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Manufacturing & Service Operations Management

Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org Dual Coproduct Technologies: Implications for Process Development and Adoption

Ying-Ju Chen, Brian Tomlin, Yimin Wang

To cite this article: Ying-Ju Chen, Brian Tomlin, Yimin Wang (2017) Dual Coproduct Technologies: Implications for Process Development and Adoption. Manufacturing & Service Operations Management 19(4):692-712. https://doi.org/10.1287/msom.2017.0637

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Dual Coproduct Technologies: Implications for Process Development and Adoption

Ying-Ju Chen,a Brian Tomlin,b Yimin Wangc a Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; b Tuck School of Business, Dartmouth College, Hanover, New Hampshire 03755; c W. P. Carey School of Business, Arizona State University, Tempe, Arizona 85287 Contact: [email protected], http://orcid.org/0000-0002-4851-0347 (Y-JC); [email protected], http://orcid.org/0000-0002-2533-1039 (BT); [email protected], http://orcid.org/0000-0002-4851-0347 (YW)

Received: December 16, 2015 Abstract. Many industries operate technologies in which multiple outputs (coproducts) Revised: September 30, 2016; December 22, are jointly produced. In some settings (“vertical”) the coproducts differ along a per- 2016; March 3, 2017 formance dimension and are substitutable. In other settings (“horizontal”) the coprod- Accepted: March 8, 2017 ucts differ in their applications and are not substitutable. In both cases, three important Published Online in Articles in Advance: attributes of a coproduct technology are its processing cost, overall yield, and coproduct September 25, 2017 split, i.e., the proportion of each output produced. For both vertical and horizontal settings https://doi.org/10.1287/msom.2017.0637 with deterministic market sizes, we characterize the optimal pricing and production deci- sions of a monopoly firm with two technologies. We establish the necessary and sufficient Copyright: © 2017 INFORMS conditions for dual activation (i.e., using both technologies) to be optimal. Dual activation is driven by differences in marginal costs across the two technologies. There is an addi- tional motive for dual activation in the horizontal setting: the desire to generate a product mix that better resembles the market mix. Building on the optimal production analysis, we characterize the optimal adoption-and-usage strategy of a firm with one incumbent technology considering a new technology. We establish the conditions for which a new technology will displace the incumbent or be used with the incumbent, and highlight some important adoption and usage differences between vertical and horizontal settings. Results are extended to the setting with uncertain market size(s).

Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2017.0637.

Keywords: operations strategy • technology management • process design

1. Introduction material that can be separated into constituent parts. Multiple devices are built on a single wafer during For example, distillation separates crude oil into var- semiconductor fabrication. The devices are then tested, ious hydrocarbon coproducts, and milling separates with the good devices oftentimes being sorted into corn into starch (which can be converted to ethanol) different quality grades (“bins”) based on their eval- and other coproducts. uated performance on some key metric, e.g., speed The production economics of a coproduct technol- for microprocessors or luminescence for light emitting ogy are driven by its cost per base unit processed diodes (LEDs). Industrial diamonds are produced by (e.g., cost per wafer start in semiconductors), its over- applying pressure and temperature to a capsule con- all yield per base unit (e.g., number of good devices taining graphite, some of which is converted to dia- generated per wafer start), and its coproduct split, that mond. The diamonds are tested and the good ones are is, the percentage of the overall yield accounted for sorted into different quality (strength) grades. These by each coproduct (e.g., the percentage of good de- are both examples of coproduct technologies whereby vices testing out to each grade; this is sometimes re- production jointly produces multiple outputs. In semi- ferred to as the “binsplit” in the semiconductor indus- conductors and diamonds, coproducts arise because try). These drivers—cost, yield, and split—are often production generates a spectrum of outputs that differ closely guarded secrets, especially in semiconductors in quality. Coproducts are a common phenomenon in and industrial diamonds, but data for the yield and a range of other industries. Chemical reactions often split have been reported for some technologies. For produce coproducts. For example, the chloralkali pro- example, the overall yield of chlorine and sodium cess eletrochemically separates brine (a salt and water hydroxide in the chloralkali process is 1.22 tonnes for mixture) into chlorine and caustic soda. The cumene every tonne of salt processed (recall the salt is mixed process oxidizes cumene to produce cumene hydroper- with water) and the split is 47% chlorine and 53% oxide, which is then decomposed into phenol and ace- sodium hydroxide (IPPC 2001). The cumene process’s tone. In other settings, nature creates a commingled overall yield of acetone and phenol is 1.20 tonnes per

692 Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 693 tonne of cumene processed (remember cumene is com- if the technologies differ in their splits. For dual activa- bined with oxygen) and the split is typically 38% ace- tion to be optimal, one technology must have a lower tone and 62% phenol (McKetta 1990). marginal cost for total product, e.g., cost per good To improve their production economics, firms devote device (slow or fast) in semiconductors, but a higher significant effort to developing new technologies that marginal cost for the higher quality product, e.g., cost improve the cost, yield, and/or split. For example, Bayer per high speed device. The fact that each technology was a finalist for the 2013 German Industry Innovation offers a different marginal-cost benefit can make dual Prize for a new chloralkali membrane-separation tech- activation attractive, but this alone does not guarantee nology that purportedly reduces the processing cost by dual activation: single activation can be optimal even 15% because of a 30% reduction in energy consump- when neither technology dominates the other from a tion (Katz 2014). CB&I and Versalis are jointly market- marginal cost perspective. ing a new cumene process technology “which improves Building on the optimal production analysis, we overall yield” by 3% but does not change the split (CBI characterize the optimal adoption-and-usage strategy 2014, pp. 1–2). Meanwhile, Shell has developed a new of a firm with an incumbent technology considering a cumene process technology that reportedly has a higher new technology. Among other implications, we estab- cost but a lower acetone split because of the produc- lish that a new technology with an identical split to the tion of an additional coproduct (Nexant 2007). In semi- incumbent will be adopted if and only if (iff) its effec- conductor manufacturing, the technology has period- tive (i.e., accounting for capacity) yield-adjusted cost ically moved to larger wafer sizes, e.g., from 200 mm is lower than the incumbent’s yield-adjusted cost and, to 300 mm wafers, which results in a higher processing if adopted, the new technology displaces the incum- cost per wafer but also a higher overall yield per wafer. bent. If the technologies have the same (effective) yield- To date, the overall yield has increased more than the adjusted costs, then the new technology is adopted cost, resulting in a lower cost per device.1 Applied Mate- iff it has a higher split of the higher-quality prod- rials undertook an LED process development project uct and it always displaces the incumbent if adopted. to increase the split of higher-quality devices (Cooke When the new technology differs in split and (effective) 2010). The Winter process (adopted by GE and DeBeers) yield-adjusted cost, then the new technology displaces

increased the overall yield and the split of higher grade the incumbent at low effective yield-adjusted costs, is diamonds but had a higher cost per capsule processed used with the incumbent at intermediate costs, and (Sung et al. 2006). is not adopted at higher costs. When market size is Upon adopting a new technology, i.e., one that dif- uncertain at the time of technology adoption, the opti- fers in the cost, yield, and/or split from its incumbent mal adoption-and-usage results for the special cases of technology, a firm faces the question of how best to identical split or identical costs remain unchanged if use the two technologies. Should it produce using both no uncertainty is resolved until after production. How- technologies or should it use one technology exclu- ever, if uncertainty is resolved between adoption and sively (presumably the new technology or why else production then, because of the risk of unused new adopt it)? In determining whether or not to adopt a capacity, it may be optimal in these special cases to new technology, the firm must evaluate how the new adopt the new technology and use it with the incum- technology will be used (exclusively or in conjunction bent technology. with the incumbent) and also consider the adoption Turning next to a horizontal market model in which cost; that is, the cost of installing the new production the two coproducts serve different markets, we again technology. These production and adoption questions characterize the firm’s optimal pricing and produc- are the central issues explored in this paper. We con- tion decisions and the resulting optimal technology sider “vertical” coproducts that provide the same func- adoption-and-usage strategy. We show that there is an tion but differ in their performance quality (e.g., speed additional motive for dual activation in the horizontal of microprocessors and strength of diamond) and “hor- setting: activating both technologies can help the firm izontal” coproducts that differ in their application and achieve a combined technology split that better resem- are therefore sold in different markets (e.g., chlorine bles the relative market sizes. Among other implica- and sodium hydroxide, acetone and phenol). tions, this gives rise to the result that—different to Focusing first on a vertical market model in which the vertical setting—if the technologies have the same the two coproducts differ along a performance dimen- (effective) yield-adjusted costs, then the new technol- sion, we characterize the optimal pricing and produc- ogy may not displace the incumbent if adopted; it can tion decisions of a monopoly firm with two technolo- be optimal to use both technologies together. gies facing a deterministic market size. We prove that The rest of the paper is organized as follows. The dual activation in which the firm produces a positive most relevant literature is discussed in Section2. quantity on both technologies can be optimal, but only The dual-technology production and pricing model is Chen, Tomlin, and Wang: Dual Coproduct Technologies 694 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS described in Section3. The vertical market setting is characterizes a two-technology, deterministic, coprod- examined in Section4 and the horizontal setting in Sec- uct production-and-pricing problem in both horizontal tion5. The model and analysis are extended to random and vertical settings and with technologies that can dif- market sizes in Section6. Conclusions are presented in fer arbitrarily in cost, yield, and split. This allows us to Section7. Proofs are contained in the appendix. Cer- study process development and adoption for coprod- tain additional materials can be found in the online uct technologies. appendix. Process development and adoption has not been a focus of the coproduct literature. Chen et al. (2013) 2. Literature Review provide some limited treatment of process improve- For the most part, the coproduct literature has focused ment in a single-technology, vertical coproduct setting, on operations decisions and to some extent on mar- and find that the firm’s profit increases as the output keting ones. Motivated by vertical coproduct settings, distribution becomes first-order stochastically larger or much of the early coproduct literature, e.g., Bitran more variable in a mean-preserving spread sense. As and Gilbert (1994), Nahmias and Moinzadeh (1997), part of their extensive exploration of downward sub- Gerchak et al. (1996), focused on the production and stitution and product withholding, i.e., offering only downward substitution decisions under split uncer- the high-quality product and not the low one, Bansal tainty.2 Other vertical coproduct papers have examined and Transchel (2014) numerically explore the impact pricing (Min and Oren 1996, Tomlin and Wang 2008, of the high-quality split on these decisions, finding Bansal and Transchel 2014) and product-line design (among other observations) that downward substitu- (Min and Oren 1996, Chen et al. 2013, Transchel et al. tion is of limited benefit when the split is low. Noting 2016). A number of papers examine procurement and that the split can improve over time in the semicon- production decisions for horizontal coproducts in the ductor industry, they relate this finding to the firm’s presence of input and/or output spot markets and operations strategy and discuss the implication that fixed coproduct splits (Boyabatlı et al. 2011, 2017; Dong downward substitution should increase over the tech- et al. 2014; Boyabatlı 2015). Although the end prod- nology’s lifecycle. Process development and adoption ucts are horizontal in Boyabatlı et al. (2011) and Dong is not the focus in either of the above papers and, there- et al. (2014), both papers allow for an intermediate pro- fore, both assume the firm has only one technology. In cessing step in which one coproduct can be converted contrast, we focus on process development and adop- into the other. Different to the above papers, Sunar tion (in both vertical and horizontal settings) by con- and Plambeck (2016) focus on how process emissions sidering a firm with two coproduct technologies that should be allocated among coproducts when emissions can differ in cost, overall yield, and split. are taxed for environmental reasons. Process development and technology adoption have Our two-technology model adds to the coproduct been studied in non coproduct settings. Viewing pro- literature, which, with some exceptions, assumes a sin- cess development as a costly action that improves a gle technology. In an extension to their single-tech- particular process attribute, e.g, cost, quality, capacity, nology production and downward substitution model, or yield, a stream of papers in operations has explored Gerchak et al. (1996) numerically examine a setting in how a firm operating a single technology (producing a which the firm has a second technology that only pro- single product) should invest in process improvement duces the low-quality product. Ng et al. (2012) develop a robust-optimization algorithm for the production efforts over time (Fine and Porteus 1989, Marcellus and downward substitution problem in the presence and Dada 1991, Li and Rajagopalan 1998, Carrillo and of multiple inputs that differ in their coproduct splits, Gaimon 2000, Terwiesch and Bohn 2001). Our paper and then test the algorithm performance in a number explores a fundamentally different question. Instead of of computational studies. Boyabatlı et al. (2011) con- exploring the improvement investment path, we con- sider spot and contract purchases of beef and assume sider a firm with two technologies that can differ in that the premium-product split is higher for contract multiple attributes so as to develop insights into tech- purchases. In their model, the firm can process both nology adoption and usage. Technology adoption and sources, but it will do so only after all its contract usage relates to technology choice, a topic that has been supply has been consumed. In essence, the motive for explored in various settings; for example, whether to processing both inputs lies entirely in the potentially invest in flexible and/or dedicated technologies, e.g., limited quantity of the more desirable input. Extending Van Mieghem (1998), and whether to invest in environ- their single-input, oil-refining model to allow the refin- mentally friendly and/or standard technologies, e.g., ery to process inputs that differ in their splits, Dong Wang et al. (2013), Drake et al. (2015). et al. (2014) establish some properties of the multiple- In closing, we note that a number of sustainability ini- input problem that allows them to efficiently search tiatives lead to coproduct type problems. For example, for the optimal input portfolio. Our paper completely recycling can create streams of commingled materials Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 695 requiring separation. The returns used in remanufac- does not purchase a product. The firm does not observe turing can vary in quality and therefore require grad- a customer’s type. Customer types are uniformly dis- ing into different categories (Ferguson et al. 2006). tributed between 0 and 1, that is, the proportion of the By-product synergy, in which waste from production of customer population with a quality valuation less than a primary product can be converted into a valuable by- or equal to θ is given by θ. Similar to Chen et al. (2013), product, (Lee 2012, Zhu et al. 2014, Lee and Tongarlak we assume that customers make their first-choice pur- 2017) creates a coproduct technology. chase decisions simultaneously, and, in the event that demand exceeds supply of a product, the firm may 3. The Production and Pricing Model (i) fill a customer’s product demand with a higher- We now present the underlying model used in the quality product at the lower-quality product’s price or paper, that being a two-product firm with two coprod- (ii) let a customer spill down to its next-choice lower- uct technologies deciding on technology production quality product. All results in the paper apply even quantities and coproduct prices. As well as being if downward fulfillment and/or spill down are not an important and relevant problem in its own right allowed. In this vertical model, the revenue function (because some firms operate a portfolio of coprod- can be obtained by tailoring Proposition 1 and Equa- uct technologies3), this production and pricing model tion (6) in Chen et al. (2013) to the case of two products, serves as our foundation for exploring technology uniformly distributed customer types, and a popula- adoption. tion of size S. We then have  + The firm produces and sells two coproducts n 1, 2   p    + 1 in a single time period. It has two technologies, labeled R Q, p p1 min Q1 Q2, S 1 ( ) − x1 A and B, each of which produces the coproducts. The +   p p   firm determines the selling price pn 0 for each prod- + 2 1 ≥ p2 p1 min Q2, S 1 − , (1) uct n and the production quantity q 0 for each tech- ( − ) − x x t ≥ 2 − 1 nology t A, B , where qt denotes the number of base ∈ { } where y +  max y, 0 . We note that the product-1 units processed, e.g., wafer starts for semiconductors or ( ) { } tonnes of salt for the chloralkali process.4 Technology t quantity Q1 influences revenue only through the total + has a marginal (per base unit) production cost of c > 0, product quantity Q1 Q2 because the products are

t vertically differentiated and product 2 is the higher- an overall yield per base unit of yt > 0, and a product-2 quality product. split of 0 αt2 1, which implies a product-1 split of  ≤ ≤  In the horizontal-market model, the firm’s products αt1 1 αt2. Therefore, a production pair q qA , qB −  P ( ) serve different applications. For example, chlorine and creates a quantity Qn q t A, B αtn yt qt of product ( ) P ∈{ } sodium hydroxide (coproducts of the chloralkali pro- n at a total cost of C q t A, B ct qt . Without loss of generality (w.l.o.g.) we( ) label the∈{ technologies} such that cess) have different purposes. Horizontal coproducts are not substitutable from a customer perspective and, αA2 αB2. Let≥ R Q, p denote the revenue that the firm gener- therefore, their demands have no cross-price elastic- ( )  ities. We adopt a linear demand function for each ates for any given product-quantity pair Q Q1, Q2  ( ) coproduct n  1, 2, parameterized as D p  S and price pair p p1, p2 . This revenue function will n n n ( ) S x p , where S > 0 is the maximum demand,( ) i.e.,− depend on the product market in which the firm oper- ( n/ n) n n ates. We assume a monopoly firm and consider two the demand at zero price, and xn > 0 is the choke price, different types of product markets: vertical and hori- i.e., the price at which demand first reaches zero. In zontal. In both cases, the salvage value of unsold prod- this horizontal model, the revenue function is given by uct is set to zero.   +  In the vertical-market model, the two coproducts serve  S1 R Q, p p1 min S1 p1 , Q1 the same basic function but differ along a quality di- ( ) − x1 + mension, e.g., semiconductor speed, for which all cus-     + S2 tomers agree that more is better (not accounting for p2 min S2 p2 , Q2 . (2) − x2 price). Product n has a quality level of xn, and prod- ucts are indexed in order of increasing quality, i.e., To summarize the firm’s pricing and production  x1 < x2. We adopt a standard customer-demand model problem, it chooses coproduct prices p p1, p2 and  ( ) from the vertical-quality, product-line design litera- technology production quantities q qA , qB to max-  ( ) ture. There is a deterministic population, size S, of imize its profit Π q, p R Q q , p cA qA cB qB, ( ) ( ( ) ) − −  infinitesimal customers that vary in their valuation of where the quantity of product-n produced is Qn q P  ( ) quality. Customers are defined by their quality valua- t A, B αtn yt qt for n 1, 2, and the revenue function tion 0 θ 1 such that a customer of type θ derives R ∈{, is} given by (1) for the vertical market and (2) a net utility≤ ≤ of θx p from purchasing a product of for(· the·) horizontal market. We note that a technology is n − n quality xn at price pn and derives a net utility of 0 if it never used (even if it is the only available technology) Chen, Tomlin, and Wang: Dual Coproduct Technologies 696 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS

if c y x + x x α , and so we assume c y < and so the revenue is constant in both product quanti- t / t ≥ 1 ( 2 − 1) t2 t / t x + x x α for all technologies throughout the ties in this region. That being the case, it is never opti- 1 ( 2 − 1) t2 paper. In Section6, we extend the model to allow for mal for the firm to set production quantities such that stochastic market sizes. the resulting product supplies lie within this revenue- saturation region. 4. Vertical Coproducts With the optimal pricing and revenue result in hand,  We first explore and characterize the optimal pric- and recalling that a production pair q qA , qB cre-  P ( ) ing and production strategy for vertical coproducts. ates a quantity Qn q t A, B αtn yt qt of product ( ) ∈{ } Building on that analysis, we then explore the result- n  1, 2, the firm’s pricing and production-quantity ing process development and technology adoption problem can be transformed to the following produc- implications. tion-quantity problem:    4.1. Pricing and Production  X X ΠV∗ max RV∗ 1 αt2 yt qt , αt2 yt qt Recall that the two products are indexed such that q 0 ≥ t A, B ( − ) t A, B ∈{ } ∈{ } product 2 is the higher-quality product, i.e., x2 > x1,  and that the revenue as a function of product prices X ct qt , (6) and product quantities is given by (1). For any non- − t A, B ∈{ } negative product quantity pair Q  Q , Q , we can ( 1 2) adapt the supply constrained, vertically differentiated where RV∗ , is given by (5) and we have used the fact  (· ·) product, optimal pricing result from Chen et al. (2013) that αt1 1 αt2, i.e., product splits sum to 1. − (specifically Theorem 1) for our setting to obtain the It is helpful to first briefly consider the setting in following optimal prices: which the firm has only one technology, and so we sup- press the technology notation t. The marginal cost of  +   1 Q1 Q2 producing good product, i.e., product 1 and product 2 pV∗ x1 max , 1 , (3) 1 2 − S combined, is the “yield-adjusted cost” c y, e.g., the   / 1 Q2 cost per semiconductor device is the processing cost p∗  p∗ + x x V2 V1 2 1 max , 1 , (4) per wafer start divided by the number of good devices

( − ) 2 − S per wafer start. Accounting for its split α2, the marginal and associated revenue cost of producing the high-quality product is c α y . /( 2 ) R∗ Q1,Q2 Proposition 1 (Single Technology). The optimal produc- V ( )  +     Q1 Q2 Q2 tion quantity and profit are  x 1 Q +Q + x x 1 Q ,  1 − S ( 1 2) ( 2 − 1) − S 2  S 1  c y   S   + qV∗ 1 / ;  Q1 Q2 < 2y α − α x x  2 2 2 2 1  (5) ( − ) 2 x  Q  S S   c y    1 + 2 + Π  +  S x2 x1 1 Q2, Q2 < Q1 Q2, V∗ x1 x2 x1 1 /  4 ( − ) − S 2 ≤ 4 ( − ) − α2 x2 x1  ( − )  x2 S  def  S, Q2 . if c y c¯ α  x x 1 α α . Otherwise  4 ≥ 2 / ≤ sat( 2) ( 2 − 1)( − 2) 2 In the region where the total product quantity is less + S x1 x2 x1 α2 c y + q∗  ( − ) − / ; than half of the market size, i.e., Q1 Q2 < S 2, the opti- V + 2 / 2y x1 x2 x1 α mal prices are set to exactly match product demands ( − ) 2 + 2 S x1 x2 x1 α2 c y with supplies, and the revenue is increasing in both Π∗  ( ( − ) − / ) . V + 2 product quantities. In the “volume saturation” region, 4 x1 x2 x1 α2 + ( − ) i.e., Q2 < S 2 Q1 Q2, the optimal prices are set to / ≤ The threshold cost c¯ α determines whether the clear product 2 but not product 1, and the total prod- sat( 2) uct quantity (1 and 2) sold is always S 2. The revenue yield-adjusted cost is low enough to justify producing is constant in the product-1 quantity but/ increasing in enough so that the resulting total product supply lies the product-2 quantity because more product 2 enables in the volume saturation region. For brevity later, we the firm to sell more of the higher-quality product 2 will say a technology operates in the volume-saturation (which commands a higher price) and less of the lower- region if c y c¯ α . / ≤ sat( 2) quality product 1. In the “revenue saturation” region, We now return to the two-technology setting. In

i.e., Q2 S 2, the firm has an ample supply of product 2 what follows, we will say a technology is activated if and it is≥ optimal/ to only sell this higher-quality prod- the firm produces a strictly positive quantity using that uct. It always sells the quantity S 2 at a price of x 2, technology. / 2/ Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 697

Definition 1 (Dominant Technology—Vertical Coprod- single technology activation is therefore optimal in this ucts). Let i A, B denote one technology and j special case. Single activation is not necessarily opti- ∈ { } the other. Technology i dominates technology j iff mal when the technologies differ in their product-2 c y c y (that is, i has a lower marginal cost for splits. Figure1 illustrates the optimal activation strat- i/ i ≤ j/ j producing good product) and c α y c α y egy in general, i.e., Theorem1(ii), as a function of the i/( i2 i) ≤ j/( j2 j) (that is, i has a lower marginal cost for producing the technologies’ yield-adjusted costs when their splits dif- high-quality product). Strict dominance occurs iff at fer. Recall that the technologies are labeled such that least one inequality is strict. αA2 > αB2. There always exists a region of yield-adjusted Intuitively, the firm should not activate a strictly cost pairs for which neither technology dominates; this dominated technology because the other technology is is the region between the two rays emanating from cheaper from both a combined product and a high- the origin. In this nondominated region, Technology A quality product perspective. More formally, any pro- is cheaper from a product-2 perspective but Technol- duction pair with a positive quantity of the dominated ogy B is cheaper from a combined product perspec- technology can be adjusted (by reducing the domi- tive. Observe that there exists a subregion within the nated technology quantity and increasing the dom- nondominated region for which dual activation is opti- inant technology quantity) in a manner that strictly mal. The area of the dual activation region is given by decreases the production cost while at the same time A  0.5x x x α α 2 (proof omitted). All else D 1( 2 − 1)( A2 − B2) weakly increasing both the supply of product 2 and held constant, this area increases as the technology split the supply of overall product (1 and 2); therefore, difference increases. Put differently, the larger the split revenue can be weakly increased and cost strictly difference between the two technologies, the more cost decreased. This is formally proved in Lemma1 in the pairs there are for which dual activation is optimal. appendix. In the weak dominance case, both technolo- The dual-activation region does not encompass the gies are effectively identical and the firm is indifferent entire nondominated region. The abbreviated intuition between them. is as follows. Different to the case when one technol- The following theorem gives the optimal technology ogy dominates, there is no intertechnology production activation strategy in general when the firm has two exchange rate that decreases the overall production

technologies. Recall that (w.l.o.g.) the technologies are cost while weakly increasing the supplies of the com- labelled such that αA2 αB2. bined products and of product 2. In other words, when ≥ Theorem 1 (Two Technologies). (i) If the technologies have neither technology dominates, production exchange  can reduce cost but only by also reducing some prod- identical product-2 splits, i.e., αA2 αB2, then it is optimal to uct supply and thus revenue. This provides a motive activate only Technology A iff cA yA cB yB and to activate only Technology B otherwise. / ≤ / to activate both technologies. However, if the yield- (ii) If the technologies differ in their product-2 splits, i.e., adjusted costs are sufficiently different then there exists

αA2 > αB2, then define an exchange rate that always reduces cost more than it decreases revenue, and so single activation can be α c  B2 A , c y x x 1 α α optimal in the nondominated region. A more detailed  α · y A/ A ≤( 2 − 1)( − B2) A2  A2 A explanation is provided in the following two para-   x + x x α2 c y  1 2 1 B2 A A graphs for the interested reader. c¯  ( ( − ) )( / )  (7) L x x x αA αB 1 αB Observe that dual activation never occurs when Tech- − 1( 2 − 1)( 2 − 2)( − 2)  1   +  − nology A’s yield-adjusted cost is lower than its vol-  x1 x2 x1 αA2αB2 ,  · ( − ) ume-saturation cost threshold, i.e., cA yA < x x  otherwise. / ( 2 − 1)·  αA2 1 αA2 but dual activation can occur at higher ( − ) α c Technology A yield-adjusted costs. Focusing on the  B2 A , c y x x 1 α α  α · y A/ A ≤( 2 − 1)( − A2) A2 low-cost case first, an optimal production pair must lead  A2 A   + to volume saturation when cA yA < x2 x1 αA2 1 αA2  x1 x2 x1 αA2αB2 cA yA / ( − ) ( − )   ( ( − ) )( / )  because Technology A’s low yield-adjusted cost justifies c¯H x x x α α 1 α (8) − 1( 2 − 1)( A2 − B2)( − A2)  1 producing a total product supply that exceeds half the   + 2  −  x1 x2 x1 αA2 , market size. Because total product volume is saturated,  · ( − )  otherwise. the optimal revenue depends only on the product-2  supply. Consider some production vector that uses both If cB yB c¯L then activate only Technology B. If c¯L < technologies. Because Technology A is cheaper than B / ¯≤ cB yB < cH then activate both Technologies A and B. If for making product 2, by an appropriate exchange of B c /y c¯ then activate only Technology A.5 B/ B ≥ H for A we can reduce the overall production cost while There always exists a dominant technology when both maintaining a constant supply of product 2 and both technologies have the same product-2 split, and ensuring the total supply still exceeds S/2 (and thus Chen, Tomlin, and Wang: Dual Coproduct Technologies 698 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS

Figure 1. Optimal Technology Activation as a Function of Yield-Adjusted Costs c y and c y A/ A B/ B 2 B  2 x + 1 B  1 x

only B A dominates B only A activate A only

B Neither dominates y

/ but activate B

2 B c B

 and 1 A B  ) 1 x – 2 Neither dominatesActivate but activate both x ( 2 B  1 A  ) 1 B dominates A x

– 2 activate B only 2 / A x 2 ( Slope = 1  B Slope = 0 0       (x2 – x1) A1 A2 (x2 – x1) B1 A2 x1 A1 + x2 A2   cA/yA x1 B1 + x2 B2

Note. Recall that Technology A has a higher product-2 split than Technology B, i.e., αA2 > αB2. maintaining the same revenue). Therefore, no produc- exchange of A for B, but we lose some revenue by doing tion vector that activates both technologies can be opti- this because we strictly reduce product-2 supply. This mal. The only question is which technology should be exchange (A for B) is attractive if the associated revenue used. Technology A is the optimal one because it has the loss is not too large. The revenue loss becomes more lower cost of producing product 2 and because its cost significant as Technology A’s production is driven lies below its saturation threshold. lower because revenue is concave in product-2 supply. At higher Technology A yield-adjusted costs, i.e., If Technology A’s yield-adjusted cost is high enough c y x x α 1 α , it cannot be optimal to relative to B’s then it can be optimal to drive Technol- A/ A ≥ ( 2 − 1) A2( − A2) activate Technology A and produce product quanti- ogy A production to zero. Therefore, dual activation ties that lie in the volume saturation region. There- is optimal when neither technology’s yield-adjusted fore, if A is activated, the revenue depends on the total cost is too high relative to other technology, but single product quantity and the product-2 quantity. Consider technology activation is optimal otherwise. The rele- some production vector that activates both technolo- vant cost thresholds depend on the product qualities x1 gies. Because Technology A is cheaper than B for mak- and x2 and the technology splits αA2 and αB2 because ing product 2, we can reduce the overall production these factors influence the marginal revenue loss and cost while maintaining a constant supply of product-2 the production “exchange rates” required to preserve by an appropriate (not one-for-one) exchange of B product-2 supply or total supply. for A, but we lose some revenue by doing this because we strictly reduce the total product supply (1 and 2). 4.2. Implications for Process Development and This exchange (B for A) is attractive if the associ- Technology Adoption ated revenue loss is not too large. The revenue loss With the optimal production strategies analyzed, we becomes more significant as Technology B’s produc- now turn our attention to the resulting implications for tion is driven lower because revenue is concave in the process development and technology adoption. total product supply. If Technology B’s yield-adjusted First let us consider a firm with a single technol- cost is high enough relative to A’s then it can be optimal ogy looking to improve this technology. It follows to drive Technology B production all the way to zero. from Proposition1 that the optimal profit decreases Alternatively, because Technology B is cheaper than A in the technology’s processing cost c, increases in the for making total product, we can reduce the overall technology’s yield y, and increases in the technol- production cost while maintaining a constant supply ogy’s product-2 split α2. Therefore, from a process de- of total product by an appropriate (not one-for-one) velopment perspective, the firm has three different and Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 699 distinct directions to pursue: cost reduction, yield in- must be applied with care to account for the fact that crease, or split-2 increase. If the firm can move one the new technology might play the role of “A” (higher of these drivers in the appropriate direction without split) or “B” (lower split). The optimal adoption-and- negatively affecting the other two drivers, then this is usage is illustrated in Figure2 and formally presented an unambiguous improvement. Oftentimes, however, as Corollary1 in the appendix. In general, there are process development may involve trade-offs between two thresholds (specified in closed form in the proof of the drivers. Let us first consider cost and yield while Corollary1) that govern the optimal strategy. The new holding the split constant. A yield-increase initiative technology is adopted iff its effective yield adjusted may result in a higher processing cost. For exam- cost is lower than the adoption threshold but the new ple, increasing semiconductor yield by moving to a technology displaces the incumbent only if its effective larger wafer size increases the yield per base-unit pro- yield-adjusted cost is sufficiently low, i.e., lower than cessed but also increases the cost per base-unit pro- the displacement threshold. In between these thresh- cessed. As the reader may have noticed, cost c and olds, the new technology is used with the incumbent yield y influence the profit only through the yield- technology. adjusted cost c y; and so a process development initia- / This general structure has some interesting special tive that simultaneously alters both cost and yield is an cases. First consider the case where the new and the improvement (at a constant split) iff the yield-adjusted  incumbent have the same product-2 splits, i.e., αI2 cost is decreased. Now, taking the split into account, αN2. In this case, the optimal adoption-and-usage strat- a process development initiative that increases the egy is very sharp: the new technology is adopted iff its split results in a superior technology (i.e., one with effective yield-adjusted cost is lower than the incum- a higher profit) only if the associated yield-adjusted bent’s yield-adjusted cost; and, if adopted, the new cost increase is not too large. In the supplement to this technology always displaces the incumbent. Next con- paper, we characterize the allowable cost increase by sider the case where the new and the incumbent have specifying the iso-profit curve that traces the yield- the same (effective) yield-adjusted costs, i.e., c y  adjusted cost needed to maintain a constant profit as I / I c + k y . Again we have a sharp characterization: the product-2 split changes over the range 0 to 1. ( N N )/ N the new technology is adopted iff its product-2 split Next let us consider a firm with a single incum-

is lower than the incumbent’s product-2 split; and, bent technology evaluating some new technology that if adopted, the new technology always displaces the differs in one or more of the cost, yield, and split incumbent. The fact that the new technology is never attributes. For example, as discussed in the introduc- used with the incumbent in either of these two spe- tion, a new technology might be an enhanced process cial cases (i.e., the adoption and displacement thresh- technology in the case of LEDs or a different cap- olds in Figure2 coincide) stems from the fact that sule technology in the case of industrial diamonds. When evaluating a new technology (i.e., should it be there is always one dominant technology if two tech- adopted?), the firm must consider the adoption cost nologies differ only in split or only in yield-adjusted and how the new technology will be used if adopted, cost. Finally, let us consider a special case in which the i.e., will it displace the incumbent or be used with the new and incumbent differ in both split and (effective) incumbent. To explore these adoption-and-usage ques- yield-adjusted cost. If the incumbent has a low enough tions, we consider a firm with one incumbent tech- yield-adjusted cost to place it in the volume satura- tion region, i.e., c¯ c¯ α , then the new technology nology, labeled I, that can adopt a new technology, I ≤ sat, I ( I2) labeled N. We assume the firm has ample capacity is adopted iff its effective marginal cost of producing for the incumbent technology but that it must install product 2 is lower than the incumbent’s marginal cost of producing product 2, i.e., c + k α y < (or retrofit) capacity if it adopts the new technology. ( N N )/( N2 N ) c α y . In other words, in this case adoption is We assume the adoption cost is linear in the capacity I /( I2 I ) driven solely by the (effective) marginal costs of pro- of Technology N installed, with kN 0 denoting the capacity cost per base unit. ≥ ducing the higher-quality product. The new technol- In this setting, the firm will install a capacity level for ogy might displace or be used with the incumbent in Technology N equal to its planned processing quantity this special case. (zero if not adopted), and therefore the adoption-and- A key implication of the optimal adoption-and- usage problem is essentially a two-technology produc- usage strategy is that there can be value to a new tech- tion problem in which the new technology’s effective nology even if it is inferior to the incumbent (where yield-adjusted cost includes the capacity cost; that is, inferior means a lower profit if used as a single tech- the new technology’s marginal cost of good product nology). This value arises because an inferior tech- + is cN kN yN . The new technology can have a lower nology can complement the incumbent by improving or( higher split)/ than the incumbent and, therefore, our the firm’s (effective) marginal cost of producing good earlier two-technology production result (Theorem1) product (1 and 2) or the high-quality product (2) such Chen, Tomlin, and Wang: Dual Coproduct Technologies 700 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS

Figure 2. Optimal Adoption-and-Usage Strategy for a Firm with One Incumbent Technology as a Function of the New Technology’s Effective Yield-Adjusted Cost

Displacement threshold Adoption threshold     D(cI, yI, I, N) A(cI, yI, I, N)

New displaces New used with New not incumbent incumbent adopted

c + k New technology’s effective yield-adjusted cost N N yN

P  that the new technology falls in the “new-used-with- t A, B αtn yt qt of product n 1, 2, we obtain the fol- incumbent” region of Figure2. The value of an infe- lowing∈{ } production-quantity problem: rior technology can be significant.6 Of course, inferior     X X technologies do not always add value. Similarly, a new Π∗H max R∗H 1 αt2 yt qt , αt2 yt qt q 0 technology that is superior to the incumbent might not ≥ t A, B ( − ) t A, B ∈{ } ∈{ }  displace it; a superior technology can lie in the “new- X used-with-incumbent” region. ct qt , (11) − t A, B ∈{ }

where R∗H , is given above. 5. Horizontal Coproducts (· ·)  + It is helpful to define sˆ2 S2 S1 S2 as the “market-2 We now turn our attention to the horizontal market split,” i.e., product 2’s portion/( of the combined) market setting in which the coproducts differ in their appli- sizes, because this will be a key driver of later results. In cation and, therefore, are not substitutable. As for particular, when technology t’s product-2 split is higher vertical coproducts, we first explore and characterize (lower) than the market-2 split, i.e., αt2 > sˆ2 (αt2 < sˆ2) the optimal pricing and production strategy and then then technology t’s output mix is misaligned with the explore the resulting process development and tech- market mix in the sense that it overproduces (under- nology adoption implications. Our primary purpose produces) product 2 from a proportional perspective. will be to highlight important differences that arise for Before addressing the two-technology case, for com- horizontal coproducts. pleteness we first present the optimal production quan- tity and profit when the firm has a single technology 5.1. Pricing and Production (and so suppress the technology t notation). Recall from Section3 that the demand function for product n  1, 2 is defined by its market size S (demand Proposition 2 (Single Technology—Horizontal Coprod- n  + ucts). Define S S1 S2. The optimal production quantity when price is zero) and its choke price xn (price at which demand first reaches zero), that the products are and profit are labeled such x1 x2, and that the revenue as a function  c y  ≤  S sˆ2 of product prices and product quantities is given by (2). qH∗ 1 / ; 2y α2 − α2x2 It follows that for any product quantities Q1 0 and 2  ≥ S   c y   Q2 0, the optimal price of product n 1, 2 is Π  + ≥ ∗H 1 sˆ2 x1 sˆ2x2 1 / 4 ( − ) − α2x2   pHn∗ xn max 1 2, 1 Qn Sn , (9) / − / if 0 α2 < sˆ2 and c α2 y x2 1 α2 1 α2 1 ≤sˆ sˆ ; are /( ) ≤ ( − ( /( − )) · and the resulting total optimal revenue is given by (( − 2)/ 2)) P2 R∗ Q , Q   R∗ Q , where  c y  H 1 2 n 1 Hn n  S 1 sˆ2 ( ) ( ) qH∗ − 1 / ; ( 2y 1 α2 − 1 α2 x1 x 1 Q S
Q , Q < S 2 − ( − )  n − n/ n n n n/   c y 2  R∗Hn Qn (10)  S + ( ) S x Q S Π∗H 1 sˆ2 x1 1 / sˆ2x2 n n 4, n n 2. 4 ( − ) − 1 α x / ≥ / ( − 2) 1 Different to the vertical-market setting, a product’s if sˆ2 < α2 1 and c 1 α2 y x1 1 1 α2 α2 price and sales volume depend only on its own quan- sˆ 1 sˆ ≤ ; and otherwise/(( − are) ) ≤ ( − (( − )/ )· ( 2/( − 2))) tity and are not influenced by the other product’s quan- + S x1 x2 x1 α2 c y tity. A product’s revenue is saturated iff its supply q∗  ( − ) − / ; H 2y x 1 α 2 1 sˆ + x α2 sˆ exceeds half its market size. With the optimal pric- 1( − 2) /( − 2) 2 2/ 2 + 2 ing and revenue result in hand, and recalling that a S x1 x2 x1 α2 c y Π∗  ( ( − ) − / )   H 2 . production pair q qA , qB creates a quantity Qn q 4 x 1 α 2 1 sˆ + x α sˆ ( ) ( ) 1( − 2) /( − 2) 2 2/ 2 Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 701

The technology’s marginal cost of producing prod- (ii) If the technologies differ in their product-2 splits, i.e., uct 2 is c α y and the marginal cost of product 1 is α > α (recall α α w.l.o.g.), then define c¯ and /( 2 ) A2 B2 A2 ≥ B2 L c 1 α2 y . If the technology output mix is misaligned c¯ as below. If c y c¯ then activate only Technology B. /(( − ) ) H B/ B ≤ L with the market mix, i.e., α2 , sˆ2, and the marginal cost If c¯L < cB yB < c¯H then activate both Technologies A and B. of the overproduced product is low, then the overpro- / 7 If cB yB c¯H then activate only Technology A: duced product is saturated. If the technology output / ≥ mix is aligned, i.e., α  sˆ , or the marginal costs are not 2 2  cA yA low, then neither product is saturated.  / 1 αB2 ,  1 αA2 ( − ) Turning now to the two-technology case, we need  −    sˆ2 1 αB2 to adapt our earlier dominance definition. In the ver-  cA yA x1 1 αA2 1 ( − ) ,  / ≤ ( − ) − 1 sˆ2 αB2 tical case—because there is one market of customers   ( − ) c¯L  2 + 2 that choose among the two products based on qual- sˆ2x1 1 αB2 1 sˆ2 x2αB2 cA yA  ( ( − ) ( − ) ) / ity valuations and prices—the relevant marginal costs  +   x1x2 αA2 αB2 αB2 sˆ2  ( − )( − ) 1 for dominance are the marginal costs of overall sup-   +  −  sˆ2x1 1 αA2 1 αB2 1 sˆ2 x2αA2αB2 , ply (i.e., products 1 and 2) and of the higher-quality  · ( − )( − ) ( − )  otherwise, product (2) supply. Because product markets are dis-  tinct in the horizontal case, dominance depends on the  cA yA  / 1 αB2 , marginal costs of each individual product.  1 αA2 ( − )  −    sˆ2 1 αA2 Definition 2 (Dominant Technology—Horizontal Coprod-  cA yA x1 1 αA2 1 ( − ) ,  / ≤ ( − ) − 1 sˆ2 αA2 ucts). Let i A, B denote one technology and j   ( − ) ∈ { } c¯H  + the other. Technology i dominates technology j iff sˆ2x1 1 αA2 1 αB2 1 sˆ2 x2αA2αB2 cA yA  ( ( − )( − ) ( − ) ) / c 1 α y c 1 α y (that is, i has a lower  +  i i2 i j j2 j  x1x2 αA2 αB2 αA2 sˆ2 /(( − ) ) ≤ /(( − ) )  ( − 1 )( − ) marginal cost for producing product 1) and ci αi2 yi   2 + 2  −  sˆ2x1 1 αA2 1 sˆ2 x2α , c α y (that is, i has a lower marginal cost/( for pro-) ≤  · ( − ) ( − ) A2 j/( j2 j)  otherwise, ducing product 2). Strict dominance occurs iff at least  one inequality is strict. if sˆ2 αB2 (Case 1), Intuitively, the firm should not activate a strictly ≤ dominated technology because the other technology    cA yA 1 sˆ2 αB2 is cheaper for each product, and so revenue can be  / αB2, cA yA x2αA2 1 ( − ) ,  αA2 / ≤ − sˆ2 1 αB2 weakly increased and cost strictly decreased by using  ( − )   2 + 2 only the dominant technology. (See appendix Lemma2  sˆ2x1 1 αB2 1 sˆ2 x2αB2 cA yA c¯  ( ( − ) ( − ) ) / for proof.) L +   x1x2 αA2 αB2 αB2 sˆ2   ( − )( − )  1 Before formally presenting the optimal activation  sˆ x 1 α 1 α + 1 sˆ x α α − , strategy when the firm has two technologies, we note  · 2 1( − A2)( − B2) ( − 2) 2 A2 B2  that the strategy bears some strong similarities to the  otherwise, vertical coproduct case: single activation is optimal if  cA yA  / 1 α , the splits are identical but dual activation can be opti-  1 α ( − B2)  − A2 mal when the splits differ and the technology yield-   sˆ 1 α   c y x 2 A2 adjusted costs are not too different (accounting for  A A 1 1 αA2 1 ( − ) ,  / ≤ ( − ) − 1 sˆ2 αA2 splits). However, in the horizontal case, the boundary c¯  ( − ) H  sˆ x 1 α 1 α + 1 sˆ x α α c y costs that specify the dual-activation region depend on  ( 2 1( − A2)( − B2) ( − 2) 2 A2 B2) A/ A  +  how the market mix compares to the technology out-  x1x2 αA2 αB2 αA2 sˆ2  ( − 1 )( − ) put mix. In particular, there are three cases: (1) both   2 + 2  −  sˆ2x1 1 αA2 1 sˆ2 x2αA2 , technologies have product-2 splits greater than the  · ( − ) ( − )  otherwise, market-2 split, i.e., both overproduce product 2; (2) the  technology product-2 splits span the market-2 split; and (3) both technologies have product-2 splits lower if αB2

c y  1 sˆ α  a product-1 perspective. The nondominated region is  A/ A c y x 2 A2  αB2, A A 2αA2 1 ( − ) larger in the horizontal setting than the vertical set-  αA2 / ≤ − sˆ2 1 αA2  ( − )   + ting because this dominance criterion is more difficult  sˆ2x1 1 αA2 1 αB2 1 sˆ2 x2αA2αB2 cA yA  ( ( − )( − ) ( − ) ) / to satisfy. Analogously to the vertical coproduct set- c¯H +   x1x2 αA2 αB2 αA2 sˆ2 ting, when neither technology dominates there may not   ( − 1 )( − )  sˆ x 1 α 2 + 1 sˆ x α2 − ,  · 2 1( − A2) ( − 2) 2 A2 exist a production exchange rate that allows the firm to   otherwise, profitably increase production on one technology while reducing production on the other. Therefore, in all three if sˆ α (Case 3). 2 ≥ A2 cases there exists a subregion for which dual activa- For technologies that differ in their splits, Figure3 tion is optimal. While the dual-activation region differs illustrates the optimal activation strategy (in general- between the cases, the region’s area is always larger in ity) for each of three cases. In each case, there always Case 2 than in Cases 1 or 3 (proof omitted). In other exists a region of yield-adjusted cost pairs for which nei- words, when the technologies’ product-2 splits span the ther technology dominates; this is the region between market-2 split, there are more cost pairs for which dual the two rays emanating from the origin. In this non- activation is optimal. The reason for this lies in an addi- dominated region, Technology A is cheaper from a tional dual-activation motive in the horizontal coprod- product-2 perspective but Technology B is cheaper from uct setting. If one technology overproduces product 2

Figure 3. Optimal Technology Activation as Function of Yield-Adjusted Costs c y and c y A/ A B/ B

≤  (b) Case 1: s2 B2

(a) Three cases B 2  2

Technology B Technology A ) A 2 product-2 split product-2 split + x

 Neither dominates B 1 1

  s  but activate A only / B2 A2 1 x A 1

Case Case Case  2 1 2 3 s A dominates B activate A only Both technologies Technology B (1 –

Both technologies B 1

Activate both A and B  1 overproduce product 2 underproduces product underproduce product 2 B x / y

as compared to the 2 as compared to the as compared to the B c market-2 split market-2 split market-2 split ) B 2

AND  1

BUT BUT s Neither dominates but activate B only

/ A1  Technology B’s product- Technology A Technology A’s B 1 /  B1 overproduces product 2 2 2 split is a closer match product-2 split is s B dominates A as compared to the a closer match market-2 split (1 – / A2 activate B only B 1 Slope =  B2  1

0 1 x Slope =  s2 = S2/(S1+ S2) x  (1 – s  / s  ) x  (1 – s  / s  ) x  + x  Market-2 split, i.e., relative market size of product 2 1 A1 2 B1 1 B2 1 A1 2 A1 1 A2 1 A1 2 A2 cA/yA

B 2 B 2  (c) Case 2:  < s <   (d) Case 3: s ≥  2 B2 2 A2 2 2 A2 + x + x B 1 B 1 )   A 2 1 1 x x

 A only 1

s Neither dominates / but activate A 1

 A dominates B ) 2 s

B 1 activate A only  2 (1 – s

A dominates B / B 1

B B Neither dominates B 2  activate A only Neither dominates / y / y 1

 but activate B only B B x but activate B only 1 c c s (1 – )

Activate both A and B B 2 Activate both A and B  B 1 2  x A1 2  s / A1 Neither dominates but activate A only /

/ ) B1

B 2 1

B A 1  1  2 s B dominates A s

activate B only / A2 Slope =  2 B dominates A

 A 2 (1 – /  A  / Slope = B2  2

B 2 B 1 activate B only s  2

x Slope =  Slope =  (1 –

x  (1 – s  / s  ) x  (1 – s  / s  ) x  + x  B 2         1 A1 2 A1 1 A2 2 A2 1 B2 2 B1 1 A1 2 A2  x (1 – s / s ) x (1 – s / s ) x + x

2 2 A2 1 A2 2 A1 2 A2 1 B2 2 B1 1 A1 2 A2 cA/yA x cA/yA

Notes. Recall that Technology A has a higher product-2 split than Technology B, i.e., α > α . Note that sˆ  1 sˆ . A2 B2 1 − 2 Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 703 and the other underproduces product 2 in relation to adoption-and-usage strategy for horizontal coproducts the market-2 split, i.e., Case 2, then activating both tech- exhibits some important differences because of the nologies can help the firm achieve a combined technol- market split and related concept of an ideal product-2 ogy output mix that better resembles the market mix. split. Recall that for vertical coproducts the new tech- nology is never used with the incumbent if they 5.2. Implications for Process Development and have the same (effective) yield-adjusted costs: the new Technology Adoption technology either displaces the incumbent or is not With the optimal production strategies analyzed, we adopted. In contrast, because there exists an ideal split now turn our attention to the resulting implications for less than one for horizontal coproducts, a new and process development and technology adoption. Some incumbent technology with the same (effective) yield- implications are similar in the horizontal and vertical adjusted costs may be used together. This is formally settings, and in what follows we focus on where the established in Corollary2 in the appendix and illus- implications diverge rather than presenting and dis- trated in Figure4. The firm wants to use the technolo- cussing all results. gies in a manner that achieves a combined product-2 First, let us consider a firm with a single technol- split as close as possible to the incumbent’s ideal split ogy looking to improve this single technology. Analo- α2∗ given in Proposition3. If the technology product-2 gous to the vertical case, it follows from Proposition2 splits span this ideal split, then the firm can exactly that the optimal profit decreases in the technology’s attain the ideal split by using both technologies in processing cost c and increases in the technology’s appropriate proportions. yield y. In fact, as before, the profit increases as the As discussed for vertical coproducts, the firm can yield-adjusted cost c y decreases. If product 1’s choke / benefit—sometimes significantly—from a new tech- price x1 is less than the technology’s yield-adjusted cost nology that is inferior to the incumbent. Because there c y, then product 1 is not economically viable in its / is an additional motive for dual activation in the hori- own right, i.e., it alone cannot justify production. Using zontal setting, that being the desire to better match the accounting terminology, we say that product 1 is a by- market-2 split when the technology product-2 splits x c y product if 1 and a coproduct, i.e., economically span the market-2 split, our numerical testing has viable in its own≤ / right, otherwise. Different to the verti- found that inferior technologies can be especially ben- cal case, the optimal profit is not necessarily increasing eficial in the horizontal setting.8 in the product-2 split α2. In fact, the profit is increas- ing in α2 iff c y x1, i.e., product 1 is a by-product. Otherwise, when/ ≥ product 1 is a coproduct, there exists 6. Uncertainty in Market Size a product-2 split less than one that attains the highest We now relax the deterministic market size assump- profit. tion and examine settings in which the size(s)—S in the vertical model and S and S in the horizontal model— (Ideal Market Product-2 Split) For any 1 2 Proposition 3 . are uncertain. We consider two possible uncertainty- given yield-adjusted cost c y, the profit is highest at a / resolution sequences. In the first sequence we assume product-2 split of α∗  1 if c y x , and at α∗  x sˆ 2 1 2 1 2 market size uncertainty is resolved after production x c y x 1 sˆ x /c y≥ + x sˆ x c y( < 1· 2 2 2 1 1 2 2 decisions are made but before prices are set. In the sec- (otherwise.− / ))/( ( − )( − / ) ( − / )) ond sequence, when considering technology adoption,

This “ideal” technology split α2∗ equals the market-2 we assume market size uncertainty is resolved after   split sˆ2 if x2 x1 or if c 0. It exceeds the market-2 capacity investment but before production and prices split otherwise, i.e., α∗ sˆ . In other words, if process are set. For reasons of space, we relegate the analy- 2 ≥ 2 development could control the split (at a given yield- sis and results for both sequences to Section A3 in the adjusted cost), then it should develop a technology online appendix and only summarize our key findings split that overshoots the market-2 split. Different to the in the main body. vertical setting, however, it should not try and push the technology split all the way to 1, i.e., create a technol- 6.1. Market Uncertainty Resolved After Production ogy that only produces product 2, unless product 1 is We first consider the sequence in which the market a by-product. size is uncertain (with a continuous distribution) at the Turning to the question of process adoption and time of production but prices are set in recourse. When usage when the firm has one incumbent technol- considering technology adoption, we assume there is ogy (labeled I) and is evaluating a new technology no uncertainty resolution between capacity investment (labeled N), the optimal adoption-and-usage strategy and production decisions. As one would anticipate, the has a similar structure to the vertical case (see Fig- firm’s optimal expected profit is lower when the mar- ure2) but the adoption and displacement threshold ket is stochastic; see Lemmas S3 and S6 in Section A3 expressions—given in Corollary S1 of the online ap- in the online appendix. We have generalized all our pendix—differ. Despite the structural similarity, the earlier analytical results for the deterministic market to Chen, Tomlin, and Wang: Dual Coproduct Technologies 704 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS

Figure 4. Different to the Vertical Coproduct Setting, New and Incumbent Technologies with the Same (Effective) Yield-Adjusted Costs May Be Used Together in the Horizontal Coproduct Setting

Case 1 Incumbent Ideal split product 2 split Incumbent technology’s ( for the incumbent’s lower than ideal product-2 split yield adjusted cost) split   12 *

New displaces New not adopted incumbent New used with incumbent

 New technology’s product 2 split N2

Case 2 Incumbent Ideal split product 2 split (for the incumbent’s Incumbent technology’s higher than ideal yield adjusted cost) product-2 split split  * 12

New displaces New used with incumbent incumbent New not adopted

 New technology’s product 2 split N2

the stochastic market. The analogue to each result can When market size is random, dual activation can be be found in Section A3 in the online appendix. We note optimal even if the yield-adjusted cost is low. This is that various expressions—such as profits, production shown in Figure5, which depicts the dual activation quantities, cost thresholds, and ideal split—that have region for a stochastic market size and a determinis- closed-form expressions in the deterministic setting are tic market size.9 For Technology A yield-adjusted costs defined by implicit functions in the stochastic setting. below 0.4 (the volume saturation yield-adjusted cost This difference aside, all our earlier production and for Technology A), dual activation is not optimal for the technology-and-adoption results and associated impli- deterministic setting but can be optimal for the stochas- cations extend (with a minor caveat) to the stochastic tic setting. The dual activation region is also shifted market setting. somewhat at higher costs. The caveat relates to the volume- and revenue-sat- Finally, we note that this saturation-cost caveat has uration product-quantity boundaries in the vertical one implication for our vertical model adoption-and- model (see (5)) and the revenue-saturation product- usage cost findings. In the deterministic setting the quantity boundary in the horizontal model (see (10)). adoption rule is particularly simple in the special case When the market size is stochastic with an infinite when the incumbent technology’s yield-adjusted cost upper support, there is always a positive probability is below its volume saturation cost: the new technology that any given production quantity will result in prod- is adopted iff it has a lower (effective) marginal cost of producing the higher quality product. Because the uct quantities that do not fall in a saturation region. As volume saturation cost does not exist in the stochastic such, in general there are not sharp cost thresholds— setting, this special case no longer exists. such as the volume saturation cost c α in Propo- sat( 2) sition1—that dictate whether an optimal production 6.2. Market Uncertainty Resolved quantity will result in saturation or not. For the verti- Before Production cal market model, this means that when market size is In settings where the capacity installation time is long random, the optimal production quantity on a low-cost relative to the production lead time, the market size technology may not lead to volume saturation in all uncertainty at the time of technology adoption is likely market size scenarios. In Theorem1 and the associated to be larger then at the time of production. In what Figure1, we established that dual activation is never follows, we assume all uncertainty is resolved between optimal if the higher-split technology’s yield-adjusted capacity investment and production. That is, technol- cost was low, i.e., below its volume saturation cost. ogy adoption occurs under uncertainty but production Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 705

Figure 5. (Color online) Optimal Technology Activation for in higher capacity levels in advance. This adopt-but- Stochastic and Deterministic Market Sizes (Vertical not-displace strategy (when splits are identical) does Coproducts) not arise if there is no uncertainty resolution between 1.5 capacity and production because there is no capacity

cL (deterministic) risk in that earlier setting: the firm’s new technology c (deterministic) H production level is the same for all market sizes and c (stochastic) L equals its capacity investment. cH (stochastic) Similarly, the earlier finding in the deterministic ver- 1.0 Activate A only tical market that the new and incumbent technology

B are never used together if technologies have identi- y /

B cal effective costs is more nuanced when market size c Activate A and B uncertainty is resolved between adoption and produc- 0.5 Activate B only tion. As with the identical split setting, it can be opti- mal to adopt the new technology and to then use it with the incumbent when the market size is large; see Propositions S2 and S4 in Section A3 in the online

0 appendix. 0 0.5 1.0 1.5 2.0 2.5 3.0 c /y A A 7. Conclusions and prices are set in recourse. As before—see Sec- Processing cost, overall yield, and coproduct split tions 4.2 and 5.2—we consider a firm with an incum- are all important drivers of a coproduct technology’s bent technology deciding whether or not to adopt a production economics. Firms, therefore, develop new new technology with a linear capacity cost. We assume process technologies that differ on one or more of that the market size(s) has a Bernoulli distribution and these dimensions so as to improve their underlying furthermore in the horizontal model assume that the economics. Examples (discussed in the introduction) product markets are both high or both low. We assume include a lower-cost chloralkali process, a higher yield cumene process, a higher split LED process, larger the incumbent technology has sufficient capacity such that it does not limit the optimal production quantity wafer semiconductor processes that increase yield if the market size is high. but increase cost, and a new diamond process that In the deterministic market setting, we established increased yield and split but also cost. The develop- for both vertical and horizontal models that the adop- ment of new technologies gives rise to the adoption tion-and-usage strategy is very sharp when the new and production questions studied in this paper: Should and the incumbent have the same product-2 splits: the firm adopt a new technology and if so should it the new technology always displaces the incumbent if be used with the incumbent technology or should it it is adopted, i.e., they are never used together. This displace it? remains true in the stochastic market setting when Focusing first on a vertical market model in which there is no uncertainty resolution between capacity the two coproducts differ along a performance dimen- investment and production. However, a more nuanced sion, we characterized the optimal pricing and produc- strategy emerges when uncertainty is resolved before tion decisions of a monopoly firm with two technolo- production. As formally established in Propositions S1 gies facing a deterministic market size. We proved that and S3 in Section A3 in the online appendix, there dual activation in which the firm produces a positive exists a range of incumbent yield-adjusted costs for quantity on both technologies can be optimal if the which the new technology is adopted but it does not technologies differ in their splits. Dual activation can displace the incumbent technology: both the new and only occur when one technology has a lower marginal incumbent are used if the market size is high but only cost for good product (i.e., both products combined) the new is used if the market size is low. The rea- but the other has a lower marginal cost for the high- son that the new technology does not necessarily dis- quality product. The fact that each technology offers a place the incumbent when adopted is that there is a different marginal-cost benefit can make dual activa- risk of unused new technology capacity. Fearing the tion attractive. However, this alone does not guarantee low-market scenario (when this probability is large dual activation: even if neither technology dominates, enough), the firm does not invest in a sufficient capac- single activation can be optimal if the technologies dif- ity of the new technology that would enable it to pro- fer sufficiently in their yield-adjusted cost. Building on duce exclusively on the new technology if the market the optimal production analysis, we characterized the size turns out to be high. The firm prefers to supple- optimal adoption-and-usage strategy of a firm with an ment new technology production with incumbent pro- incumbent technology considering a new technology. duction if the market size is high rather than invest Among other implications, we established that a new Chen, Tomlin, and Wang: Dual Coproduct Technologies 706 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS technology with an identical split to the incumbent we note that some coproduct technologies—such as will be adopted iff its effective (i.e., capacity loaded) naphtha cracking to produce ethylene and propylene— yield-adjusted cost is lower than the incumbent’s yield have the ability to control the product split (within adjusted cost and, if adopted, the new technology dis- some range) by adjusting the process settings. This places the incumbent. If the technologies have the same gives rise to interesting questions about the value of (effective) yield-adjusted costs, then the new technol- split control when product markets are uncertain. ogy is adopted iff it has a higher product-2 split and it will displace the incumbent if adopted. When the new Appendix. Proofs technology differs in split and (effective) yield-adjusted Important Note That Applies to All Proofs As presented in Section3, the firm’s pricing and produc- cost, then the new technology displaces the incumbent  tion problem is to choose coproduct prices p p1, p2 and at low effective yield-adjusted costs, is used with the  ( ) technology production quantities q qA , qB so as to maxi- incumbent at intermediate costs, and is not adopted  P( ) mize its profit Π q, p R Q q , p t A, B ct qt , where the ( ) ( ( ) ) − ∈{ }P at higher costs. We also explored the impact of mar- quantity of product-n produced is Qn q t A, B αtn yt qt ket size uncertainty. In the setting where uncertainty is for n  1, 2, and the revenue function (R ), is∈{ given} by (1) (· ·) resolved between adoption and production, we estab- for the vertical market and (2) for the horizontal market. To + lished that the risk of unused capacity can induce the ensure viability, we assume ct yt < x1 x2 x1 αt2 for all /  ( − ) firm to adopt the new technology and use it with the technologies throughout. Defining c¯t ct yt as the yield- ¯  / incumbent even when their spits are identical. adjusted production cost and qt qt yt as the yield-adjusted production quantity, the above problem is equivalent to Focusing next on a horizontal market model in  P  Π∗ maxq¯ 0, p 0 R Q q¯ , p t A, B c¯t q¯t where Qn q¯ which the two coproducts serve different markets, we P ≥ ≥ { ( ( ) ) − ∈{ } } ( ) t A, B αtn q¯t for n 1, 2. We will use this more notationally characterized the firm’s optimal pricing and produc- compact∈{ } formulation throughout the proofs. Furthermore,  tion decisions and the resulting optimal technology for expositional brevity, we may at times refer to c¯t ct yt  / adoption-and-usage decisions. Analogous to the ver- as a technology’s “production cost” and q¯t qt yt as the tical setting, we established that dual activation can technology’s “production quantity,” with the understand- be optimal when one technology does not dominate ing that these are in fact yield-adjusted costs and yield- the other, but dominance in the horizontal setting adjusted quantities. Actual production quantities given in certain results, e.g., Propositions1 and2, are obtained by the depends on the marginal cost of each individual prod-  conversion q q¯ y . uct. Importantly, we showed that there is an additional t t / t motive for dual activation in the horizontal setting: Proof of Proposition1. (Vertical Single Technology). Apply- ing (6) to the single-technology case, the firm’s problem is when the technology splits lie on either side of the mar- to maximize the profit function Π q¯  R∗ 1 α q¯, α q¯ ket split, then activating both technologies can help the ( ) V (( − 2) 2 ) − c¯q¯, where R∗ , is given by (5). Using (5), we can rewrite V (· ·) firm achieve a combined technology split that better R∗ 1 α q¯, α q¯ as R∗ q¯  r q¯ + r q¯ , where V (( − 2) 2 ) V ( ) 1( ) 2( ) resembles the market split. Among other implications,  q¯   this gives rise to the result that—different to the vertical x1 1 q¯, q¯ S 2   − S ≤ / setting—if the technologies have the same (effective) r1 q¯ ; ( )  x S yield-adjusted costs it can be optimal to use both tech-  1 , q¯ > S 2  4 / nologies together to achieve a more desirable overall     q¯α2 product split.  x2 x1 1 q¯α2, q¯α2 S 2 ( − ) − S ≤ / We hope this research opens up other lines of in- r q¯  2( ) quiry. We explored a monopoly setting, and other  x2 x1 S  ( − ) , q¯α2 > S 2. models, e.g., a perfectly competitive or oligopoly mar-  4 / ket, would shed light on the effect of competition on The first and second derivatives of the profit function Π q¯  r q¯ + r q¯ c¯q¯ are ( ) production and technology adoption. We purposely 1( ) 2( ) − excluded yield and/or split uncertainty from our  2q¯   2q¯α  S  + 2 ¯ ¯ x1 1 x2 x1 1 α2 c, q model to remove production risk as a driver of dual  − S ( − ) − S − ≤ 2  activation or technology choice. Yield and split can be dΠ q¯   2q¯α  S S ( )  2 uncertain in some coproduct settings, both at the adop- x2 x1 1 α2 c¯, < q¯ dq¯ ( − ) − S − 2 ≤ 2α2 tion and production stages, and it would be of interest   S to understand the impact of technology risk on pro-   c¯, q¯ > duction and adoption. Production learning may lead − 2α2  2   2α  S to improvements in cost, yield, and/or split over time,  + 2 ¯ x1 x2 x1 α2, q and a multiperiod model could refine some of our find-  − S ( − ) − S ≤ 2 2  d Π q¯   2α  S S ings in settings where such learning is significant. For ( )  2 2 x2 x1 α2, < q¯ example, it may be optimal to use both technologies dq¯ ( − ) − S 2 ≤ 2α2  during some transition period until the new technol-   S ogy achieves its long-run cost, yield, and split. Finally, 0, q¯ > .  2α2 Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 707

It follows that Π q¯ is concave. Furthermore, it is strictly (i) Technologies have identical splits, i.e., α  α . ( ) A2 B2 decreasing for q¯ > S 2α2 . The optimal quantity q∗ exceeds By Definition1, Technology A dominates Technology B S 2 iff dΠ q¯ dq¯ >/(0 at) q  S 2, or equivalently, c¯ if c y c y , and vice versa (because the product-2 / ( )/ / ≤ A A B B x x 1 α α . Proof of proposition statement follows by splits/ are≤ identical)./ The proposition statement then follows ( 2 − 1)( − 2) 2 application of appropriate first-order condition and subse- directly from Lemma1. quent substitution into the profit function.  (ii) Technologies differ in their splits, i.e., αA2 > αB2 Lemma 1 (Vertical Coproducts Dominance). For i A, B let because w.l.o.g. the technologies are labelled such that ∈ { }  j denote the other technology, i.e., j  A, B i. If Technology i αA2 αB2. By Lemma1, technology j A, B is not activated ≥ { }  dominates Technology j then there exists{ an}\ optimal production at optimality if it is dominated by technology i A, B j, i.e., if c¯ c¯ and c¯ α c¯ α , with at least one inequality{ }\ being vector q¯ ∗ that does not use Technology j, i.e., q¯∗  0. i j i i2 j j2 j strict≤ (see Definition/ ≤1 in/ the main body of the paper). If tech- Proof of Lemma1. (Vertical Coproducts Dominance). We first nology j A, B is dominated by technology i  A, B j ∈ { } { }\ establish that the optimal revenue RV∗ Q1, Q2 is increasing then it is optimal to activate i only and the optimal quantity + ( ) in product quantities Q1 Q2 and Q2. It is trivially true is given by Proposition1. It then remains to consider the case + for Q1 Q2 S 2 or Q2 S 2 as RV∗ Q1, Q2 is constant in in which neither technology dominates the other. Note that + ≥ / ≥ / ( + ) Q1 Q2 and Q2, respectively. For Q1 Q2 < S 2, we have +  + / ∂r1 q¯A , q¯B ∂r1 q¯A , q¯B ∂RV∗ Q1, Q2 ∂ Q1 Q2 x1 1 2 Q1 Q2 S 0. Similarly, ( )  ( ) ( )/ ( ) ( − ( )/+ ) ≥ we have ∂RV∗ Q1, Q2 ∂Q2 x1 1 2Q1 S x2 1 2Q2 S ∂q¯A ∂q¯B +( )/ ( +− / ) ( − / ) ≥   x1 1 2 Q1 Q2 S 0. For Q1 Q2 S 2 but Q2 < S 2, we 2 q¯ + q¯ ( − ( )/ ) ≥  ≥ / /  x 1 ( A B) , q¯ + q¯ S 2 have ∂R∗ Q1, Q2 ∂Q2 x2 x1 1 2Q2 S 0.   1 A B V ( )/ ( − )( − / ) ≥ − S ≤ / From the dominance Definition1, Technology i dominates  +  0, q¯A q¯B > S 2 technology j iff ci yi c j yj and ci αi2 yi c j α j2 yj ; or  / / ≤ / /( ) ≤ /( ) equivalently using the yield-adjusted cost notation, Technol- and ¯ ¯ ¯ ¯ ogy i dominates technology j iff ci c j and ci αi2 c j α j2. ∂2r q¯ , q¯ ∂2r q¯ , q¯  2 S x < 0, q¯ + q¯ S 2 Noting that α + α  1 for t  i, j, this≤ dominance/ ≤ criterion/ 1( A B)  1( A B)  −( / ) 1 A B ≤ / t1 t2 2 2 + P2 P2 ∂q¯ ∂q¯ 0, q¯A q¯B > S 2. is equivalent to c¯  α c¯  α for n  1, 2. A B i / k n ik ≤ j / k n jk / Consider any arbitrary production vector q¯ with q¯ > 0. 2  j In addition, note that ∂ r1 q¯A , q¯B ∂q¯A ∂q¯B 0. It follows that Construct a new production vector qˆ that is identical ( )/ the Hessian matrix is negative definite and hence r1 q¯A , q¯B is to q¯ except that qˆ  0 and qˆ  q¯ + c¯ c¯ q¯ . The result-  ( ) j i i j i j jointly concave in q¯A and q¯B. Similarly, for i A, B we have ˆ + ˆ ˆ ( / ) ing output quantities Q1 Q2, Q2 are component-wise

+ ( ) +  x2 x1 αi2 1 2 αA2q¯A αB2q¯B S , weakly larger than the quantities Q1 Q2, Q2 because ∂r q¯ , q¯  ( − ) − ( )/ P2 P2 P2 ( P2 ) 2 A B  + c¯  α c¯  α n  α c¯  α c¯ ( )  α q¯ α q¯ S 2 i / k n ik ≤ j / k n jk ⇒ k n ik / i ≥ k n jk / j ⇒ ∂q¯ A2 A B2 B ≤ / P2 α c¯ c¯ q¯ P2 α∀ q¯ P2 α q¯ + c¯ c¯ q¯ i  + kn ik j i j kn jk j kn ik i j i j  0, αA2q¯A αB2q¯B > S 2 P2 ( +/ P)2 ≥ ˆ +⇒ˆ ˆ ( + ( / ) ) ≥  / kn αik q¯i kn α jk q¯ j Q1 Q2, Q2 Q1 Q2, Q2 . From ⇒ ( ) ≥ ( ) + and above, RV∗ Q1, Q2 is increasing in product quantities Q1 Q2 ( ) 2  2 + and Q . Therefore, R Qˆ , Qˆ R Q , Q . The production ∂ r2 q¯A , q¯B x2 x1 2αi S < 0, αA2q¯A αB2q¯B S 2 2 V∗ 1 2 V∗ 1 2 ( )  −( − ) 2/ ≤ /  P (  ) ≥ P ( ) 2 + cost C qˆ t A, B c¯t qˆt C q¯ t A, B c¯t q¯t because C qˆ ∂q¯ 0, αA q¯A αB q¯B > S 2. ( ) ∈{ }  ( ) ∈{ } ( ) − i 2 2 / C q¯ c¯i c¯j c¯i c¯j q¯ j 0. Therefore the profit for qˆ is at least ( ) ( · / − ) In addition, note that ∂2r q¯ , q¯ ∂q¯ ∂q¯  2 x x as large as the profit for q¯ . Therefore any arbitrary feasible 2( A B)/ A B − ( 2 − 1)· αA2αB2 S. It follows that the Hessian matrix is negative production vector q¯ with q¯ j > 0 is (weakly) dominated (in the semidefinite/ and hence r q¯ , q¯ is (weakly) jointly con- sense of a (weakly) higher profit) by some feasible produc- 2( A B)  cave in q¯A and q¯B. Because Π q¯A , q¯B is a linear combination tion vector qˆ with q¯ j 0. Therefore, there exists an optimal ( )  of r q¯A , q¯B and r q¯A , q¯B , it then follows that Π q¯A , q¯B is production vector with q¯ j 0.  1( ) 2( ) ( ) jointly concave in q¯A and q¯B. The optimal activation strategy Proof of Theorem1. (Vertical Two Technologies). Applying (and production quantities) can be found by application of (6) to the two-technology case, the firm’s problem is to max- the first-order conditions subject to the nonnegative bound- Π q¯ q¯  R q¯ + q¯ imize the profit A , B V∗ 1 αA2 A 1 αB2 B , ary conditions. Detailed algebra omitted for reasons of space. q¯ + q¯ c¯ (q¯ c¯) q¯ , where(( −R ) is given( − by) (5). αA2 A αB2 B A A B B V∗ , We note that the optimal quantities in either case (i) or (ii) )) − − Π  (· ·) + Using (5), we can write q¯A , q¯B r1 q¯A , q¯B r2 q¯A , q¯B are as follows. If c¯ c¯ (c¯ c¯ ) then activate B only (A only) c¯ q¯ c¯ q¯ , where ( ) ( ) ( ) − B ≤ L B ≥ H A A − B B and the optimal yield-adjusted production quantity is given  q¯ + q¯  by the single-technology Proposition1 using technology B’s  A B + +  x1 1 q¯A q¯B , q¯A q¯B S 2  − S ( ) ≤ / (A’s) cost and split parameters. If c¯L < c¯B < c¯H then acti- r q¯ , q¯  1( A B) vate both A and B and the optimal yield-adjusted produc-  x1S +  , q¯A q¯B > S 2 tion quantities are given by q∗ , q∗  qˆ , qˆ if qˆ q¯ and  4 / ( A B) ( A B) B ≤ B  q∗ , q∗  q¯ , q¯ otherwise, where  α q¯ + α q¯  ( A B) ( A B)  A2 A B2 B ¯ + ¯  x2 x1 1 αA2qA αB2qB , λ γ µγ λ γ µγ  ( − ) − S ( ) ˆ  B A − B ˆ  A B − A  qA 2 ; qB 2 r q¯ , q¯  + λA λB µ λA λB µ 2 A B αA2q¯A αB2q¯B S 2 − − ( )  ≤ / ¯  x x S  S 1 cA αA2 x2 x1 αB2  2 1 + q¯A − ( / )/( − ) − ;  ( − ) , αA2q¯A αB2q¯B > S 2. 2 α α  4 / A2 − B2 The firm’s problem is to maximize Π q¯ , q¯ subject to q¯ 0  S αA2 1 c¯A αA2 x2 x1 ( A B) A ≥ q¯B − ( − ( )/ /( − )) and q¯ 0. 2 αA2 αB2 B ≥ − Chen, Tomlin, and Wang: Dual Coproduct Technologies 708 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS

 + 2  + 2  + + and λA x1 x2 x1 αA2, λB x1 x2 x1 αB2, µ x1 cN kN yN D c¯I , αI2, αN2 and a complementing technol- ( − ) ( − ) ( )/ ≤ ( +) x x α α , γ  S 2 x + x x α c¯ , and γ  ogy iff D c¯I , αI2, αN2 < cN kN yN A c¯I , αI2, αN2 , where ( 2 − 1) A2 B2 A ( / )( 1 ( 2 − 1) A2 − A) B ( ) ( )/ ≤ ( ) S 2 x + x x α c¯ .  ( / )( 1 ( 2 − 1) B2 − B) D c¯I , αI2, αN2 (   )  c¯I c¯I αI2 Corollary 1 (Vertical Adoption and Usage). There exist adoption  α , α 1 /  α N2 N2 ≤ − x x and displacement thresholds, A c¯ , y ,α ,α D c¯ , y ,α ,  I2 2 − 1 ( I I I2 N2) ≥ ( I I I2   + 2 + + x1 x2 x1 αN2 c¯I x1 x2 x1 αN2 αI2 1 αN2 αN2 , such that Technology N displaces Technology I if cN kN ( ( − ) ) ( − )( − )( − ) ) ( )/  + , y D c¯ ,α ,α ; Technology N is used with Technology I if  x1 x2 x1 αI2αN2 N ≤ ( I I2 N2)  ( − ) D c¯ ,α ,α < c +k y S1 2;  α N2 I ≤ sat, I ( I2)  4 ( − ) /  I2    + +  α2q¯  x2 x1 αI2c¯I x1 1 αI2 αN2 x1 c¯I c¯sat, I (A.1)  x 1 α q¯, α q¯ S 2 ( − )( ( − )) ( − ) ,  2 2 2 2  2   − S2 ≤ /  x + x x α r2 q¯  1 2 1 I2 ( ) x S  ( − )  2 2  otherwise,  , α2q¯ > S2 2.   4 / The first and second derivatives of the profit function Π q¯    ( ) where c¯t ct yt for t I, N and c¯sat, I αI2 x2 x1 r q¯ + r q¯ c¯q¯ are given as follows. / ∈ { } ( ) ( − )· 1( ) 2( ) − 1 αI2 αI2. We next consider the case where αN2 > αI2, i.e., Case 1. S 1 α S α . ( − ) 1 2 2 2 Technology I takes the role of B and N the role of A. Using /( − ) ≤ /     2 1 α2 q¯ 2α2q¯  ( − ) + ¯ Theorem1, A is activated iff c¯B > c¯L, where c¯L is given by  x1 1 α2 1 x2α2 1 c,  ( − ) − S1 − S2 − Equation (7) and is a function of c¯A, αA2 and αB2. Noting   S1 that c¯  c + k y , α  α , c¯  c¯ and α  α , and  q¯ A N N N A2 N2 B I B2 I2 dΠ q¯  ( )/   ≤ 2 1 α2 rearranging terms in (7), we obtain that N is activated iff ( )   ( − ) + dq¯  2α2q¯ S1 S2 cN kN yN A c¯I , αI2, αN2 , where A c¯I , αI2, αN2 is again  x α 1 c¯, < q¯ ( )/ ≤ ( ) ( )  2 2 given by (A.1).  − S2 − 2 1 α2 ≤ 2α2  ( − ) (ii) Theorem1 gives the optimal activation strategy for  S2  c¯, q¯ > two arbitrary technologies denoted as A and B with α α  − 2α2 A2 ≥ B2      without loss of generality. We first consider the case where  2 1 α2 + 2α2  x1 1 α2 ( − ) x2 α2, αN2 αI2, i.e., the current technology I takes the role of A  ( − ) − S1 − S2 ≤  and N takes the role of B. Using Theorem1, (i) B displaces A,  S1 2  q¯ i.e., only B is activated, iff c¯ c¯ , and (ii) B complements A, d Π q¯  ≤ 2 1 α B ≤ L ( )  ( − 2) i.e., both are activated, iff c¯ < c¯ < c¯ , where c¯ and c¯ are dq¯2  2α  S S L B H L H  x 2 1 q¯ 2 c¯  c¯   2 α2, < given by (7) and (8), respectively. Noting that A I , αA2  − S2 2 1 α2 ≤ 2α2  +   ( − ) αI2, c¯B cN kN yN and αB2 αN2, and rearranging terms  S ( )/  0, q¯ > 2 in (7) and (8), we obtain that N is a displacing technology if  2α  2 Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 709

Case 2. S 1 α > S α . Using (10), we can write Π q¯ , q¯  r q¯ , q¯ + r q¯ , q¯ 1/( − 2) 2/ 2 ( A B) 1( A B) 2( A B) − c¯ q¯ c¯ q¯ , where     A A − B B  2 1 α2 q¯ 2α2q¯  x ( − ) + x c¯  +   1 1 α2 1 2α2 1 ,  1 αA2 q¯A 1 αB2 q¯B  ( − ) − S1 − S2 −  ( − ) ( − )   x1 1 1 αA2 q¯A  S  − S1 (( − )  2    q¯ r1 q¯A , q¯B + 1 α q¯ , 1 α q¯ + 1 α q¯ S 2 dΠ q¯  ≤ 2α ( ) B2 B A2 A B2 B 1 ( )  2  ( − ) ) ( − ) ( − ) ≤ /    x1S1 dq¯ 2α2q¯ S2 S1  +   , 1 αA2 q¯A 1 αB2 q¯B > S1 2  x2α2 1 c¯, < q¯  4 ( − ) ( − ) /  − S2 − 2α2 ≤ 2 1 α1   ( − )  α q¯ + α q¯   S  A2 A B2 B +  1  x2 1 αA2q¯A αB2q¯B ,  c¯, q¯ >  − S2 ( )  − 2 1 α1   ( − ) r q¯ q¯  +     2 A , B αA2q¯A αB2q¯B S2 2  2 1 α2 + 2α2 ( )  ≤ /  x1 1 α2 ( − ) x2 α2,  x S  ( − ) − S − S  2 2 +  1 2  , αA2q¯A αB2q¯B > S2 2.  S  4 /  2 2  q¯ Π d Π q¯  ≤ 2α The firm’s problem is to maximize q¯A , q¯B subject to q¯A 0 ( )  2 ( ) ≥ dq¯2  2α  S S and q¯B 0.  x 2 α , 2 < q¯ 1 ≥   2 2 (i) Technologies have identical splits, i.e., αA2 αB2.  − S2 2α2 ≤ 2 1 α1  ( − ) By Definition1, Technology A dominates Technology B  S1  0, q¯ > . if cA yA cB yB, and vice versa (because the product-2  2 1 α / ≤ /  ( − 1) splits are identical). The proposition statement then follows directly from Lemma2. It follows that in both cases Π q¯ is concave. Furthermore, it ( ) (ii) Technologiesdiffer in their splits, i.e., αA2 > αB2 because q¯ S q¯ is strictly decreasing for > 2 2α2 (in Case 1) and for > w.l.o.g. the technologies are labelled such that α α . By /( ) A2 ≥ B2 S1 2 1 α1 (in Case 2). Proof of proposition statement then Lemma2, technology j A, B is not activated at optimality if /( ( − )) ∈ { } follows by application of appropriate first-order condition it is dominated by technology i  A, B j, i.e., if c¯ 1 α { }\ i /( − i2) ≤ and subsequent substitution into the profit function.  c¯ 1 α and c¯ α c¯ α , with at least one inequality j /( − j2) i / i2 ≤ j / j2 being strict. If technology j A, B is dominated by technol- Lemma 2 (Horizontal Coproducts Dominance). For i A, B ∈ { } ∈ { } ogy i  A, B j then it is optimal to activate i only and the let j denote the other technology, i.e., j  A, B i. If Technology i { }\ { }\ optimal quantity is given by Proposition1. It then remains to dominates Technology j then there exists an optimal production consider the case in which neither technology dominates the vector q¯ that does not use Technology j, i.e., q¯  0. ∗ ∗j other. Note that for i A, B we have ∈ { } Proof of Lemma2. (Horizontal Coproducts Dominance). From  +   2 1 αA2 q¯A 1 αB2 q¯B the dominance Definition2, Technology i dominates tech-  x1 1 αi2 1 (( − ) ( − ) ) , r q¯ q¯  ( − ) − S ∂ 1 A , B   1 nology j iff ci 1 αi2 yi c j 1 α j2 yj and ci αi2 yi ( ) + /(( − ) ) ≤ /(( − ) ) /( ) ≤ ∂q¯i  1 αA2 q¯A 1 αB2 q¯B S1 2 c j α j2 yj ; or equivalently Technology i dominates technol-  ( − ) ( − ) ≤ / /(  )   0, 1 α q¯ + 1 α q¯ > S 2 ogy j iff c¯i αin c¯j α jn for n 1, 2 using the yield-adjusted  A2 A B2 B 1 / ≤ / ( − ) ( − ) / cost notation and using α  1 α for t  i, j. t1 − t2 and Consider any arbitrary production vector q¯ with q¯ j > 0. ∂2r q¯ , q¯ Construct a new production vector qˆ that is identical to q¯ 1( A B)   +  2 except that qˆ j 0 and qˆi q¯i α jnˆ αinˆ q¯ j , where nˆ ∂q¯i ( / ) 2 arg max  α jn αin . The new quantity available in market 2 1 n 1, 2  αi2 +  { ˆ / }  ( − ) x1 < 0, 1 αA2 q¯A 1 αB2 q¯B S1 2 n 1, 2, i.e., Qn , is weakly larger than the original quan-  − S ( − ) ( − ) ≤ /  +  + 1 tity Qn because Qn qn, i qn, j αin q¯i α jn q¯ j and (by the  0, 1 α q¯ + 1 α q¯ > S 2.  A2 A B2 B 1 dominance definition) c¯i αin c¯j α jn n αin q¯i α jn q¯ j , ( − ) ( − ) / / ≤ / ⇒ ≥ 2 where the last inequality follows from the∀ fact that qˆ  q¯ + In addition, note that ∂ r q¯ , q¯ ∂q¯ ∂q¯  2x 1 α i i 1( A B)/ A B − 1( − A2)· α jnˆ αinˆ q¯ j and recalling that nˆ corresponds to the market 1 αB2 S1 < 0. It follows that the Hessian matrix is negative ( / ) ( − )/ ¯ ¯ segment with largest α jn αin ratio. It then follows directly semidefinite and hence r1 qA , qB is (weakly) jointly concave / q¯ q¯ (i A) B that R∗ Qˆ , Qˆ R∗ Q , Q . The production cost C qˆ  in A and B. Similarly, for , we also have H ( 1 2) ≥ H ( 1 2) ( ) ∈ { } c¯ qˆ  c¯ qˆ + c¯ q c¯ q¯ + c¯ q  C q¯ i i j j i α jnˆ αinˆ j i i j j because (by the x q¯ + q¯ S
( / ) ≤ ( )  2αi2 1 2 αA2 A αB2 B 2 , dominance definition) c¯i αin c¯j α jn for n 1, 2. Therefore ∂r2 q¯A , q¯B  − ( )/ / ≤ / ( )  α q¯ + α q¯ S 2 the profit for qˆ is at least as large as the profit for q¯ . There- ∂q¯ A2 A B2 B ≤ 2/ i  + fore any arbitrary feasible production vector q¯ with q¯ > 0 is  0, αA2q¯A αB2q¯B > S2 2 j  / (weakly) dominated (in the sense of a (weakly) higher profit)  and by some feasible production vector qˆ with q¯ j 0. Therefore, 2  2 + there exists an optimal production vector with q¯  0.  ∂ r q¯A , q¯B x2 2α S2 < 0, αA2q¯A αB2q¯B S2 2 ∗j 2( )  − ( i2/ ) ≤ / ∂q¯2 0, α q¯ + α q¯ > S 2. Proof of Theorem2. (Horizontal Two Technology). Apply- i A2 A B2 B 2/ ing (11) to the two-technology case, the firm’s problem is to In addition, note that ∂2r q¯ , q¯ ∂q¯ ∂q¯  2x α α S . 2( A B)/ A B − 2 A2 B2/ 2 maximize the profit Π q¯ , q¯  R∗ 1 α q¯ + 1 α q¯ , It follows that the Hessian matrix is negative semidefi- ( A B) H (( − A2) A ( − B2) B α q¯ + α q¯ c¯ q¯ c¯ q¯ , where R∗ , is given by (10). nite and hence r q¯ , q¯ is (weakly) jointly concave in q¯ A2 A B2 B)) − A A − B B H (· ·) 2( A B) A Chen, Tomlin, and Wang: Dual Coproduct Technologies 710 Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS

and q¯ . Because Π q¯ , q¯ is a linear combination of r q¯ , q¯ and c¯ α x 1 α 1 α 1 sˆ sˆ ; (ii) dΠ∗ dα < 0 B ( A B) 1( A B) /( 2 2) ≤ − ( 2/( − 2))(( − 2)/ 2) H / 2 and r q¯ , q¯ , it then follows that Π q¯ , q¯ is jointly con- if sˆ < α 1 and c¯ 1 α x 1 1 α α sˆ 1 sˆ , 2( A B) ( A B) 2 2 ≤ /(( − 2) 1) ≤ − (( − 2)/ 2)( 2/( − 2)) cave in q¯ and q¯ . The optimal activation strategy (and and otherwise (iii) dΠ∗ dα  h α f α where h α > 0 A B H / 2 ( 2) ( 2) ( 2) production quantities) can be found by application of the and f α is linear. First, consider the case of c¯ x . We ( 2) ≥ 1 first-order conditions subject to the nonnegative boundary then have c¯ 1 α x > 1 1 α α sˆ 1 sˆ and, in /(( − 2) 1) − (( − 2)/ 2)( 2/( − 2)) conditions. Detailed algebra omitted for reasons of space. In addition, f α > 0 for all 0 α 1. Therefore, dΠ∗ dα > 0 ( 2) ≤ 2 ≤ H / 2 what follows we present the optimal production quantities for all 0 α 1 and α∗ c¯  1. Next consider the case of ≤ 2 ≤ 2( ) when dual activation is optimal, i.e., when c¯ < c y < c¯ c¯ < x . In this case, f α > 0 is linear decreasing in α and L B/ B H 1 ( 2) 2 f  f  dΠ d in part (ii) of the theorem. In this case, optimal yield- α2 α2 sˆ2 > 0 and α2 α2 1 < 0. Therefore, ∗H α2 > 0  ( )| ( )| / adjusted production quantities are given by qA∗ , qB∗ qˆA , qˆB crosses zero (from the positive side) exactly once, and thus  ( ) ( ) Π c¯ is a quasiconcave function. It follows that c¯ if qˆB q¯B and qˆA qA, by qA∗ , qB∗ q¯A , q¯B if qˆB > q¯B, ∗H , α2 α2∗ ≤  ≤ ½ ( ) ( ) ( ) ( ) and by q∗ , q∗ qA , qB otherwise, where is given by the solution to the first-order condition. This ( A B) ( ½ ½ )   is obtained by solving for h α2 0, which yields α2∗ c¯ λ γ µγ λ γ µγ (+ ) ( )  B A B  A B A x1sˆ2 x2 c¯ x2 1 sˆ2 x1 c¯ x1sˆ2 x2 c¯ . This completes qˆA − , qˆB − , ( − )/( ( − )( − ) ( − )) λ λ µ2 λ λ µ2 the proof. A B − A B − S α q¯  B2 Corollary 2 (Horizontal Adoption and Usage Under Identical A 2 α 1 α 1 α α A2( − B2) − ( − A2) B2 (Effective) Costs). Let Technology I (incumbent) and Technology      c¯A αA2 N (new) have the same (effective) yield-adjusted costs, i.e., cI yI sˆ2 1 / 1 sˆ2 , / · − x − ( − ) c + k y . 2 ( N N )/ N S 1 sˆ α sˆ 1 α 1 c¯ α x (i) If αI2 < α2∗ then Technology N is adopted iff αN2 > αI2. Fur- q¯  ( − 2) A2 − 2( − A2)( − ( A/ A2)/ 2) , B thermore, Technology N displaces Technology I if αI2 < αN2 α∗ 2 αA2 1 αB2 1 αA2 αB2 ≤ 2 ( − ) − ( − ) but Technology N is used together with Technology I if αN2 > α2∗ .  S sˆ2 1 αB2 1 sˆ2 αB2 1 c¯A 1 αA2 x1 qA ( − ) − ( − ) ( − ( /( − ))/ ) , (ii) If αI2 > α2∗ then Technology N is adopted iff αN2 < αI2. Fur- ½ 2 αA2 1 αB2 1 αA2 αB2 ( − ) − ( − ) thermore, Technology N displaces Technology I if α2∗ αN2 < αI2 S 1 αA2 ≤  but Technology N is used together with Technology I if α < α∗ . qB − N2 2 ½ 2 αA2 1 αB2 1 αA2 αB2  ( − ) − ( − )   Proof of Corollary2. (Horizontal Adoption and Usage Under c¯A 1 αA2 1 sˆ2 1 /( − ) sˆ2 , Identical Cost). If the new technology is adopted, the firm will · ( − ) − x1 − build a capacity quantity for Technology N that exactly equals

and its planned processing quantity. Therefore, the effective yield- + adjusted cost for Technology N is cN kN yN . By assump- x 1 α 2 x α2 x 1 α 2 x α2  + ( )/ +  1 A2 + 2 A2  1 B2 + 2 B2 tion, cI yI cN kN yN . Define c¯ cI yI cN kN yN . Let λA ( − ) , λB ( − ) , / ( )/ / ( )/ 1 sˆ sˆ 1 sˆ sˆ Π q¯ , q¯ denote the profit if the firm processes q¯ units on − 2 2 − 2 2 ( I N ) I technology I and q¯ units on technology N. Because I and N  x1 1 αA2 1 αB2 + x2αA2αB2 N µ ( − )( − ) , have equal processing costs, i.e., c¯  c¯  c¯, the same profit 1 sˆ2 sˆ2 I N − Π q¯ , q¯ can be obtained by processing q¯  q¯ + q¯ units  S +  S + ( I N ) k N I γA x1 x2 x1 αA2 c¯A , γB x1 x2 x1 αB2 c¯B . on a (hypothetical) single technology k with the same yield- 2 ( ( − ) − ) 2 ( ( − ) − ) adjusted cost c¯ but a split of α  γα + 1 γ α , where k2 I2 ( − ) N2 Recall that the actual production quantities are obtained by γ  q¯ q¯ + q¯ . From Proposition3, there exists an “ideal”  I /( I N ) the conversion qt q¯t yt .  split α for any technology with yield-adjusted cost c¯ such that / 2∗ Proof of Proposition3. (Ideal Market Product-2 Split). Recall the optimal profit is highest at α2∗ . If αI2 and αN2 lie on oppo-  site sides of α∗ then there always exists a blending fraction γ that c¯ c y. Let Π∗ c¯, α2 denote the optimal profit for a tech- 2 / H ( ) such that the two technologies are used in proportions that nology with yield-adjusted cost c¯ > 0 and split α . Π∗ c¯, α is 2 H ( 2) given in Proposition2 and is a continuous and differentiable are equivalent to a single technology with split of α2∗ . If αI2 function of α . Differentiating with respect to α , we obtain and αN2 lie on the same side of α2∗ then the firm cannot attain 2 2 the ideal split and, moreover, should only use the technology   dΠ∗ S c¯ c¯ whose split lies closest to α∗ . This latter point follows from H  sˆ x 1 , 2 2 2 2 the fact (established in proof of Proposition3) that the opti- dα2 2 − α2x2 α x2 2 mal profit (for a given yield-adjusted cost c¯) is a quasiconcave c¯ α2 1 sˆ2 0 α2 < sˆ2 and 1 − function of the split α2. Proof of corollary statements (i) and ≤ α2x2 ≤ − 1 α2 sˆ2   − (ii) then follow.  dΠ∗ S c¯ c¯ H  ˆ 1 s2 x1 1 2 , dα2 − 2 ( − ) − 1 α2 x1 1 α2 x1 Endnotes ( − ) ( − ) 1 c¯ 1 α2 sˆ2 Devices produced on 300 mm wafer have been estimated to be sˆ2 < α2 1 and 1 − ≤ 1 α2 x1 ≤ − α2 1 sˆ2 approximately 30% cheaper than on 200 mm wafer. Future increases Π + ( − ) − in wafer size (e.g., from 300 mm to 450 mm) are predicted to deliver d ∗H S x1 x2 x1 α2 c¯   ( − ) − x x sˆ α significantly smaller savings because the lithography cost scales in dα 2 x sˆ 1 α 2 + x 1 sˆ α2 1 2( 2 − 2) 2 1 2( − 2) 2( − 2) 2 the wafer size and this cost will account for a large portion of the cost + c¯ x 1 sˆ α x sˆ 1 α , otherwise. per device. ( 2( − 2) 2 − 1 2( − 2)) 2 Our discussion focuses on the theoretical operations literature, Recall that c¯ < x + x x α by assumption; else production but we note that there is a long-established and extensive practice- 1 ( 2 − 1) 2 and profit are both zero. Now, (i) dΠ∗ dα > 0 if 0 α < sˆ focused, decision-support literature grounded in the petrochemical H / 2 ≤ 2 2 Chen, Tomlin, and Wang: Dual Coproduct Technologies Manufacturing & Service Operations Management, 2017, vol. 19, no. 4, pp. 692–712, © 2017 INFORMS 711 industry, e.g., Bodington and Baker (1990) and references therein, Chen YJ, Tomlin B, Wang Y (2013) Coproduct technologies: Prod- and the semiconductor industry, e.g., Leachman et al. (1996) and uct line design and process innovation. Management Sci. 59(12): Denton et al. (2006), and these papers reflect coproduct considera- 2772–2789. tions in places. Cooke M (2010) Combinatrion to unlock high yields and throughout in led production? 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