Production Planning under Supply and Quality Uncertainty with Two Customer Segments and Downward Substitution

Transcription

1 Production Panning under Suppy and Quaity Uncertainty with Two Customer Segments and Downward Substitution Tim Noparumpa Whitman Schoo of Management Syracuse University Syracuse, NY 1344 Burak Kazaz Whitman Schoo of Management Syracuse University Syracuse, NY 1344 Scott Webster Whitman Schoo of Management Syracuse University Syracuse, NY 1344 October 14, 011 1

2 Production Panning under Suppy and Quaity Uncertainty with Two Customer Segments and Downward Substitution This paper examines the interreationships among three forms of operationa fexibiities downward substitution, price setting, and fruit trading that are vauabe to an agricutura firm, specificay to a winemaker, operating under suppy and quaity uncertainty. The firm eases farm space to grow fruit and obtains two grades of a fruit that are used in making two different end products of differing quaity sod to two customer segments. The high-grade fruit is downward substitutabe it can be used in the production of the ow-quaity end product. This study makes three sets of contributions to the fied of suppy chain panning under random suppy and quaity. First, we show the interreationships between these three fexibiities. Contradicting earier findings, pricing fexibiity can pay a compementary roe to the downward-substitution fexibiity, increasing its utiization beyond the eves of exogenous price modes. Second, we characterize the impact of these fexibiities on the firm s vineyard ease. The addition of fruit-trading fexibiity reduces the amount of vineyard ease, however, the compementary behavior of pricing and downward substitution can create an incentive for a higher initia investment. Third, the paper demonstrates the infuence of the variation in suppy and quaity and their correation on the amount of vineyard ease, expected profit, expected amount and probabiity of downward substitution. For exampe, variation in quaity does not infuence the probabiity of fruit trading. The firm benefits most from downward substitution in the presence of imited suppy variation and significant quaity variation. Keywords: downward substitution, co-production, suppy uncertainty, quaity uncertainty, pricing, fruit trading 1. Introduction This paper investigates the interactions between three forms of operationa fexibiity downwardsubstitution, pricing and trading fexibiities for an agricutura firm that faces suppy and quaity uncertainty. Our work finds motivation from a boutique winery ocated in the State of New York, and is gaining popuarity for its Pinot Noir wines among wine connoisseurs. The firm eases vineyard in order to grow its fruit. Leasing farm space is common among agricutura businesses (see Kazaz 004, Şaşmaz and Bigiç 010, Kazaz and Webster 011), particuary among wine producers. Unike owning the and, easing farm space is economica for an agro-business because it requires a smaer initia capita investment. As expained by an executive at one of the argest wine producers (and distributors) in the word, easing farm space reduces the potentia negative effects of suppy and quaity probems on the financia performance of the business. For exampe, when the firm obtains a smaer amount of crop, or experiences quaity probems in its grapes, its return on equity is ess affected. Thus, easing farm space is ess risky for the operating environment of the wine producer. We investigate the impact of quaity uncertainty, which aong with suppy uncertainty, is one of the most common chaenges faced by a winemaker. Specificay, we examine the decisions made by the winemaker who obtains two grades of fruit crops (grapes) at the end of a growing season: high-quaity fruit and ow-quaity fruit. The amount of these two grades of crops is uncertain for two reasons. First,

3 suppy uncertainty infuences the overa amount of crop obtained, i.e., the sum of high-quaity and owquaity crops is not known prior to the growing season. Second, quaity uncertainty changes the proportion of high-quaity vs. ow-quaity grades of fruit in the amount of tota grape suppy. Thus, we formuate the probem using two random variabes: one variabe represents the randomness in suppy, corresponding to the random yied of the tota crop, and another random variabe represents the randomness in the proportion of high-quaity versus ow-quaity grapes. We make no assumptions regarding the distribution of these two random variabes. Moreover, we do not require these two random variabes to be independent, and aow them to be correated in our mode. Quaity uncertainty in the fruit suppy creates a natura segmentation for the wine producer. At the beginning of each growing season, this firm eases vineyard to grow its grapes; for the winemaker motivating our probem, this woud be Pinot Noir grapes. At the end of the growing season, the firm obtains two grades of fruit: high-quaity and ow-quaity grapes. The winemaker then produces two different types of end-product (wine). A premium wine is produced by using soey high-quaity grapes, and is marketed towards a customer segment with a higher wiingness to pay. We refer to this customer segment as the high-end market segment. One key characteristic of this market segment is that the priceeasticity of the demand function is significanty ower. 1 A reguar wine is produced for the genera pubic, popuated with simiar products with a ower seing price. We describe these consumers as the ow-end market segment. The reguar wine for the ow-end market is generay produced by using owquaity grapes. Our study investigates the infuence of and the interreationship between the foowing three forms of fexibiities that are present in the ife of a winemaker: 1. Downward substitution fexibiity: The firm can use some of its high-quaity grapes in the making of the ow-end wine. The main emphasis of the paper is the use of downward substitution, and therefore, the study focuses on identifying the conditions under which the firm benefits from this fexibiity. In the anaysis, we report on the expected amount of high-quaity crops used for the making of ow-end product as we as the probabiity of downward substitution.. Pricing fexibiity: The high-end customer segment exhibits a ow price-easticity in its demand function, and the firm determines its seing price for its premium wine sod in the high-end customer segment. Reserve wines are generay considered as premium products as they have unique tastes. For the high-end products such as reserve wines, winemakers can infuence the demand by appropriatey choosing the seing price. The firm does not have the same price- 1 Severa factors contribute to the creation of high-end segment that has consumers with ow price easticity for this winery: recent wins at severa bind-tasting competitions nationwide, a CBS Morning Show coverage for its outstanding Pinot Noir, a positive review from the second-most infuentia critic, Eric Asimov of the New York Times, and a book entited Summer in Gass by Dawson (011). 3

4 setting fexibiity for its reguar wine targeted for the ow-end market segment, which is popuated with many simiar products at a ower price eve. We specificay examine the infuence of the price-setting fexibiity in the high-end segment on the downward-substitution fexibiity. We compare our resuts from an endogenous price mode with those deveoped under a mode that uses exogenous prices. 3. Fruit-trading fexibiity: The firm can purchase additiona fruit from the open market, or se its excess fruit in the open market. This impies that, in the event of ow crop reaizations, the firm can obtain additiona high-quaity and ow-quaity grapes from other growers. Aternativey, in the event of excess fruit suppy, the firm can se its high-quaity and ow-quaity grapes in the open market. We consider the infuence of the fruit-trading fexibiity on the firm s downward substitution decisions. The paper makes three sets of main contributions. First, we show the interactions between these three forms of fexibiities. Whie earier research reports that pricing and downward-substitution fexibiities pay a substitutabe roe, our study proves that these two fexibiities show a compementary behavior. Pricing encourages the firm to downward substitute a greater amount of its high-quaity fruit and exercise it more often. Second, our study shows the impact of these three fexibiities on the firm s choice of initia vineyard ease. Whie fruit-trading fexibiity generay reduces the amount of vineyard ease, the pricing and downward substitution fexibiities can create an incentive for a arger initia investment. Third, the paper demonstrates the infuence of the variance in suppy and quaity and the correation between these two uncertainties on the firm s initia vineyard ease investment, expected profits, expected amount and probabiity of downward substitution. We show that variation in quaity does not infuence the probabiity of fruit trading, and that the firm benefits more from downward substitution under significant variation in quaity and imited variation in suppy. The paper is organized as foows: Section presents a iterature review. Section 3 introduces the mode. Section 4 examines the reationship between the downward substitution and fruit-trading fexibiities with exogenous prices in both market segments. Section 5 demonstrates the infuence of the price-setting fexibiity, Section 6 shows the impact of the three forms of operationa fexibiities on vineyard ease. Section 7 demonstrates the infuence of quaity and suppy uncertainty and their correation using numerica iustrations. Section 8 compares our mode with price-setting in the high-end Participating wineries hep estabish the fruit-trading costs through the support of the Corne University Cooperative Extension prior to the growing season. For exampe, the 010 fruit trading costs for popuar grapes are estabished as foows: igh-quaity Riesing grapes can be purchased at $1900/ton, sod at $1100/ton, whereas owquaity Riesing grapes can be purchased at $1500/ton and sod at $700/ton; high-quaity Chardonnay grapes can be purchased at $1450/ton, sod at $1050/ton, and ow-quaity Chardonnay grapes can be purchased at $100/ton, and sod at $900/ton; high-quaity Cabernet Franc grapes can be purchased at $1500/ton, sod at $800/ton, and owquaity Cabernet Franc grapes can be purchased at $100/ton, and sod at $750/ton. 4

5 segment to previous iterature that aows for price-setting in both segments. Section 9 provides concusions. A proofs are derivations are presented in an onine suppement of Appendices A and B.. Literature Review Earier research in the area of production panning has given particuar interest to soving the optima production probem under suppy uncertainty. Yano and Lee (1995) provide an extensive review on ot sizing probem with random yied. Gerchak et a. (1988) and enig and Gerchak (1990) consider a periodic review production mode with random yied and demand. They provide a detaied anaysis of a singe-period probem and show that the optima production poicy is not affected by yied variabiity. In addition to the above pubications, many studies have focused on the notion of using pricing and production recourse to mitigate suppy and demand uncertainty. Van Mieghem and Dada (1999), Petruzzi and Dada (1999), Dana and Petruzzi (001), Federgruen and eching (1999, 00) and Kocabıyıkoğu and Popescu (011) show that the producer uses production and pricing decisions to mitigate demand risk under deterministic suppy. Furthermore, Van Mieghem and Dada (1999) demonstrate that, under postponed pricing, production postponement has itte benefits to the producer. Whie many have studied the price-setting probem under demand uncertainty, few have investigated the probem under suppy uncertainty. Li and Zheng (006) is the first to consider the price-setting probem under suppy uncertainty. They investigate a singe-product periodic-review mode, where price is set at the beginning of each period, and excess demand is not ost, but backogged. Tang and Yin (007) aso examine a firm s pricing decisions under suppy uncertainty, but imit the anaysis to a inear demand function in a singe market and a discrete uniform distribution representing random suppy. Our study departs from these two studies in four ways: (1) our mode features co-production that eads to the making of two different end-products and market segmentation; () we incorporate quaity uncertainty and emphasize downward substitution; (3) unike the backogged demand feature of Li and Zheng (006), our formuation considers ost saes; and (4) we do not make restrictive assumptions regarding the demand function and distribution of uncertainty in our technica derivations. Moreover, we imit the firm s abiity to set price in one segment aone in order to refect the rea-word scenario of imited number of consumers with ow price easticity. In recent times, there has been an emergence of research that considers the option of utiizing a secondary source of suppy that aows the firm to adjust its production eve. Jones et a. (001) investigate the production panning decisions for the hybrid seed corn production under random yied and demand; they aow the firm to use an externa suppy source after the yied is reaized. Kazaz (004) extends this work by incorporating a yied-dependent cost and seing price in the oive oi industry. Kazaz and Webster (011) incorporate the price-setting and the fruit-trading fexibiities under a yied- 5

6 dependent cost structure. Our paper departs from these studies as it features: (1) a co-production system that eads to market segmentation, () quaity uncertainty, and (3) downward substitution. There is a considerabe amount of studies that investigate co-production systems. Bitran and Dasu (199) investigate the ordering poicies for mutipe items with stochastic yied and substitutabe demand using a dynamic programming formuation. Bitran and Gibert (1994) extend this work by considering the production decisions in the semiconductor industry, and provide severa practica heuristics with conditions for downward substitution decisions. Nahmias and Moinzadeh (1997) aso investigate the probem of downward substitution of randomy-graded yied by formuating a continuous review EOQtype mode. Bassok et a. (1999) consider the production panning probem under downward-substitutabe random demand in a singe period. Their study shows that a greedy aocation poicy is optima, and demonstrates the conditions under which downward substitution is beneficia. su and Bassok (1999) examine a simiar probem by incorporating random yied. Their study shows that optima soutions can be achieved by using severa methods, and, computationay, the greedy agorithm is the most efficient soution approach. One main characteristic that is common among these works in the area of coproduction is that prices are exogenous. Moreover, they ignore the infuence of a secondary source of suppy. Other studies in the area of co-production incude the work of Gerchak et a. (1996), which investigates a parae production process, where one process produces randomy-graded yied, whie the other produces ony ow-grade yied. Ӧner and Bigiç (008) consider products that cannot be substituted, but extend the economic ot scheduing mode to incude uncontroed co-production. Motivated from the beef industry, Boyabati et a. (011), study the procurement probem with fixed proportions technoogy, i.e., the proportion of high-quaity vs. ow-quaity output is fixed. They characterize optima sourcing strategies based on ong-term contracts and procuring from the spot market. Boyabati (011) extends this study on fixed proportions technoogy and investigates the procurement probem with mutipe quantityfexibe contracts, demonstrating the benefits of dua sourcing. Our paper differs from these papers as it features pricing fexibiity and random proportions. Beyond the ream of exogenous price, Bish and Wang (004) investigate the joint quantity and pricesetting probem under perfect suppy and uncertain demand for two products, and show that the firm can benefit from investment in fexibe resources. The cosest match for our study is Tomin and Wang (008) who aso examine the pricing and operationa recourse in a co-production system. They show that the producer benefits more from adopting recourse pricing poicy, i.e., deaying the pricing decision unti after a uncertainty is reaized, than from adopting a downward substitution poicy. Our current work studies a simiar probem to Tomin and Wang (008), but differs from their work in the foowing ways: 6

7 1. We study a production panning probem with co-production that aows for the utiization of the open market. We aso investigate the impact of trading fexibiities on the optima investment, downward substitution, and pricing decisions;. Our work resembes the rea-word scenario that the firm has the abiity to set the seing price ony in the high-end segment of the market as the consumers tend to be ess sensitive to changes in price. 3. Tomin and Wang (008) examine ony the infuence of quaity uncertainty, whereas our study investigates the infuence both suppy and quaity uncertainty, and shows the infuence of both suppy and quaity variation on the optima downward substitution and fruit trading decisions. 4. We do not consider the probem of demand uncertainty as it has been shown in Tomin and Wang (008) that pricing and operationa recourse dominate advance pricing and aocation decisions. 3. Probem Definition and the Mode This section presents the modeing approach used in the agricutura firm that experiences suppy and quaity uncertainty, and produces two different products to serve its two customer segments. The probem is formuated as a two-stage stochastic program. In the first stage, corresponding to the growing season, the firm determines the amount of farm space to be eased, denoted Q, at a unit cost of c in order to maximize expected profit in the presence of suppy and quaity uncertainty. At the end of the growing season, the firm reaizes two grades of fruit infuenced by two separate random variabes. Randomness in the tota crop suppy is represented with a stochasticay proportiona random variabe u, and its reaization is denoted with u defined on a support [u, u h ]. Randomness in quaity refers to the proportion of high-grade versus ow-grade fruit obtained from the eased farm space, and is described by a stochasticay proportiona variabe defined on a support [α, α h ], where α is the reaized proportion of the high-quaity fruit crop and (1 α) is the proportion of ow-quaity fruit crop. Our mode aows for correation to exist between the suppy and quaity random variabes as they foow a joint probabiity density function (pdf) g(u, α) and a cumuative distribution function (cdf) G(u, α). Thus, the first-stage objective function can be written as foows: Q0 max E Q cq E PA Q, u, (1) where PA(Q, u,α) is the optima profit from the second stage given reaizations u and α. At the end of the first stage (growing season), the firm coects two grades of fruit suppy; the reaized amount of high-quaity fruit crop is Quα and the reaized amount of ow-quaity fruit crop is Qu(1 α). Quaity uncertainty creates this natura market segmentation for the winemaker where the firm produces two versions of the fina product in order to serve two customer segments cassified as high-end and owend segments. A premium wine is produced from higher quaity grapes, targeting a high-end customer 7

8 segment that is ess sensitive to the seing price. A reguar wine is produced from the ow-quaity grapes, targeting a more price-sensitive ow-end market segment. The pressing cost of high-quaity fruit to obtain premium wine is defined as c p and the pressing cost of ow-quaity fruit to make reguar wine as c pl. At the beginning of the second stage, the winemaker makes five sets of decisions: the optima vaues of (1) the seing price of high-quaity fina product p, () the amount fruit crop (reaized suppy of highand ow-quaity fruit suppy) to be used in the production of high- and ow-quaity fina products, denoted q I and q IL, respectivey, (3) the amount of additiona high- and ow-quaity fruit to be purchased from other growers in the open market denoted q B and q BL, at unit costs of b and b L, respectivey, (4) the amount of high- and ow-quaity fruit suppy to be sod in the open market without being converted to the fina product denoted q S and q SL at unit seing prices of s and s L, respectivey, and (5) the amount of high-quaity fruit to be downward substituted for the production of ow-end product, denoted w. It is important to note that the vaues of b, s, b L and s L are avaiabe to the firm prior to the growing season (see footnote ). Due to the differences in fruit quaity, we have s > s L and b > b L. In addition, b > s, and b L > s L, which refects the fact that the firm cannot make profit from buying the fruit in the open market and immediatey seing it in the same market (i.e., no arbitrage). As a consequence of the inequaities in open market buying and seing prices, we have the foowing constraints: q I + q S + w = Quα, () q IL + q SL = Qu(1 α). (3) Constraint () states that reaized high-quaity fruit yied is aocated among interna production, open market seing, and downward substitution (i.e., it is never more profitabe to simpy discard fruit rather than seing in the open market, and it is never profitabe to buy high-quaity fruit for the purposes of downward substitution). Simiary, constraint (3) states that reaized ow-quaity fruit yied is aocated among interna production and open market seing. The demand in each customer segment is represented by D (p ) and D L, respectivey. In the high-end customer segment, we assume that the demand is price-sensitive, and is decreasing in p. We denote the inverse of demand function p (D ), and assume that the revenue function in the high-end customer segment (i.e., p (D )D )) is concave, i.e., p (D ) + p (D )D 0. The second-stage probem can be described as maximizing profit from the production and sae of the two end products for a given reaization of high- and ow-quaity fruit, Quα and Qu(1 α), respectivey. PA(Q, u, ) = p, qi, qil, qb, qbl, qs, qsl, w0 p min qi qb, D p cp qi qb b qb sqs max s.t. () & (3) plmin qil qbl w, DL c plqi qb w blqbl slqsl 8

9 = max p, qi, qil, w0 qi min{ D ( p ), Qu } qil min{ DL, Qu(1 )} wqu p cp b D p b s qi squ L pl L L L L IL L 1 L. (4) p c b D b s q s Qu b s w We deveop and anayze eeven variants of the probem in order to identify the interactions among the three forms of fexibiity. We make the foowing assumption regarding profit margins. A1: The firm makes profit from buying ow-quaity fruit, converting it into fina product, and seing the fina product, i.e., p L c pl b L > 0. Simiary, for modes in which the high-end price is exogenous, p c p b > 0. Tabe 1 provides the ist of fexibiities incuded in each of these eeven modes. Fexibiity \ Mode M1 M M3 M4 M5 M6 M7 M8 M9 M10 M11 Downward substitution Fruit trading Pricing in high-end Pricing in ow-end Tabe 1. Fexibiities incuded in each of the eeven mode variants. M1 does not feature any of the three fexibiities, and M8 is the mode described in (1) (4). M and M4 feature the downward substitution fexibiity under exogenous prices, and M6 and M8 under the pricing fexibiity in the high-end segment. M3 and M4 feature the fruit-trading fexibiity under exogenous prices, and M7 and M8 under the pricing fexibiity in the high-end segment. M9, M10 and M11 are deveoped in Section 8 in order to provide a comparison of the pricing fexibiity present ony in the high-end segment, representing the ife of a winemaker, with the hypothetica scenario when the firm can set prices in both segments; M10 corresponds to the mode of Tomin and Wang (008). Before proceeding with the anaysis of the stochastic suppy and quaity probem presented in (1) (4), we briefy examine the properties of the probem with deterministic suppy and quaity. In the deterministic variant of the probem, we repace the suppy random variabe u with its mean u and the quaity random variabe with its mean. The firm eases Q units of farm space, and reaizes highquaity crop yied of Qu and ow-quaity crop yied of Qu(1 ). Eiminating the trading and downward substitution fexibiities, i.e., q B = q S = q BL = q SL = w = 0, the firm converts its entire crop to the fina products. Assuming no demand restriction in the ow-end segment, the seing price in the highend cears the production, i.e., Qu = q I = D (p ), and the firm converts its entire crop of the owquaity fruit to the ow-end product to be sod in the ow-end segment, i.e., Qu(1 ) = q IL = D L. 9

10 Appendix B provides derivations for the optima amount of farm space to be eased and the corresponding profit under deterministic suppy and quaity. The anaysis eads to the foowing observations: (1) Expected profit under stochastic suppy and quaity is ess than that of the deterministic suppy and quaity; () Cosed-form expressions can be provided when a demand function is defined. When demand in each market segment is inear, for exampe, the optima amount of farm space and the corresponding profit under stochastic suppy and quaity decreases in the coefficient of variation, denoted cv[uα]. Under deterministic suppy and quaity, the firm engages in the ease opportunity ony when the unit easing cost is ess than the expected fruit purchasing cost. Remark 1. a) If the unit cost of easing is greater than or equa to the expected buying cost of high and ow-quaity fruit from other growers, i.e., c b Eu b E u1, then the firm reies soey on L fruit purchasing and does not ease vineyard space (Q * = 0). b) If the unit cost of easing is smaer than or equa to the expected fruit seing revenue in the open market, i.e., c s Eu s E u1, then L the firm eases as much as it can because the optima vaue of Q * approaches infinity. c) If 1 1 se u sleu c be u bleu, then Q * > 0 and is finite. Under stochastic suppy and quaity, however, the firm can invest in vineyard ease even if the expected cost of buying fruit is greater than the unit cost of easing. We next proceed with the anaysis of stochastic suppy and quaity. 4. Fruit-Trading Fexibiity and Downward Substitution (with Exogenous Pricing) In this section, we treat price in both market segments as exogenous in order to identify the reationship between fruit-trading and downward substitution fexibiities in the presence of suppy and quaity uncertainty. This is accompished with the comparison of M1 through M The Case of No Trading (Buying or Seing) of Fruit To create a benchmark for the benefits of additiona fexibiities, we begin by investigating a cassic production panning probem under suppy and quaity uncertainty, where the firm does not have the fexibiity to downward substitute or trade once the fruit yied is reaized (M1). We define the foowing regions of suppy and quaity random reaizations for a given ease amount: R1(Q) = {(u, α) : Quα D and Qu(1 α) < D L } R(Q) = {(u, α) : Quα D and Qu(1 α) D L } R3(Q) = {(u, α) : Quα > D and Qu(1 α) < D L } R4(Q) = {(u, α) : Quα > D and Qu(1 α) D L } The firm converts its entire crop yied into the fina product when the reaized high- or ow-quaity crop is ess than their respective demand, i.e., when Quα D or Qu(1 α) D L. This situation is represented by regions R1(Q) and R(Q) for the high-end fruit and R1(Q) and R3(Q) for the ow-end fruit. On the other 10

11 hand, when the reaized yied of high- or ow-quaity crop is high and is greater than the demand, i.e., Quα > D or Qu(1 α) > D L, the firm converts ony the portion of the crop that woud satisfy the demand to the fina product; these are represented by regions R3(Q) and R4(Q) for the high-end fruit and R(Q) and R4(Q) for the ow-end fruit. Using the above definition of four regions of reaized crop suppy, the optima second-stage decisions for M1 can be expressed as foows: q, q * * I IL Qu, Qu 1 if u, R1Q Qu, DL if u, RQ D, Qu 1 if u, R3Q D, D if u, R4Q L We next anayze M, which adds downward substitution fexibiity to M1. Downward substitution is beneficia ony in region R3(Q) where the firm experiences an excess amount of high-quaity fruit and an insufficient amount of ow-quaity fruit. We denote the shortage in the ow-end as, i.e., = D L Qu(1 ), and divide region R3(Q) into the foowing sub-regions: R3a(Q) = {(u, α) : D < Quα D + and Qu(1 α) < D L } R3b(Q) = {(u, α) : D + < Quα and Qu(1 α) < D L } Region R3a(Q) represents a situation in which the excess yied of high-quaity fruit is not sufficient to cover the shortages of the ow-end fina product and thus the firm converts a the excess high-quaity fruit into ow-end fina product, i.e., w * = Quα D. Region R3b(Q) represents the scenario in which there is a high yied reaization of high-quaity crop and thus the firm converts a portion of the remaining high-quaity fruit to satisfy the demand of ow-end fina product, i.e., w * = = D L Qu(1 α). Figure 1 iustrates the uses of the high-end fruit with the boundary between R3a and R3b at (D + )/Q. Using the above four regions of reaized crop suppy, the optima second-stage decisions for M are: Qu,0, Qu 1 if u, R1Q Qu,0, DL if u, RQ D,, Qu 1 if u, R3bQ D,0, D if u, R4Q * * * qi, w, qil D, Qu D, Qu 1 if u, R3a Q. (5) L 4.3 Incorporating Fruit-Trading Fexibiity (Buying q B,q BL 0 and Seing q S,q SL 0) We next incorporate the fexibiity for the firm to trade fruit in the open market without downward substitution, as featured in M3. In this scenario, it foows from assumption A1 that the firm buys fruit * from the open market when the reaized amount of internay grown fruit is ess than the demand, i.e., q B 11

12 = D Quα 0 and qbl * = D L Qu(1 α) = 0. Aternativey, when the reaized amount of fruit crop exceeds the desired demand eve, then the firm ses the unused crop in the open market, i.e., q * S = Quα D 0 and q * SL = Qu(1 α) D L = 0. Accordingy, thee optima second-stage decisions for M3 are: Figure 1. Optima downward substitution quantity under exogenous pricing. * * q, q, q * * q, q, q * I B S * IL BL SL Qu, D Qu,00 Qu 1,,0 if u, R1Q Qu, D Qu,00 if u, RQ, DL,0, D, 0, Qu D, Qu 1,,0 if u, R3Q D, 0, Qu D, if u, R4Q DL,0, Next, we anayze M4 where the firm has both the fexibiity to downward substitute and trade fruit in the open market. In this mode, the downward substitution option is ony viabe when savings from the utiization of high-quaity fruit crop in the making of the ow-enif and ony iff s < b L. Otherwise, downward substitution does not occur as it is more beneficia for the firm to se the excess crop in the open market. It is product outweighs the seing price of high-quaity crop in the open market, i.e., w * > 0 importantt to note that s < b L for the winemakers motivating our study (seee popuar grapes prices in footnote ). Therefore, to investigate the benefit from downward substitution, for the remainder of this paper, we assume s < b L. The objective functionn in (4) can be rewritten as:. 1

13 p cp b D b sqi squ L pl L L L L IL L b L smin Qu D, DL Qu 1 PAQ, u, max p c b D b s q s Qu 1 p, qi, qil0 qi min D p, Qu qilmin DL, Qu1 Simiar to the case where there is no trading option, the firm benefits from downward substitution when the reaization of high-quaity crop is high and there is an insufficient amount of ow-quaity fruit. In region R3a(Q), the excess amount of high-quaity crop is smaer than the shortage in the ow-quaity fruit, and thus, the firm benefits from downward substitution, i.e., w * = Quα D, and saves (b L s )(Quα D ) from purchasing additiona ow-quaity fruit from the open market. On the other hand, in region R3b(Q), the suppy of high-quaity crop is sufficienty high to cover the shortage in the ow-quaity fruit; specificay, w * = D L Qu(1 α) = with a resuting savings of (b L s ). Accordingy, the optima second-stage decisions for M4 can be expressed as foows: Qu, D Qu,0,0, Qu 1,,0 Qu, D Qu,0,0, DL,0, * * * * qi, qb, w, qs, D, Qu D,0,0, * * * q,, DL, DL Qu D,0 IL qbl q SL D,0,,0, Qu 1,0,0 D,0,0, Qu D, DL,0, u Q if, R1 u Q if, R u aq if, R3 u bq if, R3 u Q if, R4. (6) It is common wisdom that the introduction of an additiona form of fexibiity, as in the form of fruittrading fexibiity, woud reduce the utiization of other forms of fexibiity (e.g., downward substitution) present in the environment (e.g., Van Mieghem and Dada 1999, Jones et a. 001, Kazaz 004, and Tomin and Wang 008). owever, as shown in the foowing proposition, the additiona fexibiity to trade fruit in the open market does not infuence the probabiity of downward substitution and the expected amount of downward substitution. Thus, in the absence of the pricing fexibiity, these two forms of fexibiity neither present a substitutabe roe, nor pay a compementary roe to each other. Proposition 1. In the absence of pricing fexibiity, for a given Q, the probabiity of downward substitution and the expected amount of downward substitution does not change with the additiona fexibiity of fruit-trading in the open market. 5. The Combination of Downward Substitution, Pricing, and Fruit-Trading Fexibiities In this section, we deveop the structura properties of M5, M6, M7 and M8, where the firm has the 13

14 pricing fexibiity in the high-end segment. 5.1 Price-Setting Fexibiity in the igh-end Segment and Downward Substitution We begin our anaysis by assuming that the firm does not have the abiity to acquire or se fruit in the open market, or downward substitute its high-quaity fruit for the production of its ow-end product, which corresponds to M5, i.e. q B = q S = q BL = q SL = w = 0. Under the price-setting fexibiity in the highend market segment, the amount of high-quaity fruit reaization infuences the pricing and quantity decisions. When the reaized amount of high-quaity fruit is high, the firm has the abiity to set the profitmaximizing price and convert ony the amount of fruit that corresponds to the demand at the profitmaximizing price. On the other hand, when the high-quaity fruit reaization is imited, the firm converts a the reaized suppy into the fina product and ses at the market cearing price. In the case of ow-end product, the optima production decision foows our anaysis of the case presented in Section 4.1. In the foowing proposition, we define a threshod for the production amount in the high-end segment. The threshod, denoted TP, is the optima amount of high-end product to produce when there is no constraint on the suppy of high-quaity fruit. Proposition. The threshod for the amount of high-end product to be produced from the interna * * resource for M5 istp p c p D p. We use the threshod amount to define the foowing regions of suppy and quaity random reaizations for a given ease amount and high-end seing price: R1(Q) = {(u, α) : Quα TP and Qu(1 α) < D L } R(Q) = {(u, α) : Quα TP and Qu(1 α) D L } R3(Q) = {(u, α) : Quα > TP and Qu(1 α) < D L } R4(Q) = {(u, α) : Quα > TP and Qu(1 α) D L } In regions R1(Q) and R(Q), the firm sets the high-end price to cear the market, p (Qu). In regions R3(Q) and R4(Q), the firm has excess suppy of high-quaity fruit and sets the high-end price to se the threshod quantity. The optima second-stage quantity decisions for M5 are q, q * * I IL Qu, Qu 1 if u, R1Q Qu, DL if u, RQ TP, Qu1 if u, R3Q TP, D if u, R4Q L and the optima high-end price is p * = p (q * I ). We next investigate M6 which incorporates downward substitution in addition to the pricing fexibiity in the high-end segment. The second-stage probem in M6 can be rewritten as: 14

15 p, qi, qil0 qi min D p, Qu q min D, Qu1 p I L pl IL I L IL PA( Q, u, ) max p c q p c q min Qu q, D q IL L When Qu(1 ) D L, which corresponds to regions R(Q) and R4(Q) above, the ow-end market has sufficient suppy; there is no downward substitution and the optima decisions that appy in R(Q) and R4(Q) for M5 are aso optima for M6. To consider the case of Qu(1 ) < D L, we require a threshod quantity. Reca that TP is the optima high-end production quantity for M5 when there is no imit on high-end suppy. We simiary define a threshod production amount for M6. In particuar, TP denotes the optima high-end production amount when high-quaity fruit that is not used for high-end production gains unit profit p L c PL through downward substitution. The threshod in the presence of downward substitution is smaer than the threshod without downward substitution. Proposition 3. The threshod for the amount of high-end product to be produced from the interna D * * resource for M6 is TP p c p c D p TP. p L pl Reca that = D L Qu(1 ) is the ow-end shortage amount. We repace regions R1(Q) and R3(Q) with the foowing sub-regions: R1a(Q) = {(u, α) : Quα R1b(Q) = {(u, α) : R3a(Q) = {(u, α) : D TP D TP and Qu(1 α) < D L } D < Quα TP and Qu(1 α) < D L } D D TP < Quα TP + and Qu(1 α) < D L } R3b(Q) = {(u, α) : TP + < Quα and Qu(1 α) < D L } An interesting transition occurs between region R1b(Q) and R3a(Q). In region R1b(Q) the firm is abe to produce the optima high-end threshod quantity under downward substitution ( TP D ), then downward substitute the baance to satisfy a portion of the shortage in the ow-end segment (). In region R3a(Q), the firm has more than enough to cover aocated D TP and the shortage. owever, once the firm has D TP to high-end production and has downward substituted the quantity, the change in profit associated with aocating more voume to the high end segment is positive, and thus the firm aocates the baance of high-quaity fruit to high-end production (up to TP ; see Figure ). 3 Accordingy, the optima second-stage quantity decisions for M6 are 3 This is the optima aocation because, in the event that the tota aocated to the high end is ess than TP, the firm woud ose profit if a portion of the downward substitution amount is shifted to high-end production (foows from the definition oftp ). D 15

16 * * qi w,, q * IL Qu,0, Qu 1 D D TP, Qu TP, Qu 1 Qu,0, DL Qu,, Qu 1 TP,, Qu 1 TP,0, DL and the optima high-end price is p * = p (q * I ). In order to assess the impact of pricing fexibiity on downward substitution, we compare M6 where the firm is free to set the high-end product price with M where price is exogenous. To isoate the effect of pricing fexibiity, we set the exogenous high-end product price to the price that maximizes the high- p = end product profit when the ow-end is ignored, i.e., the exogenous high-end product price for M is p (TP ). The foowing proposition shows that pricing fexibiity in the high-end segment eads to a higher probabiity of downward substitution and a higher expected amountt of fruit utiized in the making of the ow-end product. if u, R1a Q if u, R1b Q if u, R Q if u, R3a Q if u, R3b Q if u, R4 Q (7) Figure. Optima downward substitution quantity under endogenous pricing. Proposition 4. For a given Q, the price-setting fexibiity in the high-end segment increases the probabiity of downward substitution and the expected amount of fruit downward substituted. 5. Price-Setting Fexibiity in the igh-end Segment and the Fruit-Trading Fexibiity M7 features the fruit-trading fexibiity in the presence of pricing fexibiity in the high-end segment. We begin our anaysis by anayzing the firm s abiity to buy and se fruit in the open market independenty. Simiar to the exogenous mode, the firm woud benefit from buying additiona fruit from the open market 16

17 when the fruit suppy of high- or ow-quaity crop is ow. On the other hand, when the suppy of the highor ow-quaity fruit is high, the firm can use the open market to gain additiona revenue from seing its excess fruit crop. It shoud be noted here that, because the firm does not set price in the ow-end segment, the structura properties pertaining to this segment decisions remain the same with those deveoped under exogenous price. The foowing proposition estabishes a threshod for seing high-quaity fruit in the open market, denoted TS, and another threshod for buying high-quaity fruit from the open market, denoted TB. Proposition 5. The threshod for the amount of high-quaity fruit to be sod in the open market is TS p c s D p * * p and the threshod for the amount of high-quaity fruit to be purchased in the open market is * * TB p cp b D p TS. When high-end fruit suppy is beow TB, the firm purchases up to TB in the open market. When high-end fruit suppy is above TS, the firm ses the excess in the open market. As noted above, the rues for open market buying and seing of ow-quaity fruit foow the rues for M3. This eads to six regions of suppy and random quaity random reaizations (see Figure 3). R1(Q) = {(u, α) : Quα TB and Qu(1 α) < D L } R(Q) = {(u, α) : Quα TB and Qu(1 α) D L } R3(Q) = {(u, α) : TB < Quα TS and Qu(1 α) < D L } R4(Q) = {(u, α) : TB < Quα TS and Qu(1 α) D L } R5(Q) = {(u, α) : Quα > TS and Qu(1 α) < D L } R6(Q) = {(u, α) : Quα > TS and Qu(1 α) D L }. Accordingy, the optima second-stage quantity decisions for M7 are Qu, TB Qu,0 if Qu TB q q q Qu TB Qu TS TS,0, Qu TS if Qu TS * * * I, B, S,0,0 if q, q, q * * * IL BL SL and the optima high-end price is p * = p (q I * + q B * ). 1,,0 if 1,0, if 1 Qu Qu D DL Qu D L L 17

18 Figure 3. Different regions of u, α reaization under endogenous pricing and trading in M7 (Q in the figure refers to high-quaity fruit and LQ refers to ow-quaity fruit). We next compare M7 and M3. To provide a fair comparison, we set the exogenous price of M3 in the high-end segment to be in the interva of p (TB ) and p (TS ). The foowing proposition shows thatt the firm engages in fruit trading ess frequenty in the presence off the price-setting fexibiity in the high-end segment. The resut indicates that price-setting fexibiity in the high-end segment and the fruit-trading fexibiity pay a substitutabe roe in the ife of a winemaker. Proposition 6. For a given Q, the price-setting fexibiity in the high-end segment decreases the probabiity of fruit trading and the expected amount of fruit trading. 5.3 Price-Setting in the igh-end Segment, Fruit-Trading g and Downward-Substitution Fexibiities M8, as presented in (1) (4), features a three fexibiities: price-setting in the high-end segment, fruit- market has sufficient suppy; there is no downward substitution and the optima decisions that appy in trading, and downwardd substitution. The second-stage objective function in (4) can be rewritten as: p c p s qi squ PA( Q, u, ) max pl c L qil min p, qi, qil 0 Qu qi, DL qil qi mind p, Qu qilmin DL, Qu 1 bl s min Qu qi, DL qil When Qu(1 ) D L, which corresponds to regions R(Q), R4(Q), and R6(Q) above, the ow-end R(Q), R4(Q), and R6( (Q) for M7 are aso optima for M8. D To consider the case of Qu(1 ) < D L, we require a threshod quantity. Reca thatt TP is the optima high-end production amount when high-quaity substitution. We simiary define a threshod fruit that is not used for high-end production gains unit profit p L cpl through downward production 18

19 amount for M8. In particuar, TP denotes the optima high-end production amount when high-quaity fruit that is not used for high-end production saves the open market purchase cost b L through downward substitution (the firm prefers to downward substitute over seing high-quaity fruit in the open market because s < b L ). The production threshod in the presence of downward substitution the threshod without downward substitution (TS ). TP is smaer than Proposition 7. The threshod for the amount of high-end product to be produced from the interna resource for M8 is and and * * TP p c b D p < TP, p L TB TP TS TP, D TP TP. Reca that = D L Qu(1 ) is the ow-end shortage amount. We repace regions R3(Q) and R5(Q) with the foowing sub-regions: R3a(Q) = {(u, α) : TB < Quα R3b(Q) = {(u, α) : R5a(Q) = {(u, α) : TP < Quα TP and Qu(1 α) < D L } TP and Qu(1 α) < D L } TP < Quα TS + and Qu(1 α) < D L } R5b(Q) = {(u, α) : Quα > TS + and Qu(1 α) < D L } Simiar to M5, an interesting transition occurs between region R3b(Q) and R5a(Q) (see Figure 4). In region R3b(Q) the firm is abe to produce the optima high-end threshod quantity under downward substitution with trading fexibiity ( TP ), then downward substitute the baance to satisfy a portion of the shortage in the ow-end segment (). In region R5a(Q), the firm has more than enough to cover and the shortage. owever, once the firm has aocated TP TP to high-end production and has downward substituted the quantity, the change in profit associated with aocating more voume to the high end is positive, and thus the firm aocates the baance of high- quaity fruit to high-end production. Accordingy, the optima second-stage quantity decisions for M8 are 19

20 q q, q, w, q * * * * I B S * * * IL, qbl, qsl and the optima high-end price is p * = p (q * I + q * B ). We next compare M4 and M8 in order to examine the effect of the price-setting fexibiity on the two other fexibiities. Simiar to the comparison between M and d M6, we fix the seing price in the high-end segment for M4 to be equa to the profit maximizing price p (TP ). The foowing proposition showss that in the presence of fruit-trading fexibiity, pricing fexibiity inn the high-end segment eads to a higher probabiity of downward substitution and a higher expected amount of fruit utiized in the making of the ow-end product. Qu, TB Qu,0,0, Qu 1,,0 if u, R1Q Qu, TB Qu,0,0, DL,0, if u, RQ Qu,0,0,0, Qu 1,,0 if u, R3aQ TP,0, Qu TP,0,, Qu 1, Qu TP,0 if u, R3bQ Qu,0,0,0, DL,0, if u, R4Q Qu,0,,0, Qu 1,0,0 if u, R5aQ TS,0,, Qu TS, Qu 1,0,0 if u, R5bQ TS,0,0, Qu TS, Qu 1,0, if u, R6Q Proposition 8. For a given Q, the price-setting fexibiity in the high-end segment increases the probabiity of downward substitution and the expected amount of fruit downward substituted. (8) Figure 4. Optima downward substitution quantity under endogenouss pricing and fruit-trading. Reca that when a firm does not have pricing fexibiity, fruit-trading and downward-substitution fexibiities are neither compements nor substitutes (i.e., the amount and ikeihood of downward 0

21 substitution does not change when fruit-trading fexibiity is introduced; see Proposition 1). To assess the reationship between fruit-trading and downward-substitution fexibiities in the presence of pricing fexibiity in the high-end segment, we next compare M6 and M8. Proposition 9. a) For a given Q, in the presence of price-setting fexibiity in the high-end segment, the downwardd substitution threshod with fruit-trading fexibiity is higher than the downward substitution threshod without fruit-trading segment, fruit-trading fexibiity decreases thee probabiity of downward substitution and fexibiity; b) For a given Q, inn the presencee of the price-setting fexibiity in the high-end the expected amount of fruit downward substituted. The above proposition shows that the winemaker benefitss more from downward substitution in the absence of fruit-trading fexibiity. Because the firm engages in downward d substitution at an earier reaization of high-quaity fruit in the absence of fruit-trading fexibiity, it experiences a higher probabiity of downward substitution and utiizes a greater (expected) amount of grapes for downward substitution. Furthermore, the above proposition shows that with the presence of the price-setting fexibiity, fruit-trading and downward-substitution fexibiityy pay a substitutabe roe. This resut contradicts the earier finding in the absence of the price-setting fexibiity.. Proposition 1 has shown that the fruit-trading fexibiity does not infuence the probabiity of downward substitution and the expected amount of downward substitution in the absence of price-setting fexibiity. owever, when the price- setting fexibiity is incuded in the high-end segment, Proposition 9 showss that fruit trading and downwardd substitution fexibiities pay a substitutabe roe. Figure 5. The reationship between downward-substitution, fruit-trading and price-setting fexibiities. Figure 5 provides a summary of the reationship between the three forms of fexibiities presented in this study. From Figure 5, and our anaysis in this section, it iss cear that price-setting fexibiity in the high-end segment and downward substitution fexibiity pay a compementary roe with or without the fruit-trading fexibiity. Our study proves that the winemaker benefits more by engagingg in downward substitution at an earier high-quaity crop reaization in the presence of the price-setting fexibiity. In 1

22 Section 8, we compare our mode to a mode that aows for price setting in both market segments, and anayticay demonstrate its effect on downward substitution. 5.4 The Impact of Quaity and Suppy Uncertainty This section investigates the impact of increasing variance in suppy or quaity uncertainty on the probabiity of downward substitution and fruit trading. We begin our discussion with downward substitution. Proposition 8 has estabished that price-setting fexibiity increases the ikeihood of downward substitution. In order for the firm to engage in downward substitution, the high-quaity fruit reaization has to be greater than D in modes M and M4, TP D in M6, and TP in M8. Let us denote TDS j the threshod point for downward substitution in mode j {M, M4, M6, M8}. The foowing proposition describes how the probabiity of downward substitution changes with increasing variance in either u or when the other random variabe is fixed at its mean. The proposition appies to any probabiity distribution that can be standardized, i.e., random variabe X with mean and standard deviation can be written as X = + Z where Z is the corresponding standardized random variabe with mean 0 and standard deviation 1. The cass of distributions that can be standardized incudes distributions such as norma (standardized pdf = (z) = 1 e 0.5z, z (-, )), truncated norma (standardized pdf = (z) = 1 e 0.5z / a zdz, z [-a, a]), and uniform (standardized pdf = (z) = 1 a 3, z 3, 3 ). We et u and denote the standard deviation of u and, respectivey. Proposition 10. For a probabiity distribution that can be standardized: a) When u = = 0, u > TDS j /Q, and u 1 < D L /Q for j {M, M4, M6, M8}, the probabiity of downward substitution is equa to 1, and the probabiity of downward substitution is non-increasing in u (with = 0) and in (with u = 0). b) When u = = 0, u < TDS j /Q or u 1 > D L /Q for j {M, M4, M6, M8}, the probabiity of downward substitution is equa to 0, and the probabiity of downward substitution is nondecreasing in (with u = 0). Let us denote the probabiity that the firm engages in fruit trading as P(FT > 0) when at east one of the four decision variabes reated with fruit trading q B, q S, q BL, or q SL takes a positive vaue. It is important to remind that, when quaity uncertainty is ignored as in earier pubications (e.g. Kazaz 004, Kazaz and Webster 011), the firm does not engage in fruit trading with probabiity 1 under significant suppy uncertainty. Considering the high-end fruit as the ony product in the mode, this means that fruit trading does not occur when TB < Qu < TS ; and, when the suppy random variabe shows significant variation, it is cear that 0 < P(FT > 0) < 1. owever, as shown in the foowing proposition, fruit trading occurs with probabiity 1, i.e., P(FT > 0) = 1 in M3, M4, and M7. Thus, the probabiity of fruit trading is