the strong set order sense, we know (40) is true. Hence, from (12), q˜t1 (υ, z 1 ) + q˜t2 (υ, z 2 ) = supm∈Mt1 w˜t1m (υ, z 1 ) + supm∈Mt2 w˜t2m (υ, z 2 ) ≥ w˜t1,m ≥

1 ∧m2

(υ, z 1 ) + w˜t2,m

supm∈Mt1 w˜t1m (υ, z 2 )

+

1 ∨m2

1

2

(υ, z 2 ) ≥ w˜t1m (υ, z 2 ) + w˜t2m (υ, z 1 )

supm∈Mt2 w˜t2m (υ, z 1 )

− 2 =

q˜t1 (υ, z 2 )

+

q˜t2 (υ, z 1 )

(50) − 2.

In the above, the second inequality is due to (41), the third inequality is due to (49), and the last equality is again due to (12). As  can be made arbitrarily small, this entails q˜t1 (υ, z 1 ) + q˜t2 (υ, z 2 ) ≥ q˜t1 (υ, z 2 ) + q˜t2 (υ, z 1 ),

(51)

i.e., q˜ti (υ, z)’s supermodularity in (i, z). In view of (8) and (12), we can let zt1∗ (y) ≥ zt2∗ (y). 14

Similarly, we can have an inequality almost identical to (47), but with υ + z i replaced by υ i + z. Then by similar logic, for m1 , m2 ∈ Mqt and υ 1 , υ 2 , z ∈ < with υ 1 ≤ υ 2 , w˜t1,m

1 ∧m2

(υ 1 , z) + w˜t2,m

1 ∨m2

1

2

(υ 2 , z) ≥ w˜t1m (υ 2 , z) + w˜t2m (υ 1 , z),

(52)

and hence q˜ti (υ, z) is supermodular in (i, υ). Combine the two supermodular properties with that of q˜t (·, ·), and we can conclude that q˜ti (υ, z) is supermodular in (i, υ, z). Hence, −˜ qti (−y, z) is supermodular in (i, y, z). Now consider optimization problem qti (−y, z)}. u˜it (y) = max {−˜ z∈[z t ,z t ]

(53)

We can use Theorem 2.7.6 of Topkis [21] to show that u˜it (y) is supermodular in (i, y). From (8) and (12), uit (y) = ct · y + minz∈[zt ,zt ] qti (y, z) = ct · y + minz∈[zt ,zt ] q˜ti (−y, z) = ct · y − maxz∈[zt ,zt ] {−˜ qti (−y, z)},

(54)

which is equal to ct · y − u˜it (y) according to (53). Therefore, uit (y) is supermodular in (i, −y). Consider the base-stock points Sti = inf argmin{uit (y) | y ∈ <} for i = 1, 2. By St1 ’s

definition, we have u1t (y) > u1t (St1 ) for any y < St1 . But uit (y)’s supermodularity in (i, −y)

means that

u2t (y) − u2t (St1 ) ≥ u1t (y) − u1t (St1 ) > 0.

(55)

In view of St2 ’s definition, we must have St1 ≤ St2 . This means that yt1∗ (x) = St1 ∨x ≤ St2 ∨x =

yt2∗ (x). By (7) and Theorem 2.7.6 of Topkis [21], we also know that vti (x) is supermodular in (i, −x). We have thus completed the induction process. In Proposition 3, the supermodularity of vti (x) in (i, −x) again indicates that higher

inventory levels are appreciated more under more aggressive demand outlooks. Note zt1∗ (y) ≥

zt2∗ (y) would lead to y − zt1∗ (y) ≤ y − zt2∗ (y). Note that each yti∗ (x) is increasing in x. Since

Mt1 and Mt2 are both lattices, we know from Proposition 1 that each zti∗ (y) is increasing in y and from Proposition 2 that each y − zti∗ (y) is increasing in y as well.

Now with yt1∗ (x) ≤ yt2∗ (x), we still cannot be certain about the relationship between

zt1∗ (yt1∗ (x)) and zt2∗ (yt2∗ (x)). However, it follows that

yt1∗ (x) − zt1∗ (yt1∗ (x)) ≤ yt2∗ (x) − zt1∗ (yt2∗ (x)) ≤ yt2∗ (x) − zt2∗ (yt2∗ (x)).

(56)

Therefore, under more optimistic demand forecast, we should not only bring up the orderup-to level, but also promote the average next-period inventory level. 15

A special case is when Mt1 is the singleton {1}, where 1 represents the all-one unit-rate R R function, and Mt2 contains densities m such that m · dRt ≥ 1 · dRt = Rt in the usual R stochastic sense. Another case is when Mt1 contains densities m with m · dRt ≤ Rt in the usual stochastic sense and Mt2 is the singleton {1}. For both cases, we have Mt1 ≤ Mt2 and

hence yt1∗ (x) ≤ yt2∗ (x) and yt1∗ (x) − zt1∗ (yt1∗ (x)) ≤ yt2∗ (x) − zt2∗ (yt2∗ (x)).

In the first case, Mt1 stands for the risk neutral case with demand projection Rt , and Mt2

stands for the risk averse case with optimistic/aggressive demand projections. In the second case, Mt1 stands for the risk averse case with pessimistic/conservative demand projections, and Mt2 stands for the risk neutral case. For these special cases, we have delivered the message that optimistic demand projections call for high inventory levels.

7

Concluding Remarks

For a study of risk-averse IPC using time-consistent coherent risk measures, we have identified lattice-theoretic structures of convex density sets as strong determinants of monotone trends in pricing. Moreover, we have found that a set-based optimism order between risk measures would lead to intuitive comparative statics trends on ordering and pricing policies. Future research might look at such issues as lost sales, nonzero lead times, multiple ordering stages, and unpredictable supplies.

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