Product price alignment with seller service rating and consumer satisfaction

Abstract

Empirical studies have shown strong evidence that a seller with a better customer feedback score is able to charge a higher price for his/her product or service. We study the problem faced by the seller of setting the price with the objective of maximizing expected revenue over a finite number of periods. Modeling the problem requires building a system of processes that includes: (1) the customer arrival and formation of customer reservation price; (2) the customer feedback collection and aggregation into a Seller Service Rating (SSR); and (3) determining how much to charge customers. Because of the technical difficulty in finding analytical solutions that fully reflect the closed-loop interconnections between these processes, we develop a simulation–optimization approach to solve the problem. We present a computational study and report on results of numerical experiments providing interesting insights on how the retailer could startegically align the price he charges with his service quality performance.

Appendices

Appendix A: Proofs

Proof of Proposition 1 Denote $x_{2} (r) = p_{2} - \phi r - \upsilon$ and $x_{1} (r) = p_{1} - \phi r - \upsilon$. $x_{2} (r) > x_{1} (r)$ since $p_{2} > p_{1}$. Because the failure rate of $\varepsilon$ is nondecreasing, $\frac{{f_{\varepsilon } (x_{2} )}}{{1 - F_{\varepsilon } (x_{2} )}} \ge \frac{{f_{\varepsilon } (x_{1} )}}{{1 - F_{\varepsilon } (x_{1} )}}$. But then,

$$ \begin{gathered} \frac{{\partial \left[ {\lambda_{u(r)} (p_{2} )/\lambda_{u(r)} (p_{1} )} \right]}}{\partial r} = \frac{{\partial \left[ {\frac{{1 - F_{\varepsilon } (x_{2} )}}{{1 - F_{\varepsilon } (x_{1} )}}} \right]}}{\partial r} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\phi f_{\varepsilon } (x_{2} )\left[ {1 - F_{\varepsilon } (x_{1} )} \right] - \phi f_{\varepsilon } (x_{1} )\left[ {1 - F_{\varepsilon } (x_{2} )} \right]}}{{\left[ {1 - F_{\varepsilon } (x_{1} )} \right]^{2} }}\,\,\,\,\,\, \ge \,\,0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \end{gathered} $$

(A-1)

Proposition 1 implies that customer demand is less sensitive to a price increase when SSR is high. We note that popular distributions, such as exponential, normal, uniform, gamma and Weibull distributions with shape parameter larger than or equal to 1, which are frequently used to model economic uncertainty have the nondecreasing failure rate property (Lariviere & Porteus, 2001). We assume, henceforth, that the distribution of $\varepsilon$ has this property.

Proof of Lemma 1Because customer ratings are mutually independent,

$\theta_{i} = E\left( {\frac{{\sum\limits_{\ell = 1}^{K} {w_{\ell } } }}{K}} \right) = \frac{1}{K}E\left( {\sum\limits_{\ell = 1}^{K} {w_{\ell } } } \right) = E(w(i))$ where $w(i)$ is an arbitrary CRS provided by a customer when the SSR is equal to i. But then, $\theta_{i} = E(w(i)) = \sum\limits_{j = 1}^{h} {jZ_{ij} }$ so that

$$\begin{aligned} & \theta_{i} = \sum\limits_{j = 1}^{h} {z_{ij} } + \sum\limits_{j = 2}^{h} {z_{ij} } + ... + \sum\limits_{j = h - 1}^{h} {z_{ij} } + z_{ih}\\ & \le \sum\limits_{j = 1}^{h} {z_{kj} } + \sum\limits_{j = 2}^{h} {z_{kj} } + ... + \sum\limits_{j = h - 1}^{h} {z_{kj} } + z_{kh} = \theta_{k},\end{aligned} $$

(A-2)

where the inequality follows from Assumption 1.

Proof of Proposition 2 It is sufficient to show that $m_{12}$ is nondecreasing in $z_{12}$. For notational convenience, let $x = z_{12}$ in what follows. For fixed x,

$$ m_{12} (x) = \sum\limits_{\ell = q}^{K} {\left( \begin{gathered} K \hfill \\ \ell \hfill \\ \end{gathered} \right)} \,x^{\ell } (1 - x)^{K - \ell } ,\,\,{\text{where }}q = \left\{ \begin{gathered} \frac{K}{2}\,,\,\,if\,\,K\,\,is\,\,\,even \hfill \\ \left\lceil \frac{K}{2} \right\rceil ,\,\,\,\,if\,\,K\,\,is\,\,\,odd \hfill \\ \end{gathered} \right. $$

(A-3)

$$ \frac{{\partial m_{12} (x)}}{\partial x} = \sum\limits_{\ell = q}^{K} {\left\{ {\,\,\ell \,\left( \begin{gathered} K \hfill \\ \ell \hfill \\ \end{gathered} \right)x^{\ell - 1} (1 - x)^{K - \ell } - (K - \ell )\,\left( \begin{gathered} K \hfill \\ \ell \hfill \\ \end{gathered} \right)x^{\ell } (1 - x)^{K - \ell - 1} } \right\}\,} $$

(A-4)

For $\ell$ < K, note that

$$ (K - \ell )\left( \begin{gathered} K \hfill \\ \ell \hfill \\ \end{gathered} \right)x^{\ell } (1 - x)^{K - \ell - 1} = (\ell + 1)\,\left( \begin{gathered} K \hfill \\ \ell + 1 \hfill \\ \end{gathered} \right)x^{\ell } (1 - x)^{K - \ell - 1} $$

(A-5)

and with the exception of the single first term in ( A-4), all other terms cancel so that

$$ \frac{{\partial m_{12} (x)}}{\partial x} = \,\,q\left( \begin{gathered} K \hfill \\ q \hfill \\ \end{gathered} \right)x^{q - 1} (1 - x)^{K - q} $$

(A-6)

But since $0 < x \le 1$, it follows that (A-6) is nonnegative and so $m_{12}$ is nondecreasing in $z_{12}$.

Proof of Proposition 3 The proof is by induction. Note that $V_{0} (c,2) = \,V_{0} (c,1)\, = \,0.$ Thus we assume the result is true for all periods k, 0 $\le$ k $\le$ t-1 and show it is true for time period t.

First, we note that

$$\begin{aligned} & V_{t - 1} (c - x,1)m_{21} \left( x \right) + V_{t - 1} (c - x,2)m_{22} \left( x \right)\\ & \ge V_{t - 1} (c - x,1)m_{11} \left( x \right) + V_{t - 1} (c - x,2)m_{12} \left( x \right)\end{aligned} $$

(A-7)

since $\begin{gathered} V_{t - 1} (c - x,2)\left( {m_{22} \left( x \right) - m_{12} \left( x \right)} \right) + V_{t - 1} (c - x,1)\left( {m_{21} \left( x \right) - m_{11} \left( x \right)} \right) \ge \hfill \\ V_{t - 1} (c - x,1)\left( {m_{22} \left( x \right) - m_{12} \left( x \right)} \right) + V_{t - 1} (c - x,1)\left( {m_{21} \left( x \right) - m_{11} \left( x \right)} \right) \ge 0 \hfill \\ \end{gathered}$.when r_t = 1, let $p*$ be the price that maximizes Eq. (11). With r_t = 2 it follows from Eq. (2) that there exists $p^{\prime} > p*$ where $\lambda (p^{\prime},2_{{}} )$ = $\lambda (\,p*,1_{{}} )$, so that $k_{t} (p^{\prime},2) = k_{t} (\,p*,1)\,\,.$

Let $h(c,\,2,\,p^{\prime})$ be (11) evaluated at $r_{t}$ = 2 and $p^{\prime}$. Thus we have.

$\begin{gathered} h(c,\,2,\,p^{\prime}) {=} \left\{ {\mathop {\text{e}} \nolimits^{{ - k_{t} (p^{\prime},2)}} \sum\limits_{x = 0}^{c} \frac{{[k_{t} (p^{\prime},2)]^{x} }}{x!}\left[ {p^{\prime}(x {-} c) {+} (V_{t - 1} (c {-} x,1)m_{21} (x)+ V_{t - 1} (c {-} x,2)m_{22} (x)} \right] {+} p^{\prime}c} \right\} \ge \hfill \\ \left\{ {\mathop {\text{e}} \nolimits^{{ - k_{t} (\,\,p*,1)}} \sum\limits_{x = 0}^{c} \frac{{[k_{t} (\,\,p*,1)]^{x} }}{x!}\left[ {p^{\prime}(x - c) + (V_{t - 1} (c - x,1)m_{11} (x)\, + \,V_{t - 1} (c - x,2)m_{12} (x)} \right] + p^{\prime}c} \right\} \ge \hfill \\ \left\{ {\mathop {\text{e}} \nolimits^{{ - k_{t} (\,p*,1)}} \sum\limits_{x = 0}^{c} \frac{{[k_{t} (\,\,p*,1)]^{x} }}{x!}\left[ {\,\,p*(x - c) + (V_{t - 1} (c - x,1)m_{11} (x)\, + \,V_{t - 1} (c - x,2)m_{12} (x)} \right] + \,\,p*c} \right\} \hfill \\ = V_{t} (c,1). \hfill \\ \end{gathered}$ The result now follows since $V_{t} (c,2)\,\,\, \ge \,\,$$h(c,\,2,\,p^{\prime})$.

Appendix B: Example of FBP policy

Suppose $\Omega = \{ 1,2\}$ and c = 2 units. The dynamic programming problem has six possible states: $(c,r) = \{ (0,1),(0,2),(1,1),(1,2),(2,1),(2,2)\}$. Let the initial state be $(c,r) = (2,1)$. Assume $K = 2$. Also, let the total number of review points be $T = 3$ and consider the average arrival rate to be $a = 5$. Assume that the CSR transition probability matrix $Z_{2 \times 2} = \{ z_{ij} \}$ is $Z = \left( {\begin{array}{*{20}c} {0.4} & {0.6} \\ {0.3} & {0.7} \\ \end{array} } \right).$ For this Z, the SSR transition matrix is $M = \left[ \begin{gathered} .16\,\,\,\,\,\,.84 \hfill \\ .09\,\,\,\,\,\,.91 \hfill \\ \end{gathered} \right]$. In addition, suppose the consumer valuation on service rating is $\phi = 10$, and let the intrinsic value of the product be $\upsilon = 20$. Finally, assume the random term of reservation price follows a normal distribution with mean zero and variance 4, i.e., $\varepsilon \sim N(0,4)$. We solve the problem and show the price paths between periods and the depletion of inventory in Fig. 

Fig. 6

The FBP policy

6.

The expected revenues are calculated using Eq. (11) as follows:

$\begin{gathered} V1\left( {1,1} \right)\, = \,.V1\left( {1,2} \right)\, = \,0.964\, \times \,29.16\, = \,28.10; \hfill \\ V1\left( {2,1} \right)\, = \,0.906\, \times \,2x28.36\, + \,0.075\, \times \,28.36\, + \,0\, = \,53.52; \hfill \\ \end{gathered}$

$\begin{gathered} V2\left( {1,1} \right)\, = \,0.720\, \times \,31.32\, + \,28.10x.280\, = \,30.42;V2\left( {2,1} \right)\, = \,2\, \times \,30.61x.567\, \hfill \\ + \,0.284x\left( {28.10\, + \,30.61} \right)\, + \,0.149\, \times \,53.52\, = \,59.36;{\text{ and}}, \hfill \\ \end{gathered}$

$V3\left( {2,1} \right)\, = \,0.296\, \times \,2x31.56\, + \,0.367x\left( {31.56\, + \,30.42} \right)\, + \,0.337\, \times \,59.36\, = \,61.44$

Consider nodes 1–6. The shaded circles represent either all available units are sold or it is the end of the season. Each unshaded circle represents the states (c, r) at the beginning of each period and the expected revenue with t periods to go. At the start of the selling season (which is t = 3 in Fig. 6), we price the product at $31.56$. All units are sold in period 1 when the demand is 2 units or more with a probability 0.296 (according to (8)). If we sell 1 unit with a probability of 0.367, then SSR is not updated (i.e. r = 1 since items sold is less than K = 2 and thus $m_{11}$ = 1 according to (9)). The price at the beginning of period 2 (i.e. t = 2) is lowered to 31.32. If we sell 1 unit in period 2 with probability 0.720 then selling ends. If we do not sell in period 2, then the price is lowered further to 29.16 at the beginning of period 3 (t = 1). The states (c,r) and the corresponding prices represented by the remaining nodes are interpreted in a similar way. The total expected profit of this FBP policy is given by V₃(2,1) = $61.44. (See detailed calculations at the bottom of the figure.)

Appendix C: How the average price and revenue are computed.

In this appendix we provide the details how we computed the average price and average revenue using y = 0.2 and SSR = 1 as example. Consider the results shown in Table

Table 10 Detailed Results for y = 0.2, Case 5 and initial SSR = 1

10 for y = 0.2 with current SSR = 1. The recorded price in each period is a weighted average of prices over the 10 replications, where weights are determined by quantities sold in the various runs. By way of further explanation, for a given case, let $p_{\eta t}$ be the price and $Q_{\eta t}$ be the quantity sold in replication $\eta$ corresponding to period t. The recorded “quantity sold” for period t is $\sum\limits_{\eta } {Q_{\eta t} } /10$, whereas the recorded price is $\sum\limits_{\eta } {Q_{\eta t} } p_{\eta t} /\sum\limits_{\eta } {Q_{\eta t} }$. Finally, at the bottom of each column, the “Weighted Average Price” aggregates the period prices using proportions of quantity sold. The “Expected Revenue” is an aggregation of period prices multiplied by the quantities sold.

See Table 10.