Optimal product family design with platform modularity and component sharing under uncertain environment

2.1 Mass customization

Since the late 1980s, the analysis and implementation of mass customization (MC) systems have received extensive attention from researchers and practitioners. Mass customization (MC) refers to a significant production paradigm, which focuses on providing a high volume of various customized products and services in a large market with near-mass production efficiency [29]. This paradigm attempts to combine the benefits of craft production in the pre-industrial economy and the mass production in the industrial economy. It has become a necessary condition for many companies to surviving in an increasingly diversified and highly competitive markets [33]. Revolutionary progress in technologies such as information technology, flexible manufacturing systems, and rapid prototyping makes it possible to transform the production paradigm from mass production to mass customization [31].

2.2 Platform modularity

Under MC, platform modular design has been a cost-effective approach to achieve the goal of highly differentiated products with standardized modules and a common platform [20]. Early research on product platform modularity regards product platforms as common modules or design variables and aimed to identify a standard set of components or variables to form a platform for developing a product family. Researchers have found that platform modularity has several advantages. For example, [23] investigates serial production systems with uncertain product quality, and finds integrated modularity has economic advantage on supply and production planning decisions. Furthermore, [20] integrates the sales, product modular design, and production to minimize the costs and completion time. They propose a modular configuration evaluation for companies to select optimal decisions with different costs and time.

Some researchers have expanded the concept of production cost to reflect product preference or quality, for each module behaves differently in terms of its preference. The connection between production costs and module preference generates several preference-cost functions, where the module preference depends essentially on the production costs [13, 32]. For example, [32] considers manufacturers’ joint decisions about platform-based development and product performance. They characterize the preference-cost function by a linear model, which is also adopted in our paper. Following their research, [13] considers a joint technology and integration of manufacturing under conditions of mass customization. They use the same linear preference-cost function and find that the subsystem-based royalty method is the optimal business model when the degree of consumers’ diversity is low. [30] adopts Kanos model to classify customer requirements qualitatively and proposes a multi-objective model to solve the product configuration. Besides the above extensions of platform modularity, many researchers also take into account other considerations, such as commonality and product performance [10], commonality and modularity in product platform development [14], and the tradeoff between customer utility and mass production [13].

Among these considerations, product performance is assumed not to be affected by the platform strategy. However, abuse of platform strategy will bring a series of problems. When the platform is introduced, the performance of the platform components is the same for all products, which leads to over-designing low-end products or inadequate design of high-end products. The above-mentioned product platform research ignores the product performance, assuming the more components in the platform, the better platform configuration. In this work, optimal product preference and profits are both considered to improve both efficiency and effectiveness of a product platform.

2.3 Component sharing

The literature relating to component sharing is broad and diverse. It has been studied from a multi-project management perspective [17], from the perspective of software or component reuse [2] and the supply chain management [7]. There are many different criteria that manufacturers have employed to determine whether they should use the component sharing strategy. For example, [24] considers component sharing in a vehicle manufacturer’s vertical production process with consumers’ valuation on luxury, volume, and economy brands. Their results show that a volume brand shares components with an economy brand outperform a luxury brand with a volume brand. [1] investigates components’ sets with sensitive information. With the proposed framework, the optimal component configuration can facilitate sharing and protect identified sensitive information. However, little research has focused on the impact of component sharing on product preference. Our research combines the platform modularity with component sharing and intends to provide guidance to the manufacturers when maximizing profit when taking product preference into consideration.

2.4 Uncertainty theory

The uncertain programming was founded by Liu [3], and it provides a way to solve mathematical programming involving uncertain variables. For example, Liu [4] has already used uncertain programming in the study of the machine scheduling problem with uncertain processing times, the vehicle routing problem with uncertain travel times, the project scheduling problem with uncertain duration times, etc. As extensions of uncertain programming theory, Liu and Chen [6] proposed the uncertain multi-objective programming and uncertain goal programming. Following that, uncertain programming has led to fruitful results in both theory and practice.

This paper contributes to the above literature in the following ways. First, prior research on the theoretical model of product configuration considers the platform an effective way to reduce costs but neglects it hurts the consumer’s utility. Second, we contribute to the platform literature by considering the significant role uncertainties play in a product strategy choice. Our analysis provides guidelines for the manufacturer who plans to design and develop new products.

4.1 Platform-based approach (A1)

From the above analysis, we only consider the case of P < 2A, and the platform delivers performance at q _p  = q _l to get the optimal solution.

Proposition 1. The optimal performance levels ( q h , 1 * , q l , 1 * ) are q h , 1 * = [ ( Q h - E [ ( Q h - D h ) + ] ) v h γ ( A + ( Q h E [ g ] - E [ g ( Q h - D h ) + ] ) Cu 1 ) ] 1 γ - 1 ,(13) q l , 1 * = [ ( Q l - E [ ( Q l - D l ) + ] ) v l - ( Q h - E [ ( Q h - D h ) + ] ) ( v h - v l ) γ ( P - A + ( Q l E [ g ] - E [ g ( Q l - D l ) + ] ) C ) ] 1 γ - 1 .(14)

Proof 1. Referring to the total expected profit E [Π] ₁, the first order partial derivatives with respect to q _h , q _l can be obtained as follows, respectively, ∂ E [ Π ] ∂ q h = ( Q h - E [ ( Q h - D h ) + ] ) v h - Q h CE [ g ] γ u 1 q h γ - 1 + CE [ g ( Q h - D h ) + ] γ u 1 q h γ - 1 ) - A γ q h γ - 1 , ∂ E [ Π ] ∂ q l = ( E [ ( Q h - D h ) + ] - Q h ) ( v h - v l ) + ( Q l - E [ ( Q l - D l ) + ] ) v l - Q l CE [ g ] γ q l γ - 1 + CE [ g ( Q l - D l ) + ] γ q l γ - 1 + γ ( A - P ) q l γ - 1 .

The corresponding second order derivatives of the equivalent objective function E [Π] ₁ can be found D = ∂ 2 E [ Π ] ∂ q h 2 = ( ( E [ g ( Q h - D h ) + ] - Q h E [ g ] ) u 1 C - A ) γ ( γ - 1 ) q h γ - 2 < 0 , E = ∂ 2 E [ Π ] ∂ q l 2 = ( ( E [ g ( Q l - D l ) + ] - Q l E [ g ] ) C + A - P ) γ ( γ - 1 ) q l γ - 2 < 0 , F = ∂ 2 E [ Π ] ∂ q h q l = 0 . It is obvious that DF - E² > 0 and D < 0, so the E [ Π ] 1 * is the maximum value.

Lemma 1. Under the platform strategy, the optimal performance of high-end products q h , 1 * decreases with the under design coefficient u₁ while the optimal performance of low-end products q l , 1 * does not depend on u₁.

As stated in Lemma 1, when consumers’ utility is concerned, the performance level of high-end products decreases with the under design coefficient. In addition, as the unit costs of low-end products do not depend on the under design coefficient, the optimal performance of low-end products is not affected by it.

Lemma 2. Under the platform strategy, the optimal performance of high-end products q h , 1 * decreases with the product fixed cost parameter A while the optimal performance of low-end products q l , 1 * increases with A (A < P < 2A).

As to the fixed cost parameter, the optimal performance of low-end products increases with the product fixed cost parameter. When P < 2A, the platform delivers a performance at q _p  = q _l . When the upper and lower bounds of the platform increases (i.e., A increases), on the one hand, the fixed cost of the platform becomes larger; on the other hand, the performance of the platform may increase, which benefits the low-end products produced on it. In addition, as the fixed cost of low-end products Aq l β gets close to the platform fixed cost Pq p β , the manufacturer is more motivated to make products on the platform to gain more economies of scale, which may increase the manufacturer’s profit. However, for the high-end products, manufacturers want more components of high-end products to be produced on the platform to spread over the increased platform fixed costs. Therefore, the optimal performance of high-end products decreases with the product fixed cost parameter.

Next, we derive the optimal profit of the manufacturer.

Proposition 2 The optimal profit E [ Π ] 1 * . is E [ Π ] 1 * = γ − 1 γ × { [ ( ( Q h − E [ ( Q h − D h ) + ] ) v h ) γ γ ( A + ( Q h E [ g ] − E [ g ( Q h − D h ) + ] ) C u 1 ) ] 1 γ − 1 + [ ( ( Q l − E [ ( Q l − D l ) + ] ) v l − ( Q h − E [ ( Q h − D h ) + ] ) ) γ γ ( P − A + ( Q l E [ g ] − E [ g ( Q l − D l ) + ] ) C ) ] 1 γ − 1 ( v h − v l ) γ γ − 1 } + E [ ( Q h − D h ) + ] s h + E [ ( Q l − D l ) + ] s l .(15)

Proof 2. The expected profit E [Π] ₁ of the platform-based approach can be expressed as E [ Π ] 1 * = ( Q h - E [ ( Q h - D h ) + ] ) v h q h - ( A + ( Q h E [ g ] - E [ g ( Q h - D h ) + ] ) u 1 C ) q h γ + [ ( Q l - E [ ( Q l - D l ) + ] ) v l - ( Q h - E [ ( Q h - D h ) + ] ) ( v h - v l ) ] q l - ( P - A + ( Q l E [ g ] - E [ g ( Q l - D l ) + ] C ) ) q l γ + E [ ( Q h - D h ) + ] s h + E [ ( Q l - D l ) + ] s l .

Let a 1 = ( Q h - E [ ( Q h - D h ) + ] ) v h ,(16) b 1 = ( A + ( Q h E [ g ] - E [ g ( Q h - D h ) + ] ) u 1 C ) ,(17) a 2 = ( Q l - E [ ( Q l - D l ) + ] ) v l - ( Q h - E [ ( Q h - D h ) + ] ) ( v h - v l ) ,(18) b 2 = ( P - A + ( Q l E [ g ] - E [ g ( Q l - D l ) + ] C ) ) ,(19) then q h , 1 * = [ a 1 γ b 1 ] 1 γ - 1 , q l , 1 * = [ a 2 γ b 2 ] 1 γ - 1 . Thus E [ Π ] 1 * can be simplified as follows, E [ Π ] 1 * = a 1 q h - b 1 q h γ + a 2 q l - b 2 q l γ + E [ ( Q h - D h ) + ] s h + E [ ( Q l - D l ) + ] s l = a 1 [ a 1 γ b 1 ] 1 γ - 1 - b 1 [ a 1 γ b 1 ] γ γ - 1 + a 2 [ a 2 γ b 2 ] 1 γ - 1 - b 2 [ a 2 γ b 2 ] γ γ - 1 + E [ ( Q h - D h ) + ] s h + E [ ( Q l - D l ) + ] s l = a 1 γ γ - 1 [ γ b 1 ] 1 γ - 1 - 1 γ a 1 γ γ - 1 [ γ b 1 ] 1 γ - 1 + a 2 γ γ - 1 [ γ b 2 ] 1 γ - 1 - 1 γ a 2 γ γ - 1 [ γ b 2 ] 1 γ - 1 + E [ ( Q h - D h ) + ] s h + E [ ( Q l - D l ) + ] s l = γ - 1 γ × { [ a 1 γ γ b 1 ] 1 γ - 1 + [ a 2 γ γ b 2 ] 1 γ - 1 } + E [ ( Q h - D h ) + ] s h + E [ ( Q l - D l ) + ] s l .

As shown in Proposition 4.1 and 4.1, the optimal performance levels ( q h , 1 * , q l , 1 * ) depend on customers’ demands and valuation, which supplements the classical assumption wherein the product performance is solely driven by the consumers valuation. This is because increasing customer demands would increase manufacturers’ profit margin and induce them to increase product performance willingly. As to the production quantity, on the one hand, a higher retail price (due to a higher optimal performance level) gives the manufacturer incentives to produce more; on the other hand, a higher optimal performance level leads to a higher variable cost, which in turn, results in a bigger loss for the unsold products. The trade-off between these two driving forces leads to a balance for the optimal performance levels of high-end and low-end products.

In the component sharing approach, the fixed cost is Bq cs β with the q _cs  (q _cs  ≤ q _l ) unit performance of component sharing where A < B < P. The under design level coefficient is u₂ (0 < u₂ ≤ 1), and the economies of scale uncertain coefficient is k. Assume k has inverse uncertain distribution Φ k - 1 ( α ) . Let E [ k ] = ∫ 0 1 Φ k - 1 ( 1 - α ) d α .

The objective function is given by: E [ Π ] 2 = ( Q h - E [ ( Q h - D h ) + ] ) v h q h - ( A + ( Q h E [ k ] - E [ k ( Q h - D h ) + ] ) u 2 C ) q h γ + [ ( Q l - E [ ( Q l - D l ) + ] ) v l - ( Q h - E [ ( Q h - D h ) + ] ) ( v h - v l ) ] q l - ( B - A + ( Q l E [ k ] - E [ k ( Q l - D l ) + ] C ) ) q l γ + E [ ( Q h - D h ) + ] s h + E [ ( Q l - D l ) + ] s l .(20)

Similar to the analysis of the platform-based approach, when q _cs  = q _l we get the optimal solution.

Proposition 3. The optimal performance levels ( q h , 2 * , q l , 2 * ) and profit E [ Π ] 2 * are derived from the first order optimality conditions as q h , 2 * = [ ( Q h - E [ ( Q h - D h ) + ] ) v h γ ( A + ( Q h E [ k ] - E [ k ( Q h - D h ) + ] ) Cu 2 ) ] 1 γ - 1 ,(21) q l , 2 * = [ ( Q l - E [ ( Q l - D l ) + ] ) v l - ( Q h - E [ ( Q h - D h ) + ] ) ( v h - v l ) γ ( B - A + ( Q l E [ k ] - E [ k ( Q l - D l ) + ] ) C ) ] 1 γ - 1 ,(22) E [ Π ] 2 * = γ - 1 γ × { [ ( ( Q h - E [ ( Q h - D h ) + ] ) v h ) γ γ ( A + ( Q h E [ k ] - E [ k ( Q h - D h ) + ] ) Cu 2 ) ] 1 γ - 1 + [ ( ( Q l - E [ ( Q l - D l ) + ] ) v l - ( Q h - E [ ( Q h - D h ) + ] ) ( v h - v l ) ) γ γ ( B - A + ( Q l E [ k ] - E [ k ( Q l - D l ) + ] ) C ) ] 1 γ - 1 } + E [ ( Q h - D h ) + ] s h + E [ ( Q l - D l ) + ] s l .(23)

The optimal results in Proposition 3 are similar to Proposition 1 and 2. The high customers’ demands and valuation push up both the optimal performance levels ( q h , 2 * , q l , 2 * ) and the manufacturers profitability. However, the increase of the unit costs due to higher performance levels of high-end and low-end products may dampen the manufacturer’s profit, which may consequently decrease stocking quantity.

We next derive the impact of the under design coefficient and fixed cost parameters on the optimal performance levels ( q h , 2 * , q l , 2 * ) and the manufacturers profitability.

Lemma 3. Under the component sharing,

(1) the optimal performance of high-end products q h , 2 * decreases with the under design coefficient u₂ while the optimal performance of low-end products q l , 2 * does not depend on u₂.

(2) the optimal performance of high-end products q h , 2 * decreases with the product fixed cost parameter A while the optimal performance of low-end products q l , 2 * increases with A (A < B < P < 2B).

The result in Lemma 3 is similar to that of Lemma 1 and 2. We next derive the optimal decisions when both platform modularity and component sharing are used in THE production process.

If the manufacturer decides to use both platform modularity and component sharing to develop two products, the objective function is: E [ Π ] 3 = s h E [ ( Q h − D h ) + ] + ( Q h − E [ ( Q h − D h ) + ] ) ( ( q h − q l ) v h + q l v l ) − Q h C q h γ ( u 1 E [ g ] + u 2 E [ k ] ) + C q h γ ( u 1 E [ g ( Q h − D h ) + ] + u 2 E [ k ( Q h − D h ) + ] ) + s l E [ g ( Q l − D l ) + ] + ( Q l − E [ g ( Q l − D l ) + ] ) q l v l − Q l C q l γ ( E [ g ] + E [ k ] ) + C q l γ ( E [ g ( Q l − D l ) + ] + E [ k ( Q l − D l ) + ] ) − A ( q h γ + q l γ ) − ( P − 2 A − B ) q p γ − ( B − 2 A − P ) q c s γ .(24)

Lemma 4. For platform q _p , if P > 2A + B, q _p  = 0, there is no platform; otherwise, q _p  = q _l .

For component sharing q _cs , for B < 2A + P then q _cs  = q _l .

Lemma 4 implies that component sharing can always provide a positive economic benefit, while the platform is dependent on its manufacturing and designing costs.

Proposition 4. The optimal performance levels ( q h , 3 * , q l , 3 * ) and profit E [ Π ] 3 * are derived from the first order optimality conditions as

q h , 3 * = [ ( Q h − E [ ( Q h − D h ) + ] ) v h γ ( A + C ( Q h ( u 1 E [ g ] + u 2 E [ k ] ) − u 1 E [ g ( Q h − D h ) + ] − u 2 E [ k ( Q h − D h ) + ] ) ) ] 1 γ − 1 ,(25) q l , 3 * = [ ( Q l − E [ ( Q l − D l ) + ] ) v l − ( Q h − E [ ( Q h − D h ) + ] ) ( v h − v l ) γ ( − 3 A + C ( Q l ( E [ g ] + E [ k ] ) − E [ g ( Q l − D l ) + ] − E [ k ( Q l − D l ) + ] ) ) ] 1 γ − 1 ,(26) E [ Π ] 3 * = γ − 1 γ × { [ ( ( Q h − E [ ( Q h − D h ) + ] ) v h ) γ γ ( A + C ( Q h ( u 1 E [ g ] + u 2 E [ k ] ) − u 1 E [ g ( Q h − D h ) + ] − u 2 E [ k ( Q h − D h ) + ] ) ) ] 1 γ − 1 + [ ( ( Q l − E [ ( Q l − D l ) + ] ) v l − ( Q h − E [ ( Q h − D h ) + ] ) ( v h − v l ) ) γ γ ( − 3 A + C ( Q l ( E [ g ] + E [ k ] ) − E [ g ( Q l − D l ) + ] − E [ k ( Q l − D l ) + ] ) ) ] 1 γ − 1 } + E [ ( Q h − D h ) + ] s h + E [ ( Q l − D l ) + ] s l .(27)

When both platform modularity and component sharing are considered in the production process, the optimal performance levels ( q h , 3 * , q l , 3 * ) are dependent on economies of scale and the under design coefficients in both platform and component sharing.

Lemma 5. Under platform and component sharing, the optimal performance of high-end products q h , 3 * decreases with the under design coefficients u₁ and u₂ while the optimal performance of low-end products q l , 2 * does not depend on u₁ and u₂.

Note that the optimal performance of high-end products depends on both under design coefficients in platform and component sharing, which means that it is more sensitive to the under design.

When the manufacturer decides to develop two products individually, there is no platform or component sharing, so q _p  = 0, q _cs  = 0. Therefore, the economies of scale E [g] =1, E [k] =1, and the under design coefficient u₁ = 1, u₂ = 1.

Proposition 5. The optimal performance levels ( q h , 4 * , q l , 4 * ) and profit E [ Π ] 4 * are derived from the first order optimality conditions as

q h , 4 * = [ ( Q h − E [ ( Q h − D h ) + ] ) v h γ ( A + ( Q h − E [ ( Q h − D h ) + ] ) C ) ] 1 γ − 1 ,(28) q l , 4 * = [ ( Q l − E [ ( Q l − D l ) + ] ) v l − ( Q h − E [ ( Q h − D h ) + ] ) ( v h − v l ) γ ( A + ( Q l − E [ ( Q l − D l ) + ] ) C ) ] 1 γ − 1 ,(29) E [ Π ] 4 * = γ − 1 γ × { [ ( ( Q h − E [ ( Q h − D h ) + ] ) v h ) γ γ ( A + ( Q h − E [ ( Q h − D h ) + ] ) C ) ] 1 γ − 1 + [ ( ( Q l − E [ ( Q l − D l ) + ] ) v l − ( Q h − E [ ( Q h − D h ) + ] ) ( v h − v l ) ) γ γ ( A + ( Q l − E [ ( Q l − D l ) + ] ) C ) ] 1 γ − 1 } + E [ ( Q h − D h ) + ] s h + E [ ( Q l − D l ) + ] s l .(30)

When the manufacturer develops two products individually, the optimal performance levels are solely driven by the consumers valuation and demands, where the customer utility is the maximum without any compromise.

In this section, we compare the performance of four approaches analyzed in sections 4.1-4.4. First, we present the following comparison.

Lemma 6. When economies of scale of the platform are greater than or equal to that under component sharing (k ≤ g), the optimal performance level of low-end products under platform strategy is lower than that under component sharing strategy, that is, q l , 1 * < q l , 2 * .

Naturally, because the platform fixed cost is higher than that of component sharing, that is, B < P, when economies of scale of the platform are greater than or equal to that under component sharing, the optimal performance level of low-end products under the platform strategy is lower than that under component sharing strategy. This directly affects average consumers utility in terms of low-end product performance.

Lemma 7. The optimal performance level of high-end products in A3 is lower than that under A1 and A2, that is, q h , 3 * < q h , 1 * , q h , 3 * < q h , 2 * .

Lemma 7 shows that the approach combining the platform and component sharing dampens the utility of the high-spending consumer group, which may consequently decrease manufacturers’ profitability. Hence, unless economies of scale are large significantly in A3, the manufacturer is reluctant to adopt A3, which may hurt the surplus of the high-spending consumer group.