## Price-Setting Models for Innovative Products

Modern organizations have abundant choices in selecting the right product supply chain. Functional products are items that are purchased regularly, are widely available from multiple sources, and have long life cycles. Alternatively, innovative products are new to the marketplace, have unpredictable demand, and usually have short life cycles.

Read this article. Researchers suggest that products belong to either a functional or innovative category, where innovative products are much more challenging to price. What innovative products do you own, and how do you know you are getting the value you paid for?

### Newsvendor Models with the OSDT

In this section, the decision-making procedure for a retailer of innovative products is introduced. Before the selling season, the retailer must decide the order quantity q. For one unit of the innovative product, the wholesale price is $W$, and the retail price is $R$ and $R > W$. If there is an excess, the unit salvage price is $S_o > 0 (S_o < W)$. The unit opportunity cost is $S_u > 0$ for the shortage. The retailer's profit function is as follows:

$r(x,q)= \left\{ \begin{array}{ll} (R−W) x−(W−S_o)(q−x); x < q \\ (R−W)q−S_u(x−q); x ≥ q \end{array} \right.$,

(1)

The demand function of the innovative product is described by a random variable $X$ and the probability mass (density) function is $f(x)$.

Definition 1. Given the probability function $f(x)$ for random vector and $π:Ω→[0,1]$ be a function satisfying

$max{π(x):x∈Ω}=1$,

(2)
$π(x)$ is the relative likelihood degree  of $x$ if $π(x_1) > π(x_2)$ for $f(x_1) > f(x_2)$ and $π(x_1)=π(x_2)$ for $f(x_1)=f(x_2)$. For a retailer, his/her satisfaction levels towards profits are represented by the satisfaction function.

Definition 2.
The satisfaction function is a strictly increasing function of the profit $r$,

$u : G → [0,1]$,

(3)

where $G$ is the set of profit $r$.

Obviously, the relative likelihood degrees and the satisfaction levels can be utilized to describe the relative position of the probabilities and the payoffs, respectively.

Usually, because of the short life cycle of the innovative product, there is only one chance given to the retailer to determine his/her order quantity and a unique demand will appear accordingly. Therefore, before ordering products, the retailer has to meditate which demand should be factored in. We take into account four types of demands (scenarios) for each order quantity with contemplating the likelihood degrees and the satisfaction levels, that is the demands with the higher satisfaction and likelihood (Type I), the lower satisfaction and higher likelihood (Type II), the higher satisfaction and lower likelihood (Type III), the lower satisfaction and likelihood (Type IV). It is intuitively acceptable that active, passive, daring and apprehensive retailers are inclined to take into account Type I, Type II, Type III and Type IV demands, respectively. Therefore, we call Type I, Type II, Type III and Type IV demands active, passive, daring and apprehensive focus points, respectively (shown in Table 1). Which kind of focus point is taking into account reflects the personality of the retailer under demand uncertainty.

Table 1. Four different focus points.

 satisfaction higher lower likelihood higher active focus point passive focus point lower daring focus point Apprehensive focus point

Following operators are introduced to characterize the focus points.

Definition 3. Given a vector $[z_1, z_2,⋯,z_n]$, $lower[z_1, z_2,⋯,z_n]$ and $upper[z_1, z_2,⋯,z_n]$ are defined as follows:

$lower[z_1, z_2,⋯,z_n] = [∧_{i=1},⋯,_n z_i, ∧_{i=1},⋯,_n z_i,⋯, ∧_{i=1},⋯, _n z_i]$,

(4)

$upper[z_1, z_2,⋯,z_n] = [∨_{i=1},⋯, _n z_i, ∨_{i=1},⋯, _n z_i,⋯,∨_{i=1},⋯, _n z_i]$,

(5)

$lower [z_1, z_2,⋯,z_n]$ and $upper [z_1, z_2,⋯,z_n]$ represent the lower and upper bounds of $[z_1, z_2,⋯,z_n]$. For instance, for a statex, the relative likelihood degree is 0.3 and the satisfaction level is 0.8, which is represented as $[0.3,0.8]$ .$lower[0.3,0.8]=[0.3,0.3]$ and $upper[0.3,0.8]=[0.8,0.8]$ represent that $x$ has at least 0.3 relative likelihood degree and 0.3 satisfaction level and $x$
has at most 0.8 relative likelihood degree and 0.8 satisfaction level.

In the following, we introduce how to obtain these four types of focus points.

Active focus point: For an order quantity $q$, the active focus point is

$x^{∗}_{1}(q) ∈ \overset{argmax}{x} \text{ lower}[π(x),u(x,q)]$.

(6)

Example 1. There are four demands $x_1, x_2, x_3$ and $x_4$. Their probabilities are 0.05, 0.15, 0.5 and 0.3 so that the corresponding relative likelihood degrees are 0.1, 0.3, 1.0 and 0.6, respectively. For an order quantity $q$ whose $[π(x),u(x,q)]$ are, for instance, $[0.1,0.6], [0.3,0.2], [1.0,0.3]$ and $[0.6,0.8]$, respectively. $\overset{max}{x} lower [π(x),u(x,q)]$ is $max([0.1,0.1], [0.2,0.2], [0.3,0.3], [0.6,0.6]) = [0.6,0.6]$ which corresponds to $x_4$. Thus, $\overset{argmax}{x} lower [π(x),u(x,q)]$ is $x_4$. Clearly $x_4$ is the demand with a higher likelihood degree and satisfaction level.

Passive focus point: For an order quantity $q$, the passive focus point is

$x^{∗}_{2} (q) ∈ \overset {argmin}{x} upper [1−π(x),u(x,q)]$.

(7)

Apprehensive focus point: For an order quantity $q$, the apprehensive focus point is

$x^{∗}_{3} (q) ∈ \overset {argmin}{x} upper[π(x),u(x,q)]$.

(8)

Daring focus point: For an order quantity $q$, the daring focus point is

$x^{∗}_{4}(q) ∈ \overset{ argmin} {x} upper[π(x),1−u(x,q)]$.

(9)

Comments: Equations (6)–(9) are from four bi-objective optimization problems as follows: $\overset{max}{x} π(x),\overset{max}{x}u(x,q)$; $\overset{max}{x}π(x),\overset{min}{x}u(x,q)$; $\overset{min}{x}π(x)$,$\overset{min}{x}u(x,q)$ and $\overset{min}{x}π(x)$, $\overset{max}{x}u(x,q)$. From Equations (6) to (9), there is no other $[π(x), u(x,q)]$ satisfies $π(x)>π(x^{∗}_{1}(q))$ and $u(x,q)>u(x^{∗}_{1}(q),q)$; or $π(x)>π(x^{∗}_{2}(q))$ and $u(x,q)$; or $π(x)$ and $u(x,q)$; or $π(x)$ and $u(x,q)>u(x^{∗}_{4}(q),q)$. It means that $x^{∗}_{1}(q)$, $x^{∗}_{2}(q)$, $x^{∗}_{3}(q)$ and $x^{∗}_{4}(q)$ are Pareto optimal solutions of the above four bi-objective optimization problems which are used to seek for the demands with the higher likelihood and satisfaction, the higher likelihood and lower satisfaction, the lower likelihood and satisfaction and the lower likelihood and higher satisfaction, respectively. In other words, for any $q$ no demand can cause an even higher satisfaction with an even higher likelihood than its active focus point $x^{∗}_{1}(q)$; no demand can provide an even lower satisfaction with an even higher likelihood than its passive focus point $x^{∗}_{2}(q)$; no demand can lead to an even lower satisfaction with an even lower likelihood than its apprehensive focus point $x^{∗}_{3}(q)$; no demand can generate an even higher satisfaction with an even lower likelihood degree than its daring focus point $x^{∗}_{4}(q)$.

Advantages in phenomena explanation: Let us consider the following anecdotal evidence. In September 2014, Apple® released iPhone 6 and iPhone 6 Plus, but the Chinese market was left out the first wave of countries. The iPhone 6 was sold for as much as 10 times the U.S. price in Chinese black market, due to the delayed release. There were many scalpers trying to buy and resell the iPhone 6 in this risky and fragile market. Grothaus observed that some of the scalpers treat it as a "gamble" and just took into account the scenario that they can make profits and "feed their family". This kind of phenomena in an innovative product market can be explained by the behavior of a daring retailer. Even though some scenario may occur with a low likelihood, the high gain lures him/her to take action. On the other hand, this kind of phenomena is very hard to be explained by lottery-based models, including expected utility models, value at risk models or conditional value at risk models. The reason is that the expression of risk preferences in these models rely on the framework of weighting average, which ignored the importance of some unique and irreplaceable scenario (focus point) in the progress of decision-making.

For an order quantity $q$, multiple demands may be considered as one type of focus point, the sets of the above mentioned four types of focus points are denoted as $X_1(q)$, $X_2(q)$, $X_3(q)$ and $X_4(q)$, respectively.

In newsvendor models, the focus point is regarded as the retailer's most focused demand, and the retailer chooses the order quantity that will lead to the best outcome (highest satisfaction level) in case the focus point (focused demand) really happen. Therefore, the following optimal order quantities are obtained.

$q^{∗}_{1} ∈ \overset{argmax}{q} \quad \overset{max}{x^{∗}_{1} (q) ∈ X_1(q)} \quad u(x^{∗}_{1} (q),q)$,

(10)

$q^{∗}_{2} ∈ \overset{argmax}{q} \quad \overset{max}{x^{∗}_{2} (q) ∈ X_2(q)} \quad u(x^{∗}_{2} (q),q)$,

(11)

$q^{∗}_{3} ∈ \overset{argmax}{q} \quad \overset{max}{x^{∗}_{3} (q) ∈ X_3(q)} \quad u(x^{∗}_{3} (q),q)$,

(12)

$q^{∗}_{4} ∈ \overset{argmax}{q} \quad \overset{max}{x^{∗}_{4} (q) ∈ X_4(q)} \quad u(x^{∗}_{4} (q),q)$,

(13)

We call $q^{∗}_{1}$, $q^{∗}_{2}$, $q^{∗}_{3}$ and $q^{∗}_{4}$ optimal active, passive, apprehensive and daring order quantities, respectively. It should be noted that the optimal orders are adopted only based on the satisfaction levels of the focus points. A numerical example is given for the easy understanding of the decision procedure.

Example 2. A fashion store is scheduled to order a kind of newly designed fashion. For a unit, retail price $R$, wholesale price $W$, salvage price $S_o$ and opportunity cost $S_u$ are all set, for example, as 10, 7, 1 and 4 (1000RMB), respectively. The profit of the store is

$r(x,q) = \left\{ \begin{array}{ll} 9x−6q, & x < q \\ 7q−4x, & x ≥ q \end{array} \right.$

(14)

Suppose that the set of demand values is $D={350,450,550,650,750}$ so that the set of order quantities is $D={350,450,550,650,750}$. Their probabilities are 0.085, 0.135, 0.386, 0.282, and 0.112, respectively. Using (2), we can calculate the relative likelihood degrees of them (shown in Table 2).

Table 2. The relative likelihood degrees of demands.

Demands 350 450 550 650 750
likelihood degrees 0.22 0.35 1.00 0.73 0.29

Following Equation (14), the profits (1000yuan) are obtained for each order quantity (see Table 3). For simplification, the satisfaction function is $u(r)=\dfrac{r+1350}{3600}$, which is profit's linear function with $u(−1350)=0$ and $u(2250)=1$. The corresponding satisfaction levels are shown in Table 4.

Table 3. Profits for each order quantity.

 Demands 350 450 550 650 750 Orders 350 1050 650 250 −150 −550 450 450 1350 950 550 150 550 −150 750 1650 1250 850 650 −170 150 1050 1950 1550 750 −1350 −450 450 1350 2250

Table 4. Satisfaction levels obtained for order quantities.

 Demands 350 450 550 650 750 Orders 350 0.67 0.56 0.44 0.33 0.22 450 0.50 0.75 0.64 0.53 0.42 550 0.33 0.58 0.83 0.72 0.61 650 0.17 0.42 0.67 0.92 0.81 750 0.00 0.25 0.50 0.75 1.00

Let us analyze the case of the order quantity 450. The relative likelihood degree and satisfaction level [1.00, 0.64] on demand 550 is undominated by the ones of other demands, that is to say, demand 550 causes the relatively high satisfaction and likelihood. Hence, demand 550 is regarded as the active focus point of the order quantity 450. Since there is no other demand can simultaneously cause higher relative likelihood degree and lower satisfaction level than demand 650, demand 650 is regarded as the passive focus point of 450. Similarly, demand 750 and demand 450 are regarded as the apprehensive and daring focus point of 450, respectively. In addition, we can obtain focus points for other order quantities (see Table 5).

Table 5. Focus points of order quantities.

Order Quantities
350 450 550 650 750
Active 550 550 550 650 650
Passive 650 650 450 450 550
Apprehensive 750 750 350 350 350
Daring 350 450 750 750 750

In step 2, the optimal order quantities are chosen on the basis of satisfaction levels of focus points. The satisfaction levels for each order quantity with different types of focus points is easily calculated (see Table 6). Using (10–13), we get the optimal active, passive, apprehensive and daring order quantities, that is 650, 550, 450, and 750, respectively.

Table 6. Satisfaction levels for focus points.

Order Quantities
350 450 550 650 750
Active 0.44 0.64 0.83 0.92 0.75
Passive 0.33 0.53 0.58 0.42 0.50
Apprehensive 0.22 0.42 0.33 0.17 0.00
Daring 0.67 0.75 0.61 0.81 1.00

The newsvendor models with the OSDT provided a fundamental alternative to analyze the supply chain management problems for the innovative product, such as the product pricing, channel coordination and contract design in the supply chain. In the following, we'll focus on the price-setting newsvendor problem for the innovative product.