## 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?

### Price-Setting Newsvendor Models Based on OSDT

In the following model, a manufacturer who produces and sells an innovative product in the monopoly market is considered. Before selling season, the manufacturer produces $q$ units at unit cost W and setup cost is assumed to be zero. We consider the following widely used linear inverse demand function.

$x=b−aR$,

(15)

where $b > 0$ shows the limit demand when the retail price equals to $0$, and $a > 0$ represents the decreasing of demand when the retail price increases by one unit. We call a as the price sensitivity of market demand. The demand's uncertainty is described by the parameter b with probability density function $f(b)$. The profit function is with the retail price and the production quantity as the decision variables. With considering (1), it can be expressed as:

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

(16)

If one considers Definitions 1 and 2, then we have the relative likelihood function of $b$, i.e., $π(b)$ and the satisfaction function, i.e., $u(R,b,q)$. Similar to the newsvendor model, the following types of focus points are considered.

Active focus point: For retail price $R$ and production quantity $q$, the active focus point is

$b1(R,q) ∈ \overset{argmax}{x} lower[π(b),u(R,b,q)]$,

(17)

$b_1(R,q)−aR$ is the focused demand value with the relatively high likelihood degree and satisfaction level for the production quantity $q$.

Passive focus point: For retail price $R$ and production quantity $q$, the passive focus point is

$b_2(R,q) ∈ \overset{argmin}{x} upper[1−π(b),u(R,b,q)]$,

(18)

$b_2(R,q)−aR$ is the focused demand value with the relatively high likelihood degree and relatively low satisfaction level for the production quantity $q$.

Apprehensive focus point: For retail price $R$ and production quantity $q$, the apprehensive focus point is

$b_3 (R,q) ∈ \overset{argmin}{x} upper[π(b),u(R,b,q)]$,

(19)

$b_3(R,q)−aR$ is the focused demand value with the relatively low likelihood degree and satisfaction level for the production quantity $q$.

Daring focus point: For retail price $R$ and production quantity $q$, the daring focus point is

$b_4(R,q) ∈ \overset{argmin}{x}upper[π(b),1−u(R,b,q)]$,

(20)

$b_4(R,q)−aR$ is the focused demand value with the relatively low likelihood degree and relatively high satisfaction level for a production quantity $q$.

The sets of the four types of focus points of the retail price $R$ and production quantity $q$ are denoted as $B_1(R,q)$, $B_2(R,q)$, $B_3(R,q)$ and $B_4(R,q)$, respectively. The optimal production quantities for the manufacturers are

$q_1(R) ∈ \overset{argmax}{q} \overset{max}{b_1(R,q) ∈ B_1(R,q)} \quad \quad u(R,b_1(R,q),q)$,

(21)

$q_2(R) ∈ \overset{argmax}{q} \overset{max}{b_2(R,q) ∈ B_2(R,q)} \quad \quad u(R,b_2(R,q),q)$,

(22)

$q_3(R) ∈ \overset{argmax}{q} \overset{max}{b_3(R,q) ∈ B_3(R,q)} \quad \quad u(R,b_3(R,q),q)$,

(23)

$q_4(R) ∈ \overset{argmax}{q} \overset{max}{b_4(R,q) ∈ B_4(R,q)} \quad \quad u(R,b_4(R,q),q)$,

(24)

From Equations (17)–(24), we can see that for a fixed $R$, the profit functions of the active, passive, apprehensive and daring manufacturers are $r(R,b_1(R,q_1(R)),q_1(R))$, $r(R,b_2(R,q_2(R)),q_2(R))$, $r(R,b_3(R,q_3(R)),q_3(R))$ and $r(R,b_4(R,q_4(R)),q_4(R))$, respectively, which are named as active, passive, apprehensive and daring profit functions, respectively. Because they are the functions of single variable $R$, for simplicity, we use $r_1(R)$, $r_2(R)$, $r_3(R)$ and $r_4(R)$ in the following parts. For each type of manufacturer, the optimal retail price is which to maximize his/her profit function.

$R^{∗}{1} ∈ \overset{argmaxr1}{R} (R), q^{∗}_{1} ∈ q^1 (R^{∗}_{1})$,
(25)

$R^{∗}{2} ∈ \overset{argmax r_2}{R} (R), q^{∗}_{2} ∈ q^2 (R^{∗}_{2})$,
(26)
$R^{∗}{3} ∈ \overset{argmax r_3}{R} (R), q^{∗}_{3} ∈ q^3 (R^{∗}_{3})$,
(27)

$R^{∗}{4} ∈ \overset{argmax r_4}{R} (R), q^{∗}_{4} ∈ q^4 (R^{∗}_{4})$,
(28)

$R^{∗}_{1}$, $R^{∗}_{2}$, $R^{∗}_{3}$ and $R^{∗}_{4}$ are optimal retailer prices for active, passive, apprehensive and daring manufacturers, respectively. They are named as optimal active, passive, apprehensive and daring retail prices, respectively.