# Supply Chain Model for Product Repulsion

Subrata Saha, Izabela Ewa Nielsen, Ilkyeong Moon

► To cite this version:

Subrata Saha, Izabela Ewa Nielsen, Ilkyeong Moon. Supply Chain Model for Product Repulsion. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2022, Gyeongju, South Korea. pp.543-549, 10.1007/978-3-031-16407-1_64 . hal-05386135

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# Supply Chain Model for Product Repulsion

Subrata Saha<fnref n="1" />(✉), Izabela Ewa Nielsen<fnref n="1" />, and Ilkyeong Moon<fnref n="2" />,<fnref n="3" />

**Abstract.** This study investigates the effect of product deletion strategy on the profitability of a supply chain while the manufacturer sells both high-and-low quality products. We formulate models to explore optimal supply chain decisions consisting of one manufacturer and one retailer in four scenarios based on the manufacturer's decision to continue both products, only low-quality products, and only high-quality products, or introduce a new product with adjusted quality. The result reflects that a manufacturer can maximize profit by offering only low-quality products or products with adjusted quality based on consumer valuation. If the manufacturer modifies the quality of existing products, total selling quality might be high even if its quality is low when consumers' preferences remain unchanged. Continuation of both products or only high-quality products does not always ensure higher profits.

**Keywords:** Product elimination · Manufacturing · Quality · Supply chain

## 1 Introduction

Introducing a new version of a similar type of product or removing an existing one over time is recognized as one of the critical strategic measures by which firms continuously renew themselves to survive in rapidly changing business environments [1, 10]. For example, Samsung Galaxy S20, S20 Plus, and Ultra have been discontinued<fnref n="1" /> and replaced with some exclusive model such as Samsung Galaxy F series for some specific countries like India<fnref n="2" />. Similar measures are practiced in

<footnote n="1">¹ Department of Materials and Production, Aalborg University, 9220 Aalborg East, Denmark</footnote>

<footnote>saha@m-tech.aau.dk, izabela@mp.aau.dk</footnote>

<footnote n="2">² Department of Industrial Engineering, Seoul National University, Seoul 08826, Republic of Korea</footnote>

<footnote>ikmoon@snu.ac.kr</footnote>

<footnote n="3">³ Institute for Industrial Systems Innovation, Seoul National University, Seoul 08826, Republic of Korea</footnote>

<footnote n="1">¹ www.samsung.com/my/smartphones/galaxy-a-series/?product1=sm-a725fzkhxme&product2=sm-a526blvhxme&product3=sm-a326blvhxme.</footnote>

<footnote n="2">² en.wikipedia.org/wiki/Samsung_Galaxy_F_series.</footnote>
household appliances or consumer electronics products, where firms have been discontinued as well as introduced new versions to maintain a stable relationship with their consumers [7].

The rapid rate of technological innovation and frequent changes in consumer preferences reduce the lifecycles of products such as personal computers, cellular phones, and new telecommunication services. Product deletion as a critical product management decision equally contributes to strategic organizational development as many others. Various organizations such as Heinz, Procter & Gamble, Kraft, and Unilever shortlist and cull their product portfolios regularly [12]. Product up-gradation, deletion, or replacement with the adjusted quality decision have immense potential to improve overall profitability [9]. A strategic decision to delete products can be used as a “competitive weapon” in critical market situations. For example, a strategic product may be discontinued merely due to its undesirable sales volume or profit, putting meaningful business relations at risk [6, 11].

This study presents an analytical model that helps the manufacturers decide when product deletion is favorable. In a two-period supply chain interaction consisting of a single manufacturer and a single retailer supply chain setting, the manufacturers offer both products in the first period. In the second period, the manufacturer decides whether to offer only the high-quality, low-quality or both products. Consequently, we derive the analytical solutions of the optimal wholesale prices, product quality, and product prices for the possible four scenarios. The detailed discussion on the effects of parameters on the profit and product quality can provide a comprehensive overview of marketing and operational aspects they need to consider. We believe that the solutions and corresponding thresholds based on the parameters are beneficial for managers to understand the dynamics and help make product deletion strategies.

The following are the critical contributions of the study: there is extensive empirical and analytical literature on the effect of the strategic introduction of the new product. However, the analytical evaluation of product deletion is sparse. Our analysis clearly demonstrates that the manufacturer's strategic product deletion decision can ensure higher profits for both members. It is sometimes preferable to offer only a product with an adjusted quality instead of continuing with both products.

## 2 Model Description

We consider a one-to-one supply chain wherein a manufacturer offers both high and low-quality ($i = H, L$) products to a retailer in the first period and sets wholesale prices $w_H$ and $w_L$, respectively. The product qualities for the low and high-quality products are $\theta$ and $t\theta$, $t > 1$, respectively, and the unit cost for producing a low-quality product with quality $\theta$ is $\theta^2$, and the that of a high-quality product is $\eta t^2 \theta^2$, ($t > 1$) where $\eta$ is the efficiency for the manufacturer in adopting new technology. The quadratic cost function considered in this study is extensively used in the literature [1,3–5]. Followed by the manufacturer decisions,
the retailer sets retail prices for low and high-quality products as $p_L$ and $p_H$, respectively.

We assume that consumers are heterogeneous on product quality ($\theta$) and are uniformly distributed over $[0, b]$, where $b$ represents the degree of market potential among consumers for the product category; therefore, the surplus from purchasing the product with quality $\theta$ and retail price $p$ for the consumers whose valuation is $U$ is $U\theta - p$, and consumers make purchase decisions based on both retail price and quality [2]. Each consumer buys at most one unit of either product. Particularly, the consumer's utility is $t\theta U - p_H$ if a unit of the high quality product is bought, $\theta U - p_L$ if a unit of the low quality product is bought, and zero if neither products is bought. We assume the manufacturer offers both product in the first period, the demand function is obtained as follows: $d_L = \frac{p_H - p_L}{t\theta - \theta} - \frac{p_L}{\theta}$ and $d_H = b - \frac{p_H - p_L}{t\theta - \theta}$, $t > 1$. List of notations are presented in Table 1 below:

<table><caption>**Table 1.** List of notations</caption><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td colspan="2"><strong>Subscript</strong></td></tr><tr><td><i>i</i></td><td>Denotes product type, <i>i</i> &isin; {H, L}</td></tr><tr><td><i>j</i></td><td>Denotes scanarios, <i>j</i> &isin; {BHL, OL, OH, AH}</td></tr><tr><td colspan="2"><strong>Decision variable</strong></td></tr><tr><td><i>w</i><sup><i>j</i></sup><sub><i>i</i></sub></td><td>Wholesale price of <i>i</i><sup><i>th</i></sup> product</td></tr><tr><td><i>p</i><sup><i>j</i></sup><sub><i>i</i></sub></td><td>Retail price of <i>i</i><sup><i>th</i></sup> product</td></tr><tr><td><i>t</i><sup><i>j</i></sup></td><td>Quality level of the upgraded product relative to that of the existing product (<i>t</i><sup><i>j</i></sup> &gt; 1)</td></tr><tr><td>&pi;<sup><i>j</i></sup><sub><i>i</i></sub></td><td>Profits for the retailer</td></tr><tr><td>&pi;<sup><i>j</i></sup><sub><i>m</i></sub></td><td>Profits for the manufacturer</td></tr><tr><td><i>Q</i><sup><i>j</i></sup><sub><i>i</i></sub></td><td>Total quantity</td></tr><tr><td colspan="2"><strong>Parameters</strong></td></tr><tr><td><i>b</i></td><td>Market potential</td></tr><tr><td>&eta;</td><td>Quadratic quality cost factor coefficient</td></tr><tr><td>&theta;</td><td>Quality of the existing product</td></tr></tbody></table>

In the second period, the manufacturer has four options: (i) offers both products (Scenario BHL), (ii) only offers a low-quality product (Scenario OL), (iii) offers only a high-quality product (OH), (iv) offers an adjusted high-quality product (Scenario AH). Based on the assumption, profit functions for the retailer ($\pi_r$) and manufacturer ($\pi_m$) are obtained as follows:

$$ 
\begin{aligned}
\pi_r(p_L, p_H) &= (p_L - w_L)d_L + (p_H - w_H)d_H \\
\pi_m(w_L, w_H, t) &= (w_L - \theta^2)d_L + (w_H - \eta t^2 \theta^2)d_H
\end{aligned}
\qquad (1) 
$$
If the manufacturer offers only low-quality products, then the consumers do not have any alternative, and the profit function for the manufacturer and the retailer are obtained as follows:

$$ \pi_r = (p_L - w_L) \left(b - \frac{p_L}{\theta}\right) \qquad \pi_m = (w_L - \theta^2) \left(b - \frac{p_L}{\theta}\right) \qquad (2) $$

Finally, if the manufacturer offers only high-quality products, the profit function for the manufacturer and the retailer are obtained as follows:

$$ \begin{aligned} \pi_r &= (p_H - w_H) \left(b - \frac{p_H}{t\theta}\right) \\ \pi_m &= (w_H - \eta t^2 \theta^2) \left(b - \frac{p_H}{t\theta}\right) \end{aligned} \qquad (3) $$

In the following section, we present optimal decisions for all scenarios.

## 3 Model Solutions and Discussions

We assume that the manufacturer decides wholesale prices and quality level for a high-quality product and that the downstream retailer decides market prices. In addition, the manufacturer acts as a Stackelberg leader, and the retailer is the follower. We use backward induction to find the optimal decision. Proposition 1 summarizes the optimal decision in Scenario BHL.

**Proposition 1.** Optimal decision in Scenario BHL is obtained as follows:

$$ \begin{aligned} \pi_r^{BHL} &= \frac{2b^3 + 15b^2\eta\theta + 18b\eta\theta^2 - 48b\eta^2\theta^2 + 27\eta\theta^3 - 144\eta^2\theta^3 + 128\eta^3\theta^3 + 2(b^2 - 4b\eta\theta - 12\eta\theta^2 + 16\eta^2\theta^2)\sqrt{X}}{432\eta} \\ \pi_m^{BHL} &= \frac{2b^3 + 15b^2\eta\theta + 18b\eta\theta^2 - 48b\eta^2\theta^2 + 27\eta\theta^3 - 144\eta^2\theta^3 + 128\eta^3\theta^3 + 2(b^2 - 4b\eta\theta - 12\eta\theta^2 + 16\eta^2\theta^2)\sqrt{X}}{216\eta} \\ v_L^{BHL} &= \frac{\theta(3b + \theta)}{4} v_H^{BHL} = \frac{5b^2 - \eta\theta^2(3 - 8\eta) + 19b\eta\theta + (2b\eta + 5b)\sqrt{X}}{36\eta}; Q_L^{BHL} = \frac{(b + (3 - 8\eta)\theta) + 2\sqrt{X}}{12} \\ Q_H^{BHL} &= \frac{2b - 4\eta\theta - \sqrt{X}}{6}, w_L^{BHL} = \frac{\theta(b + \theta)}{2}, w_H^{BHL} = \frac{2b^2 + 7b\eta\theta - \eta\theta^2(3 - 8\eta) + 2(b + \eta\theta)\sqrt{X}}{18\eta}, v_L^{BHL} = \frac{b + 4\eta\theta + \sqrt{X}}{6\eta\theta}. \end{aligned} $$

**Proof:** Because $\frac{\partial^2 \pi_r}{\partial p_L^2} = \frac{-2}{\theta(t-1)} < 0$, $\frac{\partial^2 \pi_r}{\partial p_H^2} = \frac{-2t}{\theta(t-1)} < 0$ and $\frac{\partial^2 \pi_r}{\partial p_L^2} \cdot \frac{\partial^2 \pi_r}{\partial p_H^2} - \left(\frac{\partial^2 \pi_r}{\partial p_L \partial p_H}\right)^2 = \frac{4}{(t-1)^2} > 0$, the retailer's profit function in Eq. (1) is concave in $p_L$ and $p_H$. Solving first-order conditions, the retail prices are obtained as follows:

$$ p_L = \frac{w_L + b\theta}{2} \qquad p_H = \frac{w_H + bt\theta}{2} \qquad (4) $$

Based on the retailer response, the profit function for the manufacturer is obtained as follows:

$$ \pi_m = \frac{(tw_L - w_H)(w_L - \theta^2)}{2\theta - 2t\theta} + \frac{1}{2} \left( b + \frac{w_L - w_H}{(t-1)\theta} \right) (w_H - \eta t^2 \theta^2) \qquad (5) $$
Note that the profit function for the manufacturer is a function of three variables, and after solving the first-order equations, possible optimal solutions are obtained as follows:

$$
\begin{align*}
\{w_{1L}, w_{1H}, t_1\} &\equiv \left\{ \frac{\theta(b+\theta)}{2}, \frac{b^2 + \eta\theta^2 - b(\eta\theta + \sqrt{b^2 - 4b\eta\theta + 4\eta\theta^2})}{2\eta}, \frac{(b - \sqrt{b^2 - 4b\eta\theta + 4\eta\theta^2})}{2\eta\theta} \right\} \\
\{w_{2L}, w_{2H}, t_2\} &\equiv \left\{ \frac{\theta(b+\theta)}{2}, \frac{b^2 - b\eta\theta + \eta\theta^2 + b\sqrt{b^2 - 4b\eta\theta + 4\eta\theta^2}}{2\eta}, \frac{b + \sqrt{b^2 - 4b\eta\theta + 4\eta\theta^2}}{2\eta\theta} \right\} \\
\{w_{3L}, w_{3H}, t_3\} &\equiv \left\{ \frac{\theta(b+\theta)}{2}, \frac{2b^2 + 7b\eta\theta - \eta\theta^2(3 - 8\eta) - 2(b - \eta\theta)\sqrt{X}}{18\eta}, \frac{b + 4\eta\theta - \sqrt{X}}{6\eta\theta} \right\} \\
\{w_{4L}, w_{4H}, t_4\} &\equiv \left\{ \frac{\theta(b+\theta)}{2}, \frac{2b^2 + 7b\eta\theta - \eta\theta^2(3 - 8\eta) + 2(b + \eta\theta)\sqrt{X}}{18\eta}, \frac{b + 4\eta\theta + \sqrt{X}}{6\eta\theta} \right\}
\end{align*}
$$

where, $X = b^2 - 4b\eta\theta - 4\eta(3-4\eta)\theta^2$. One can verify that the first two solutions are not valid because, in this scenario, the selling quantity of high-quality products becomes zero. To verify concavity, we compute the Hessian matrix, and the value of principle minors for the fourth solution is valid because $|H_1|_{\{w_{4L}, w_{4H}, t_4\}} = \frac{-t}{(t-1)\theta} < 0$; $|H_2|_{\{w_{4L}, w_{4H}, t_4\}} = \frac{1}{(t-1)\theta^2} > 0$; and $|H_3|_{\{w_{4L}, w_{4H}, t_4\}} = \frac{2\eta^2\theta Y}{(b-2\eta\theta+\sqrt{X})^4}$, where $Y = 25b^4 - 80b^3\eta\theta - 6b^2\eta(24 - (35-39\eta)\theta^2 + 8\eta^2\theta(9(3+\eta)\theta) + (9-\eta)(15+8\eta))\theta^3 + 88b\eta^3\theta^3(25b^3 + 30\theta b^2)\eta + 8(\eta^2 - \theta(3-2\eta)\theta^2)\sqrt{X}$. Therefore, the fourth solution is valid if $Y < 0$. In summarizing, we obtain the above optimal decision □.

Note that the derivations of the optimal solutions for the other three scenarios are similar; therefore, we omitted the proof. We present the optimal decision for the rest of the scenario in Table 2.

<table><caption>**Table 2.** Optimal decision in Scenario OL, OH, and AH</caption><thead><tr><th>Scenario</th><th>Scenario OL</th><th>Scenario OH</th><th>Scenario AH</th></tr></thead><tbody><tr><td>w<sub>L</sub><sup>j</sup></td><td>θ(b+θ)/2</td><td>—</td><td>—</td></tr><tr><td>w<sub>H</sub><sup>j</sup></td><td>—</td><td>(b+4ηθ+√X)(7b+4ηθ+√X)/72η</td><td>2b<sup>2</sup>/9η</td></tr><tr><td>t<sup>j</sup></td><td>—</td><td>b+4ηθ+√X/6ηθ</td><td>b/3ηθ</td></tr><tr><td>P<sub>L</sub><sup>j</sup></td><td>θ(3b+θ)/4</td><td>—</td><td>—</td></tr><tr><td>p<sub>H</sub><sup>j</sup></td><td>—</td><td>(b+4ηθ+√X)(19b+4ηθ+√X)/144η</td><td>5b<sup>2</sup>/18η</td></tr><tr><td>Q<sub>L</sub><sup>j</sup></td><td>b-θ/4</td><td>—</td><td>—</td></tr><tr><td>Q<sub>L</sub><sup>j</sup></td><td>—</td><td>5b-4ηθ-√X/24</td><td>b/6</td></tr><tr><td>π<sub>r</sub><sup>j</sup></td><td>(b-θ)<sup>2</sup>θ/16</td><td>(4ηθ-5b+√X)<sup>2</sup>(b+4ηθ+√X)/3456η</td><td>b<sup>3</sup>/104η</td></tr><tr><td>π<sub>m</sub><sup>j</sup></td><td>(b-θ)<sup>2</sup>θ/8</td><td>(4ηθ-5b+√X)<sup>2</sup>(b+4ηθ+√X)/1728η</td><td>b<sup>3</sup>/54η</td></tr></tbody></table>

Product introduction or deletion is a critical strategic activity of any manufacturer to be alive in the market. Some of the essential factors that affect the success of introduction or deletion decisions include product complexity, aspiring regulatory bodies, shrinking product release cycles, market potential,
competitor's aggressiveness, complex value chain, etc. [7,8]. The product's value
advantage, price, technological sophistication, and innovativeness are common
product characteristics that affect the strategic decision. However, analytical
research addressing the particular effect of earlier product deletion is relatively
sparse. While researchers put lots of emphasis on the issues of how upgraded
product conceptualization and development processes affect success, less atten-
tion has been given to exploring the strategic effect of a deletion. Note that
we proposed a simple solution, and the decision in the first and second periods
are independent. Now the graphical representation for the profit functions, sales
volume, and quality are presented in Fig. 1.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Maximum (a) manufacturer's profit, (b) retailer's profit, (c) new product quantity sold, (d) product quality in Scenario BHL (red); OL(green); OH (black); and OHA (yellow): $\theta = 0.3$, $b \in (0, 1)$, $\eta \in (0.5, 1.2)$ (Color figure online)</figcaption></figure>

It is clear from Fig. 1 that the manufacturer can offer only low-quality prod-
ucts if *b* is low or high-quality products with adjusted quality. If *b* is low, con-
sumers are not willing to pay a higher price; therefore, offering a high-quality
product or both products might not be profitable, and the results also reflect
that fact. Note that the white region represents the area where the manufacturer
cannot offer both products or only a new product. In that case, the manufacturer
can offer only low-quality products or products with modified quality. Overall,
the key insights obtained from the study reflect that the manufacturer can sell
a maximum amount of products with adjusted quality, which might also lead to
higher profits instead of continuing both products. As noted in the introduction,
some manufacturers offer region- or country-specific products by adjusting qual-
ity specifications due to low-purchase power, less quality sensitivity, etc. Our
results also support the practice.

4 Conclusion and Future Research Direction

In a constantly changing business world, technologies change, customer needs
change, competitors change, and environmental regulations change. If the prod-
uct specification remains the same, the manufacturer will cease to exist with
all these changes. In this regard, researchers mostly focus on exploring factors
that influence the success of new product introduction; however, product deletion strategy is not explored comprehensively. Therefore, we formulate a stylized two-period supply chain model to study the impact of product deletion strategy.

Only a handful of articles have related product deletion decisions to supply chain management. We proposed a simple model in this study and found that the manufacturer's product decision can benefit the supply chain members and sometimes help them expand the sales volume. Although product deletion is a difficult strategic choice that is affected by multiple factors, therefore, in the future, we will develop a more robust model in a stochastic environment. Moreover, the model outcome needs to be verified with the actual date and feedback to evaluate the feasibility, reliability, and accuracy.

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