Capacity, pricing and production under supply and demand uncertainties with an application in agriculture

European Journal of Operational Research 275 (2019) 1037–1049

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

Capacity, pricing and production under supply and demand uncertainties with an application in agriculture Amirmohsen Golmohammadi a, Elkafi Hassini b,∗ a b

Faculty of Management, Laurentian University, Canada DeGroote School of Business, McMaster University, Canada

a r t i c l e

i n f o

Article history: Received 4 December 2017 Accepted 17 December 2018 Available online 22 December 2018 Keywords: Operational Research in agriculture Uncertain yield uncertain supply Lot-sizing capacity planning

a b s t r a c t We consider capacitated joint lot-sizing and pricing problems when supply and a price-sensitive demand are uncertain. In cases where there is excess capacity, the decision maker has the option to rent her capacity. We model the case where price and production quantity are determined before the yield is realized (joint decisions) as well as the case where the price is determined after the yield is known (sequential decisions). In the joint decisions case, we introduce the concept of expected demand fill rate elasticity and characterize the conditions in which a one-sided production and pricing policy is optimal. We extend the results to the case when there is a fixed production cost and show that a two-sided production and pricing policy is optimal. We also investigate the conditions under which it is optimal for the decision maker to rent her capacity. In addition, we look at the case in which the decision maker is risk averse and analyze the effect of the risk attitude on the optimal price and production quantity. We extend our model to the multi-period case and the case where there is a lead time to acquire new capacity. We show that a one-sided production, pricing and capacity planning policy is optimal in this situation as well. We apply our model to the agricultural sector and present numerical examples using data from the California almond industry. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The models in this paper can be applied in both manufacturing and agricultural settings. Despite the importance of the agricultural sector in reducing poverty (Cervantes-Godoy & Dewbre, 2010; Ligon & Sadoulet, 2008) and its contribution to economic development (Diao, Hazell, Resnick, & Thurlow, 2007; Vogel, 1994), there are many operations management aspects of the agricultural sector that require further investigation (Lowe & Preckel, 2004). Indeed, despite support from policy makers, such as the Canadian government’s “Business Risk Management Programs,” (Martin & Stiefelmeyer, 2011), the agribusiness industry still has a lot to learn and apply from the fields of demand and production management (Taylor, 2006). In our study of the agribusiness supply chain we have observed three major trends. First, while it is recognized that integration of the internal cross-functional operations is a prerequisite for coordination of the external supply chain operations (Braunscheidel & Suresh, 2009; Stevens & Graham, 1989), the agribusiness supply

∗

Corresponding author. E-mail addresses: [email protected] (A. Golmohammadi), [email protected] (E. Hassini). https://doi.org/10.1016/j.ejor.2018.12.027 0377-2217/© 2019 Elsevier B.V. All rights reserved.

chain is lagging on this front when compared with other sectors such as the manufacturing and retail supply chains. In particular, we think integration of production planning and sales is one of the important areas that would benefit from a systematic joint optimization. Agricultural supply chains used to suffer from excess stock as the overall global production often surpasses the demand. However, during the last few years the rate of demand growth is accelerating and quickly closing the gap between supply and demand (Trostle, 2008). This phenomenon creates a competitive market further stressing the importance of sales management. Second, while farmers are traditionally considered to be price takers, there is a growing movement that calls for farmers to move toward being price makers (e.g., Boersma, 2015; Cox, 2010; Hogan, 2015; Wiesemeyer, 2010). There is also a trend of consolidation in the agribusiness industry. Big companies have started buying farmlands and this is causing a shift in power where farmers can play an important role in setting prices (The Economist, 2015). Finally, in the agribusiness supply chain there are two major players: farmers and handlers. Handlers are the firms that purchase the agricultural products from the farmers and sell them to the wholesalers and/or retailers. There is now an understanding that farmers and handlers should cooperate and share risks, especially the yield risk which is now mainly the responsibility of farmers only.

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The main characteristic of agricultural production is the uncertainty of yield. Based on a United States Department of Agriculture (USDA) report, for a typical agricultural product, the coefficient of variation of year to year yield is about 0.3 (Roberts, Osteen, & Soule, 2004). The uncertainty in the demand side further exacerbates uncertainty in the agribusiness supply chain. Even staple foods such as potatoes have uncertain demand (Taylor, 2006). To mitigate the effects of these uncertainties, farmers often rent out the farm to the handler to avoid dealing with the demand and supply uncertainties (Kunkel, Peterson, & Mitchell, 2009). Thus, the handler bears the uncertainty of supply and the spot market demand. Although the farmer benefits from reducing the production risks (especially that of demand), she may lose the opportunity of making higher profits by selling later in the season at high prices. Thus, the farmer faces a trade-off between gaining control over prices and assuming the costs of mismatch between demand and supply due to their uncertainties. The aim of this paper is therefore to develop efficient production and pricing policies for a farmer to optimally manage uncertainty in both supply and demand. The farmer may choose to rent out the farm in the case where there is not enough demand for internal production. To capture these features we develop a single period model and extend it to cases where there is a fixed production cost or penalty cost. The basis of our model is a newsvendor problem with pricing under uncertainty of demand and supply. We propose a unifying framework for this problem. We model the case where price and production quantity are determined before the yield is realized (joint decisions) and extend our model to the case where the price is determined after the yield is known (sequential decisions). The joint decisions situation is common in make-to-order production systems. The sequential pricing case is also applicable in manufacturing systems, but may be more so in the agribusiness sector. In the joint decisions case, we introduce the concept of Expected Demand Fill Rate (EDFR) elasticity as an extension of the Lost Sales Rate (LSR) elasticity, developed by Kocabiyikoglu and Popescu (2011) when only the demand is uncertain. We extend our results to the multi-period case where the farmer may decide to change her production capacity by acquiring new land or selling a portion of the existing land. In addition, we investigate the problem under the case where the decision maker is risk averse and analyze how the decision maker’s attitude toward risk affects the optimal production and pricing decisions. Previously it has been demonstrated that in the presence of supply uncertainty, it is difficult to find a proper structure for the optimum price (Feng, 2010; Li & Zheng, 2006). The difficulty stems from the complexity that results from having to jointly decide about price and quantity in the presence of joint uncertainty in demand and supply. For example, when only demand is uncertain there have been different approaches in analyzing the resulting joint ordering and pricing problem. One of these approaches is to transform the problem through a change of variables to one that is easier to analyze (e.g., see Petruzzi & Dada, 1999). Another approach is to develop conditions on the demand distribution (e.g., increasing failure rate as in Petruzzi & Dada, 1999; and increasing generalized failure rate in Lariviere & Porteus, 2001) or problem decision variables (e.g., LSR elasticity in Kocabiyikoglu & Popescu, 2011). In this study we show that by conditioning over EDFR elasticity a proper structure for the optimum price can be found. To summarize, one of our major contributions is the extension of the LSR elasticity to EDFR elasticity. We show, in the mathematical proofs, the usefulness of this extension in establishing operational policies. A second contribution is the application of dynamic pricing and capacity management to the important, but perhaps less mainstream, agribusiness sector. Finally, we apply the EDFR elasticity concept to the single newsvendor model and offer a unifying analysis framework in the presence of uncertainties in both

demand and supply. Our results offer a more general framework, that encompasses, among others, the results found in Petruzzi and Dada (1999) and Kocabiyikoglu and Popescu (2011). The paper is organized as follows: in Section 2 we review the most relevant papers to our study. In Section 3 we provide the description of the problem and analyze the single period model. In Section 4 we look at the multi-period case. Section 5 is devoted to the case where the pricing and lot-sizing decisions are made sequentially. Section 6 considers the case in which there is a leadtime for production and we conclude the paper in Section 8. 2. Literature review There is a growing body of literature on lot-sizing under uncertain supply and demand. Yano and Lee (1995) reviewed studies that have considered the lot-sizing problem with uncertain yield. However, there is a relatively smaller portion of the literature that is devoted to joint lot-sizing and pricing under uncertain yield and demand. Golmohammadi and Hassini (2014) reviewed this literature and proposed a framework for classifying it. He (2013), Surti, Hassini, and Abad (2013), Xu and Lu (2013), Kazaz and Webster (2015), Li, Li, and Cai (2015), and Lu, Xu, Chen, and Zhu (2018) investigated the single period joint ordering and pricing under uncertain demand and supply. He (2013) assumed that the lot-sizing and pricing decisions are made sequentially. They assumed a linear demand function and a stochastically proportional yield, and found conditions under which the optimum solution is unique. Li et al. (2015) investigated a variation of the study of He (2013) in which the demand function is iso-elastic and its uncertainty is multiplicative. They focused on how variability of supply and demand affects the sequence of the decisions. Surti et al. (2013) examined the case in which the ordering and pricing decisions are made simultaneously as well as sequentially. They also looked at cases where demand may be deterministic allowing them to obtain insights into the impact of supply uncertainty. Kazaz and Webster (2015) examined the effect of risk averseness on the newsvendor problem with pricing and focused on the case that the uncertainty of yield is stochastically proportional and the expected value of yield rate is equal to one. They compared the optimal prices and production quantities in risk neutral and risk averse cases when either demand or supply are deterministic. Xu and Lu (2013) considered the possibility of using emergency orders to satisfy any demand shortage after the realization of yield and demand. In the single period case of our study, we look at the case where the decisions are made simultaneously and consider generalized demand and yield functions. Lu et al. (2018) investigated the production and pricing decisions of a company that produces two substitutable products. They assumed that the yield of one of the products is uncertain and demand is a linear function of price. They focused on developing an efficient algorithm to solve the problem. All the aforementioned studies, focused on the case where yield is stochastically proportional and demand has a specific structure. In our study we consider more general functions for supply and demand and also, we extend our study to the multi-period case. In addition, we provide more general insights about the characteristics of the problem. Some studies explored the joint lot-sizing and pricing problem in a multi-period case. Yan and Liu (2009) and Zhu (2013) considered the case where supply uncertainty is caused by disruption in the supply process. In both studies the demand function is assumed to have additive uncertainty and the unsatisfied demand is backlogged in each period. Chao, Chen, and Zheng (2008) assumed that the uncertainty of supply lies in the production capacity. They focused on the case where the demand has a linear form with additive uncertainty and capacity is stochastically increasing

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and concave. They proved that the base-stock list-price policy is optimal. Their work has been extended by Feng (2010) where it is assumed that the demand has a general form, but the uncertainty is additive, and there is no limitation on the uncertainty of capacity. It is shown that a re-order point policy is optimal for the replenishment strategy. Another extension to the case of a multisupplier case was done by Feng and Shi (2012). They showed that a base-stock list-price policy is optimal when the suppliers are reliable, however this result cannot be extended to the uncertain supply case. Yang, Chen, and Zhou (2014) extended the study of Feng (2010) to the case in which the orders can only be made in batches. They provided the conditions under which a base-stock list-price policy is optimal. Li and Zheng (2006) investigated the case where the uncertainty of supply is stochastically proportional to the yield. They also found that although a base-stock list-price policy is optimal for the certain yield case (Federgruen & Heching, 1999), this result does not hold when supply is uncertain. Chen, Feng, and Seshadri (2013) extended the previous study to a dual supplier scenario. They showed that the optimal ordering policy is characterized by a threshold below which it is not optimal for the firm to order. They also extend their findings to the case in which the pricing and lot-sizing decisions are made sequentially. Chao, Gong, and Zheng (2016) investigated a variation of the study of Chen et al. (2013) in which one of the suppliers is reliable. They showed that the firm uses a combination of base-stock and reorder point policies to replenish its inventory. All the above mentioned multi-period studies assumed that the unsatisfied demand is back-ordered. This assumption relatively simplifies the theoretical analysis of the problem. Also their results suggest that, under uncertainty of yield, finding a structured pricing policy is more cumbersome in comparison to a deterministic yield case. In this study we assume that the unsatisfied demand is lost which is a more realistic assumption in several environments and in particular that of the agricultural products market. By introducing the concept of expected demand fill rate (EDFR) elasticity we were able to find a condition in which a one-sided production, pricing and capacity planning policy is optimal. There are a few papers that consider lot-sizing and pricing problems in an agricultural environment. Jones, Lowe, Traub, and Kegler (2001) considered optimization of hybrid seed corn production. Kazaz (2004) looked at the special case of the production of olive oil. Both studies considered two stage models and assumed that the price is given. In addition to production quantity decision, Tan and Çömden (2012) included time of seeding in their model. They considered a multi-period problem in which the uncertainty of yield is normally distributed. The problem of land and irrigation allocation in a single period case is studied by Huh and Lall (2013). Tan and Çömden (2012) and Huh and Lall (2013) did not consider pricing in their models. Kazaz and Webster (2011) considered joint lot-sizing and pricing in an agricultural environment when the demand is deterministic but yield is uncertain. They assumed that the firm can either produce its own supply or purchase it from other farmers. The purchasing decision, as well as the pricing decision, happen after the realization of the production yield. Our study extends theirs in different ways. First, we consider a multi-period context. Second, our model is not restricted to an agricultural environment and we consider the scenario where the decision about the price happens before the realization of the yield and simultaneously with the production decision. Finally, in our model the farmer has a chance to rent out her farm to mitigate the risk of demand and supply uncertainties. Our work is also related to studies in the area of joint production and capacity planning. For example, Porteus, Angelus, Wood et al. (20 0 0) consider a finite horizon capacity planning problem where demand is uncertain. They assumed that the firm can only expand the capacity and does not have the option to shrink it in

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the case that they predict a low future demand. They also assumed that the unsatisfied demand is lost at a penalty cost. The capacity expansion cost includes both fixed and variable costs. They proved that the (s, S) policy is optimal to increase capacity up to St in period t, whenever the cumulative capacity is below st . Angelus and Porteus (2002) looked at a similar model where they added production decisions and the firm can shrink its capacity. They considered two cases depending on whether inventories are carried over to the next period or not. In both cases the optimal capacity policy has two thresholds; capacity is expanded if its level is below first threshold, remain unchanged if between two thresholds, and shrunk if it is above the second threshold. The optimal production policy has a critical fractile form when no inventory is carried over and a base-stock policy when inventory is carried over. In addition to incorporating pricing and uncertain yield, in our paper we assume that the firm can rent out the additional unused capacity. 3. Problem description and modelling We consider joint lot-sizing and pricing problems faced by a farmer. We assume that the size of the farmer’s land is limited and represented by z. In each period, the farmer has to make two decisions. The first one is the size of the land, q, that she wants to cultivate at a unit production cost c. The remaining land, z − q, will be rented by the farmer at a unit rate r. The demand for renting the farm in each period is represented by X, which varies in [k, k] and is random with ψ and  as its p.d.f. and c.d.f., respectively. Since the production yield is highly volatile in agricultural environments, the production quantity in each period is a random variable U(q, Y) where Y is the uncertain factor of the yield, with p.d.f. g and c.d.f. G, and varies in [l , l ]. The farmer’s second decision is to determine the price of its products, p ∈ [ p, p]. We consider a price sensitive and uncertain demand per period, D(p, V), where V is the random factor of demand with p.d.f. φ and c.d.f.  and varies in [v, v] Due to long lead times and large market sizes in the agribusiness supply chain, it is reasonable to assume that unsatisfied demand is lost. We assume that there is a penalty cost, b, for each unit of unsatisfied demand and that the excess production is salvaged at a rate h for each unit of overproduction where h ≤ c ≤ p. Accordingly the expected profit function of the farmer is as follows:

f ( p, q ) = pE [min {D( p, V ), U (q, Y )}] + hE [max {U (q, Y ) − D( p, V ), 0}] − bE [max {D( p, V ) − U (q, Y ), 0}] + rE [min {z − q, X }] − cq

(1)

In Eq. (1), the first term represents the sales revenue, the second term is the salvage revenue from the excess production, the third term is the unsatisfied demand penalty cost, the fourth term represents the revenue from renting out the excess capacity, and the last term is the production cost. With some calculations, we can rewrite the objective function as follows:

f ( p, q ) = R( p, q ) + hE [U (q, Y )] − bE [D( p, V )] + rE [min {z − q, X }] − cq

(2)

where

R( p, q ) = ( p − h + b)E [min {D( p, V ), U (q, Y )}]

(3)

and the profit maximization problem can be modelled as follows:

(z ) = max p,q

s.t.

f ( p, q ) q ≤ z.

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For analytical tractability we consider the following assumptions that are common in the literature (e.g., Petruzzi & Dada, 1999; Kocabiyikoglu & Popescu, 2011; Pang, 2011). Assumption 1. We assume that U(q, Y) is an increasing function of Y and q. In addition we assume it is a continuous, twice differentiable and concave function of q. A special case that satisfies Assumption 1 is the stochastically proportional yield where U (q, Y ) = Y q. Assumption 2. Consider d ( p, v ) as the deterministic demand and τ ( p, v ) = pd ( p, v ) as the revenue function when demand is deterministic. We assume d ( p, v ) is a decreasing function of p and an increasing function of v. Also we assume τ ( p, v ) is a concave function of p for all the values of v. Let u(q, y) represent the deterministic form of the yield function. In order to facilitate the theoretical analysis of the problem, we introduce v( p, u(q, y )) and y(q, d ( p, v )) as follows:

d ( p, v( p, u(q, y ))) = u(q, y )

(4)

u(q, y(q, d ( p, v ))) = d ( p, v ).

(5)

If the observation of the uncertain part of demand takes the value of v( p, u(q, y )) then the supply matches the demand and the farmer will not experience any overproduction or underproduction. This is a generalized form of the stocking factor introduced by Petruzzi and Dada (1999). Similarly, if the observation of the uncertain part of yield takes the value of y(q, d ( p, v )), supply and demand will be equal. One of the goals of this study is to find the optimum production and pricing policies with regard to the current available land. To this end we introduce two classes of policies. Definition 1 (One-Sided Production and Pricing Policy). A production and pricing policy, (q∗ , p∗ ) is one-sided if given z, the available acreage, there exist z° such that q∗ = min{z, z◦ }, p∗ = max{P (z ), P (z◦ )}, where P(z) is a non-increasing function of z. In the one-sided production and pricing policy, the amount of optimal production, q∗ , is characterized by a threshold, z°. In this policy, if the available acreage is less than the threshold z°, the farmer puts all the available land under cultivation. If the available acreage is more than the threshold z° then the farmer puts z° units of land under cultivation and rents out the rest of the land. The behaviour of the optimal price is similar to the list-price policy. In this policy, the optimal price is a decreasing function of q∗ . If the available land is less than z°, then as the size of the available land increases, the optimal price decreases due to the fact that the optimal production quantity increases. We use the function P(z) to represent the decreasing behaviour of the optimal price with respect to the available acreage. The shape of P(z) is dependent on the structure and the parameters of the problem. When the available acreage exceeds the threshold z°, increasing it will not change the price. The latter remains constant at P(z°), since the optimal production quantity is also constant at z°. Definition 2 (Two-Sided Production and Pricing Policy). A production, pricing and capacity planning policy, (q∗ , p∗ ), is two-sided if given z, the available acreage, there exist two threshold values, n and N, where

q∗ =

⎧ ⎨0

z ⎩ N

⎧

if z ≤ n ⎨P (n ) if n < z < N , p∗ = P (z ) ⎩ if N ≤ z P (N )

and P(z) is a non-increasing function of z.

if z ≤ n if n < z < N if N ≤ z

In the two-sided production and pricing policy, the optimal production quantity is characterized by two thresholds, n and N. When the available acreage is less than the lower threshold, n, the farmer will not produce anything and will rent out all the available acreage. If the available acreage is between the lower threshold and the upper threshold, the farmer will put all the available land under cultivation. If the available acreage is more than the upper bound, then the farmer puts N unit of land under cultivation and rents out the rest of the available acreage. Similar to the one-sided policy, in this policy, the optimal price is a decreasing function of the optimal production quantity q∗ . When the available land is less than n, since q∗ is constant, the optimal price is constant and equal to P(n). When the available acreage is between the lower and upper thresholds, as the size of the available land increases, the optimal price decreases due to the fact that the optimal production quantity increases. We use the function P(z) to represent the decreasing behaviour of the optimal price with respect to the available acreage. The shape of P(z) is dependent on the structure and the parameters of the problem. When the available acreage exceeds the threshold N, increasing it will not change the price. The latter remains constant at P(N), since the optimal production quantity is also constant at N. We will provide comments on the intuition behind these policies in the context of the models that we discuss later in the paper. 3.1. Expected demand fill rate elasticity The concept of LSR elasticity was first introduced by Kocabiyikoglu and Popescu (2011) in the context of a newsvendor problem with pricing under deterministic yield. Some other studies, such as de Vericourt and Lobo (2009), Chen, Zhou, and Chen (2011) and Pang (2011), have also used conditioning over LSR elasticity to obtain some properties for their objective functions. Here we generalize this concept by incorporating the uncertainty of yield. For any values of p and q, we define s(p, q) as the expected amount of demand that is filled by slightly changing the values of p and q. In other words, s(p, q) represents the slope of the demand satisfaction function in terms of p and q. We call s(p, q) the expected demand fill rate (EDFR) and it is equal to





¯ v( p, U (q, Y ))) , s( p, q ) = E Uq (q, Y )(

(6)

in which Uq (q, Y) is the derivative of U(q, Y) with respect to q and ¯ . ) = 1 − (. ). The term ( ¯ v( p, U (q, Y ))) defines the probability ( of demand exceeding supply. Let w = p − h + b be the unit lost sale cost. EDFR elasticity determines the variation of EDFR with respect to variation of w for a given quantity and it can be defined as

( p, q ) =

−ws p ( p, q ) , s( p, q )

(7)

where sp (p, q) is the first derivative of s(p, q) with respect to p. EDFR elasticity can be considered as the generalization of the lost sale rate elasticity introduced by Kocabiyikoglu and Popescu (2011). For the case that U (q, Y ) = q the EDFR elasticity and LSR elasticity are equal. 3.2. Single period model with no penalty cost In this section we assume that the producer does not tolerate any penalty cost for unsatisfied demand (b = 0). This assumption is reasonable when the market size is much larger than an individual producer’s capacity. Note that the structure of our problem is that of a newsvendor problem with pricing under uncertainty of demand and supply. Consequently the following results can be applied in more general situations of style goods and perishable items where both demand and supply can be random.

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As it was mentioned earlier, de Vericourt and Lobo (2009), Chen et al. (2011) and Pang (2011) applied conditioning over LSR elasticity to find some nice structural results for the newsvendor problem with pricing when yield is certain. Uncertainty of yield limits the application of LSR elasticity in our problem. However, we show in Proposition that conditioning over the EDFR elasticity enables us to find some interesting structural results for our problem.

Table 1 Behaviour of z° with respect to the parameters of the problem.

Proposition 1. Let p∗ (q) and q∗ (p) be the optimal price and quantity for a given q and p, respectively. We have: (a) For all p and q satisfying (p, q) ≥ 1/2, f(p, q) is jointly concave in p and q. (b) For all q satisfying (p∗ (q), q) ≥ 1/2, f(p∗ (q), q) is concave in q. (c) For all p satisfying (p, q∗ (p)) ≥ 1/2, f(p, q∗ (p)) is concave in p.

According to Theorem 1, if the mentioned condition is satisfied, we put all the available land under cultivation unless z > z°. In this case, q∗ = z◦ and the rest of the land is available for renting. When z ≤ z°, the optimum price decreases as the available land increases. When z passes z°, the optimum price remains constant and it is equal to P(z°). As indicated in Theorem and Definition, the threshold z° characterizes the behaviour of the optimal production quantity and price in the one-sided production and pricing policy. In the next proposition, we look at the behaviour of this threshold with regard to the parameters of the problem.

The proof of Proposition and subsequent proofs are included in the Online Supplement. Proposition provides the conditions that guarantee the uniqueness of the optimal solution. Under those condition we can use first order conditions to find the optimal price p and land size q. In Proposition we offer a different condition that guarantees the unimodality of the objective function. Proposition 2. If (p, q) is increasing in q, then (p∗ (q), q) ≥ 1. Proposition simplifies the analysis of the problem. In the case that we have a more specific structure for the problem, more detailed results can be found based on this proposition. In Corollary we consider the case where uncertainty of yield is additive and demonstrate that when V and Y have special distributions the objective function is unimodal. Corollary 1. Let U (q, Y ) = u(q ) + Y . Then: (a) If V is IFR (has an increasing failure rate) and d p /dv is decreasing in v then f(p, q) is unimodal. (b) If Y is DFR (has a decreasing failure rate) then f(p, q) is unimodal. The uniform and Erlang distributions are examples of distributions that have an increasing failure rate. The Weibull distribution with a scale parameter β satisfying 0 < 1/β < 1 is DFR. The exponential distribution is both IFR and DFR. In addition, in Appendix B, we provide examples where EDFR elasticity is an increasing function of q. Propositions and provide conditions that guarantee the uniqueness of the solution. In order to provide a better insight about the optimum solution, we need to characterize the behaviour of the optimum land size with respect to z. Proposition 3. (a) The optimum expected profit is an increasing function of z. (b) The optimum value of q is an increasing function of z. According to Proposition, the farmer benefits from having a larger land at the beginning of the period. This is understandable when we see that in our context the size of land plays the role of production capacity. As a result more available land means more opportunity for production and renting out the farm. Consider q˜(z ) as the optimum production quantity which is a function of available land. Then according to Proposition, part (b), q˜(z ) is a nondecreasing function of z. Finding similar structural properties for the optimal price would require more limiting conditions. The next Theorem provides conditions in which the one-sided production and pricing policy is optimal. Theorem 1. If (p, q) ≥ 1, for all the values of p and q, then the onesided production and pricing policy is optimal.

z°

p↑

h↑

c↑

r↑

¯ ↑ 

¯ ↑

↑

↑

↓

↓

↑

↓

Proposition 4. Consider z° as the threshold in the one-sided production and pricing policy. Table 1 shows the behaviour of z° with respect to the parameters of the problem. According to Proposition, as the market becomes more attrac¯ ), the farmer tive (higher selling price and stochastically higher  tends to put a larger portion of the land under cultivation. Consequently, z° increases. In addition, a more attractive renting option or an increase in production costs leads the farmer to reduce the amount of production and results in a lower value for z°. 3.2.1. Produce or rent? The first decision that the farmer has to make is to decide whether to start production or rent out all the available farm. The farmer can avoid any risk that is related to production or demand by renting out all the available farm. In this situation the farmer does not have to worry about the optimum price and quantity. However starting her own production may provide a situation where the farmer can increase her profit by setting a proper price and putting enough land under cultivation. In the next proposition we provide the conditions under which it is not optimal for the farmer to start her own production. Proposition 5. It is not optimal to produce if and only if for all the values of p the following inequality holds:

¯ z ). ( p − h )s( p, 0 ) + hE [Uq (0, Y )] ≤ c + r(

(8)

If u(0, y ) = 0, ∀y, then (8) can be simplified to

¯ z ). pE [Uq (0, Y )] ≤ c + r (

(9)

Condition (8) is intuitive; it states that when it is optimal to not produce, the marginal increase in revenue should not exceed the sum of the marginal production cost and the marginal decrease in revenue with regard to renting the farm. It is interesting to note that when u(0, y ) = 0, ∀y, condition (9) is not dependent on the salvage value. Although the price in the secondary market affects the production quantity, it has no effect on the decision of whether or not to start production. 3.2.2. Expected value analysis In the pursuit of tractability, studies have avoided to explicitly model uncertainties by assuming either demand (Silver, 1976) or supply (Petruzzi & Dada, 1999) is deterministic. As indicated in our farming example in the introduction, this approach leads to suboptimal outcomes where the shortage and overage costs may be underestimated. We are particularly interested in studying the impact of ignoring uncertainty on the farmer’s production decision. ¯¯ z ) be the optimal expected profit when yield is ¯ z ) and ( Let (

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certain and the optimal expected profit when demand is certain, respectively. Denoting the overproduction rate under the certain demand case by ξ ( p, q, v ) = P r {U (q, Y ) ≥ d ( p, v )}, we can establish the following result in Lemma. ¯ z) ≥ Lemma 1. (a) If u(q, y) is concave with respect to y, then ( (z ). ¯¯ z ) ≥ (z ). (b) If d ( p, v ) is concave with respect to v, then ( (c) If

dv ( p, v )ξv ( p, q, v ) ≤ −1 dvv ( p, v )ξ ( p, q, v )

∀ p, q, v,

(10)

¯¯ z ) ≥ (z ). then ( (d) If d ( p, v ) is convex with respect to v and

dv ( p, v )ξv ( p, q, v ) ≥ −1 dvv ( p, v )ξ ( p, q, v )

∀ p, q, v,

(11)

¯¯ z ) ≤ (z ). then ( Lemma 1 shows that using a certain yield model, when u(q, y) is concave, will result in overestimation of the profit. It also shows that, under some condition, the certain demand case may result in either overestimation or underestimation of the profit. Misleading values of the expected profit can also cause the farmer to deviate from the optimal decision. In Proposition we investigate how ignoring uncertainty can affect the production decision. To that end ¯ z) we let f¯( p, q ) and f¯¯ ( p, q ) be the functions that correspond to ( ¯¯ z ), respectively. and ( Proposition 6. Assuming that u(0, y ) = 0, ∀y we have (a) If u(q, y) is concave with respect to y, then f¯q ( p, 0 ) ≥ fq ( p, 0 ). (b) If d ( p, v ) is concave with respect to v or,

dv ( p, v )ξvt ( p, q, v ) ≤ −1 dvv ( p, v )ξ ( p, q, v )

∀ p, q, v,

then f¯¯q ( p, 0 ) ≥ fq ( p, 0 ). (c) If d ( p, v ) is convex with respect to v and

dv ( p, v )ξv ( p, q, v ) ≥ −1 dvv ( p, v )ξ ( p, q, v )

, ∀ p, q, v,

then f¯¯q ( p, 0 ) ≤ fq ( p, 0 ). According to Proposition part (a), under some conditions, by using certain yield model, the farmer may start production while it is optimal to just rent out all the available farm. The same situation may happen when the farmer is using a certain demand model (Proposition part (b)). From part (c), we deduce that the farmer may rent out all the available farm while it is better to have some production. 3.3. Single period model with penalty cost Here, we assume that every unit of unsatisfied demand would be penalized. Adding penalty cost complicates the analysis of the problem. For the sake of simplicity, in this section we assume that the uncertainty of demand has an additive form so that D( p, V ) = d ( p) + V . The random variable V is the random factor of demand and without loss of generality we assume E[V ] = 0. Consider p(d) as the reverse function of d(p). Since d(p) is a decreasing function of p, then there is a one to one relation between demand and price. Based on the upper and lower bounds of the price, we can define the upper and lower bounds for d accordingly. Assume that d ∈ [d, d]. Therefore, instead of the demand function, we can use price as a function of demand in the form of P (d, V ) = p(d ) + V . Using the price function, the profit function is as follows:

f (d, q ) = R(d, q ) + hE [U (q, Y )] − bd + rE [min {z − q, X }] − cq (12)

where

R(d, q ) = ( p(d ) − h + b)E [min {d + V , U (q, Y )}].

(13)

To study the value function we define the lost sale rate elasticity in the deterministic yield case as follows:

(d, q ) = −

( p( d ) − h + b ) φ ( q − d ) . ¯ q − d) pd (d )(

In the following proposition we establish conditions under which the profit function is concave. Proposition 7. If  (d, q) ≥ 1 and pdd (d ) + pd (d )d ≤ 0, ∀d, q, then f(d, q) is jointly concave with respect to d and q. In Proposition 7 we used the same conditions that are used in the study of Pang (2011) to prove the optimality of the (s, S) policy. Proposition 7 bridges the results of the certain yield case to the uncertain yield case. Note that in Proposition 7 we assumed that d(p) has a general structure. It is easy to show that for some more restricted demand functions, such as the linear case, the conditions mentioned in Proposition 1, part (a), are sufficient to prove the concavity of f(p, q) in the presence of a penalty cost. We also note that there are some situations in which EDFR elasticity is more than 0.5 while LSR elasticity is less. An example of this situation is provided in Appendix B of online supplement. Therefore we cannot claim that EDFR elasticity conditions imply those of the LSR elasticity. In the next theorem we show when a one-sided policy is optimal. Theorem 2. If the conditions of Proposition 7 are satisfied and (d, q) ≥ 1 for all the values of d, q then a one-sided production and pricing policy is optimal. In the presence of a penalty cost we require more restricted conditions to prove the optimality of the one-sided production and pricing policy. The structure of the result is the same as the one that is described for Theorem. In terms of the production decisions, the model with a penalty cost shows almost the same behaviour as that with no penalty costs. We note that the results of Theorems and generalize previous results in the literature (such as de Vericourt & Lobo, 2009; Chen et al., 2011; Pang, 2011) where supply was assumed to be deterministic. As in those studies, it is not easy to relate the condition in Theorem and to the probability distribution of demand and yield. Remark 1. It is not optimal to start production, if and only if for all the values of p the following inequality is satisfied:

¯ z ). ( p(d ) − h + b)s( p, 0 ) + hE [Uq (0, Y )] ≤ c + r(

(14)

If u(0, y ) = 0, ∀y, then (14) can be is simplified to

¯ z) ( p(d ) + b)E [Uq (0, Y )] ≤ c + r(

(15)

The proof is analogous to that of Proposition 5. Similar to the case of no penalty cost model, we note that when u(0, y ) = 0, ∀y, the salvage value has no effect on the decision of starting production. However, we see that the penalty cost plays a role. Having a larger penalty cost motivates the farmer to start production to avoid the costs of unsatisfied demand. 3.4. Single period model with fixed production cost In some situations the farmer may have to acquire some facilities and equipment for starting production. We consider M as the fixed production cost that accounts for the cost of acquiring these facilities. We define λ(q):



λ (q ) =

0 1

if q = 0 , if q > 0

A. Golmohammadi and E. Hassini / European Journal of Operational Research 275 (2019) 1037–1049 Table 2 Behaviour of n and N with respect to the parameters of the problem.

N n

1043

p↑

h↑

b↑

c↑

r↑

¯ ↑ 

¯ ↑

Since the analysis of the problem under risk aversion can be cumbersome, in this section, we focus on a standard newsvendor problem with pricing under uncertain supply and demand. Therefore, the random profit function will be defined as follows:

↑ ↓

↑ ↓

↑ ↓

↓ ↑

↓ ↑

↑ ↓

↓ ↑

f˜( p, q ) = pE min {D( p, V ), U (q, Y )} − cq

We define profit function as follows:

f ( p, q ) = R( p, q ) + hE [U (q, Y )] − bE [D( p, V )] + rE [min {z − q, X }] − cq + Mλ(q ).

(16)

In the presence of fixed production costs, production becomes less profitable. As a result we expect the farmer to follow a different policy. Theorem 3. Let b = 0. If U (0, Y ) = 0 and (p, q) ≥ 1, ∀p, q, then the two-sided production and pricing policy is optimal. According to Theorem, when we have a fixed production cost it is optimal to not start production unless the available land has a certain level n. Also if z is greater than n but less than N all the available land will go under cultivation. The corresponding price will decrease as z increases. If z > N, then the optimal size of the land for cultivation is N and the rest will be rented out. As a result we keep the optimal price constant for all the values of z greater than N. In the presence of a penalty cost, the conditions for optimality of two-sided production, pricing and capacity planning policy become more restrictive. The following remark addresses these conditions. Remark 2. Let b > 0. If the conditions of Theorem 2 are satisfied, then a two-sided production and pricing policy is optimal. The proof is analogous to Theorem 3. Remark 2 extends Theorem 3 when there is a penalty cost. In this situation the farmer will not start production until she is certain that the expected profit from production can compensate for the fixed cost. Two threshold values, n and N, characterize the behaviour of the optimal price and production in the two-sided production and pricing policy. In the next proposition, we analyze the behaviour of these two values with respect to parameters of the problem. Proposition 8. Consider N and n as the upper and the lower threshold values in the two-sided production and pricing policy. Table 2 shows the behaviour of N and n with respect to the parameters of the problem. According to Proposition, as the market becomes more attractive, the farmer relies more on the production option. Also, a higher penalty cost and salvage value leads the farmer to produce more. On the other hand, when the renting option becomes more attractive and a production cost increases, the farmer tends to reduce the amount of production. 3.5. Risk averse decision maker The attitude of the firm toward risk is one of the main factors that can affect its optimal decisions. Accordingly, analyzing the effects of the firm’s risk attitude on its optimal strategies has been the subject of several studies (Bruno, Ahmed, Shapiro, & Street, 2016; Merzifonluoglu, 2015; Rubio-Herrero & Baykal-Gürsoy, 2018; Sawik, 2016; Wu, Bai, & Zhu, 2017; Zhou, Li, & Zhong, 2018). Risk averseness is considered as one of the pillars of logical decision making (Grechuk & Zabarankin, 2014). Hence, in this section, we focus on the case in which the decision maker is risk averse.

To model the risk aversion, we use utility function B(x) where B (x) > 0 and B (x) ≤ 0. Then, the objective function in this situation will be A( p, q ) = E[B( f˜( p, q ))]. Also, we define R+ ( p, q, Y ) = pU (q, Y ) − cq as the random profit when demand surpasses the supply. First, we look at how risk averseness affects the uniqueness of the solution. Proposition 9. (a) A(p, q) is concave with respect to p. (b) A(p, q) is concave with respect to q. According to Proposition 9, Part (a), the objective function is concave with respect to p. Consequently, we can use the first order condition to find the optimal price with respect to q and subsequently, solve the problem to find the optimal production quantity. According to Part (b), we have a similar case for the optimal production quantity. Therefore, for any value of price, there is a unique order quantity that maximizes the objective function. Consider p∗ (q) as the optimal price with respect to q and q∗ (p) as the optimal production quantity with respect to p in the risk neutral case. Similarly, consider p°(q) as the optimal price with respect to q and q°(p) as the optimal production quantity with respect to p in the risk averse scenario. In the next proposition, we look at how risk averseness affects the optimal pricing and production strategies of the decision maker. Proposition 10. Assume U (q, Y ) = u(q ) + Y . (a) If E[B (R+ ( p, q∗ ( p), Y )) ≤ 1], then q°(p) ≤ q∗ (p). (b) If τ p (p∗ (q), ) ≤ 0, ∀ ∈ [v, v( p∗ (q ), U (q, l ))] and E[B (R+ ∗ ( p (q ), q, Y )) ≤ 1], then p°(q) ≤ p∗ (q). The joint uncertainty of supply and demand and the general structure of the supply and demand functions makes the analysis of the problem complex. Hence, for Proposition 10, we focus on the case where U (q, Y ) = u(q ) + Y . In Part (a) of Proposition, we show that if E[B (R+ ( p, q∗ ( p), Y )) ≤ 1], then a risk averse decision maker tends to produce less in comparison to a risk neutral decision maker. The proposed condition is satisfied for several types of utility functions. For instance, for an exponential utility function where B(x ) = 1 − e−κ x , if κ ≤ 1, then the condition is satisfied. In this situation, the risk of overproduction leads the decision maker to produce less in comparison to risk neutral case. In Part (b), we observe a similar behaviour with respect to the optimal price. If the proposed conditions are satisfied, then the risk averse decision maker is sensitive to risk of overproduction and consequently, she tends to charge a lower price for the produced items. 4. Multi-period case In the past few years, larger companies have started to invest in agricultural production (The Economist, 2015). In addition, there is evidence that show farmers actively look for real estate options to buy and sell lands (Dyck, 2017). In such situation, one aspect that significantly impacts the production and pricing strategies is the capacity of production. In this section, we extend our results to the multi-period finite horizon case where the farmer can change the production capacity in each period. Similar to the single period case, the farmer has to decide about the size of the land that she wants to put under cultivation in addition to the price of products. Also here we assume that the farmer can change capacity of production in each period. According to the

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demand forecast, the farmer might want to increase her production capacity or sell a portion of the land in order to maximize her profit. We consider at and ot as the amount of land that is acquired and sold in period t, respectively, and we assume that the land is cultivable in the next period. Consider mt as the available amount of land at the end of period t. Note that mt = zt + at − ot . We assume at ≤ at and ot ≤ ot where at represents the upper bound on the amount of land that the farmer can acquire, and ot represents the upper bound on the amount of land that the farmer can sell in period t. In the case that farmer cannot acquire any land in a particular period we set at = 0. Similarly, in the case that the farmer cannot sell lands in a particular period we set ot = 0. The unit purchase cost of acquiring new farm land in period t is represented by δ t . The selling price of land in period t is ϱt To simplify the exposition, in the sequel we will omit the time index, from the decision variables, whenever their omission does not create confusion to the reader. Similar to the single period case, and based on the nature of the agricultural environment, here we assume that the unsatisfied demand is lost. Also due to inventory issues and the perishable nature of agricultural products, we assume that the farmer does not keep any inventory and would sell the excess production in a secondary market. Finally, we assume that demand random variables Dt are independent. Considering α as the discount factor, a dynamic programming formulation of our problem is

f t (z ) =

t

J (z, p, q, a, o)

max 0 ≤ q ≤ z, 0 ≤ p, a ≤ a, o ≤ o

in which



t



t

t

t

t

− b E D ( p, V ) − U (q, Y ) t

t





t

− ct q + rt E min z − q, X t

max{m , z − o }



− ct q

+ α E f t+1 (U (q, Y t ), z + a − o) .



(17)

if zt ≤ mt t

if mt < zt < m t

+

+

Theorem 4. (a) Let bt = 0 and Mt = 0 ∀, t. If t (p, q) ≥ 1/2, ∀t then ft (z) is concave for all the values of t. (b) Let bt = 0 and Mt = 0 ∀, t. If t (p, q) ≥ 1, ∀t, then one-sided production and pricing policy is optimal. Also for the optimal capacity planning policy, consider two constants mt and mt in each period where mt ≥ mt . Then m∗t is defined as follows:

t



− δ t a + t o + rt E min z − q, X t

In the next Theorem we extend the results that we have in Propositions 1 and 7 and Theorems 1–3 to the multi-period case.

t





+

(18)

zt







f T (z ) = δ T z.

⎩

J (qˆ, z, p, q, a, o)

+ ht E qˆ − Dt ( p, V t )

At the end of the planning horizon, we assume that the unit value of the available land is δ T so that

m∗t =

t

max ≤ q ≤ z, 0 ≤ p a ≤ a, o ≤ o





− δ t a + t o + Mt λ(q ) + α f t+1 (z + a − o).

⎧ t ⎨min{mt , zt + a }

f t (qˆ, z ) =

Jt (qˆ, z, p, q, a, o) = pE min Dt ( p, V t ), qˆ

+ h E U (q, Y ) − D ( p, V ) t

In some cases the farmer may be able to postpone the pricing decision until after the yield is realized. In this situation at the beginning of the period, the farmer decides about the size of the land that she wants to put under cultivation. At the end of the season, after realization of yield, the farmer decides about the price of the products. Similar to Section 3.2 we assume that unsatisfied demand is lost, all the excess inventory is salvaged and there is no penalty cost. Considering qˆ as the amount of yield at the beginning of the period, the recursive formulation of the problem is as follows:

in which





5. Sequential decision making



Jt (z, p, q, a, o) = pE min Dt ( p, V t ), U t (q, Y t ) t

single period model, under a similar conditions. The capacity planning policy is fairly simple. If the available land is more than mt , the farmer sells a portion of the land and keeps mt units of the farm land unless selling the land is limited. If z < mt , the farmer buys mt − z units of land unless the upper bound on at limits the acquisition of new land. Also, when mt ≤ z ≤ mt , the farmer will not make any farm trade. From now on, we refer to the proposed policy in Theorem 4 as one-sided production, pricing and capacity planning policy.

(c) Let bt > 0 and Mt = 0 ∀, t. If  t (d, q) ≥ 1 and ptdd (d ) + ≤ 0, ∀d, q, t, then ft (z) is concave for all the values of t. (d) Let bt > 0 and Mt = 0 ∀, t. If  t (d, q) ≥ 1, t (d, q) ≥ 1 and ptdd (d ) + ptd (d )d ≤ 0, ∀d, q, t, then one-sided production and pricing policy is optimal. Also the capacity planning policy is similar to part (b). (e) Let bt = 0 and Mt > 0 ∀, t. If U t (0, Y t ) = 0 and t (p, q) ≥ 1, ∀t, then the two-sided production and pricing policy is optimal. Also the capacity planning policy is similar to part b. ptd (d )d

According to Theorem 4, in the multi-period case, the optimal production and pricing policy has a similar structure to the

(19)

Postponing the pricing decision until after realization of yield helps the farmer to make more informed decisions. Here we look at the conditions that guarantee the concavity of the objective function. Consider t ( p, qˆ ) as the LSR elasticity introduced by Kocabiyikoglu and Popescu (2011). Proposition 11. If t ( p, qˆ ) ≥ 1/2, ∀t, then f t (qˆ, z ) is concave for all the values of t. Postponing the pricing decision until after realization of yield simplifies the analysis of the model. Thus we only need to condition over LSR elasticity to establish the concavity of the value function. The farmer determines the optimal price in each period based on the observed yield. However the conditions in Proposition cannot guarantee the monotonicity of the optimal price with respect to observed yield. The next proposition addresses the optimal production, pricing and capacity planning. Proposition 12. If t ( p, qˆ ) ≥ 1, ∀t, then in each period there exist z˜t , mt and mt where q∗t = min{zt , z˜t },

t

if m ≤ zt





m∗t =

⎧ ⎨min{mt , zt + at }

zt ⎩max{mt , zt − ot }

if zt ≤ mt t if mt < zt < m t t if m ≤ z

and optimal price is a decreasing function of qˆ. Proposition 12 characterizes the optimal policy when the decisions are made sequentially. The conditions mentioned on this proposition are similar to the studies of de Vericourt and Lobo (2009) and Chen et al. (2011). In this situation the optimal lotsizing and capacity planning policies are similar to the one-sided production, pricing and capacity planning policy. In addition, in order to adjust the production and demand, the farmer reduces the optimal price as the yield increases.

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6. Gradual maturity of trees Based on the nature of the crop, the farmer may need to wait for a period of time until the tree starts bearing fruit. For instance, it takes three years for some type of olive trees to start bearing fruit. In this section, we incorporate this phenomenon into our model. Here we assume that it takes I periods for the trees to mature and start production. In order to consider a more general case, we assume that the farmer can buy farmlands with trees of different ages. For instance, at the beginning of a period, the farmer may buy z1 units of land with one year old trees and z3 units of land with three year old trees. This assumption increases the state set of the problem. Assume that zi represents the amount of land with age i that the farmer owns at the beginning of a period. Also assume that z represents the amount of land that is ready for cultivation with all the trees mature enough to bear fruit. We assume that the farmer can buy and sell lands with different ages in each period. Intuitively we assume that the lands with more mature trees have a higher price. Consider Zt = (zt1 , zt2 , . . . , ztI−1 , zt ) as the set of land in different stages of maturity, owned by the farmer at the beginning of period t. Also consider ati as the amount of land with age i that the farmer buys at the end of period t and At = (at0 , at1 , . . . , atI−1 , at ). Similarly we consider oti as the amount of land with age i that the farmer sells at the end of period t and Ot = (ot1 , ot2 , . . . , otI−1 , ot ). δit represents the buying price of the t , δ t ). Considland with age i in period t and t = (δ0t , δ1t , . . . , δI−1 ering it as the selling price of the land with age i in period t and t , t ), the dynamic programming formulation of t = (1t , . . . , I−1 the problem is as follows:

f t (Z ) =

max

t

0≤q≤z,0≤,,p

in which

J (Z, p, q, , )







Jt (Z, p, q, , ) = pE min Dt ( p, V t ), U (q, Y t )



+ ht E U (q, Y t ) − Dt ( p, V t )



+





− ct q − t A + t O + rt E min z − q, X t



+ α f t+1 (a0 , z1 + a1 − o1 , . . . , zI−2 + aI−2 −oI−2 , zI−1 + aI−1 − oI−1 + z + a−o). (20) The farmer can use buying and selling lands in order to manage the capacity. If the farmer predicts an increase in demand in the near future, she may prefer to buy farmlands that are closer to maturity. On the other hand if the farmer experiences a gradual and steady increase in the demand, she may prefer to buy lands that are in earlier stages of maturity, in order to decrease the land acquisition cost. In the next proposition we look at the optimal policy. Proposition 13. (a) If t (p, q) ≥ 1/2, ∀t, then ft (Z) is concave for all the values of t. (b) If t (p, q) ≥ 1, ∀t, then one-sided production, pricing and capacity planning policy is optimal. Proposition 13 shows that the results in Section 4 can be extended to the case where the farmer has to wait for maturity of the trees. Note that when the farmer does not require to wait for the maturity of trees, there exists two thresholds that determine the optimal capacity planning policy. However here, since the farmer can trade farmlands in different maturity stages, for each stage there are two thresholds that define the optimal land acquisition policy. Note that the above result can easily be extended to the case where the farmer incurs a variable cost for each unit of land that she is keeping based on the age of the land.

Fig. 1. Difference between dynamic and static pricing with respect to initial asset.

7. Numerical analysis In this section, we explore some of the characteristics of the models using numerical analysis. In order to implement a reasonable range of parameters in our example, we use the California almond industry’s data. Given the availability of data, we consider the model with no penalty and fixed costs. We assume that the planning horizon is 6 years (T = 6) with a discount factor of 0.95. According to Freeman, Viveros, Klonsky, and Moura (2012), the almond yield varies between 1600 lb to 2800 lb per acre in California. We assume that the uncertainty of yield has a stochastically proportional form and it is uniformly distributed between 1600 lb and 2800 lb. Also Freeman et al. (2012) estimated the overall cost of production, land renting rate and land purchasing rate to be 3675$/acre, 349$/acre and 70 0 0$/acre, respectively. We assume that the salvage value is 1.09$/lb which is the production cost per pound without considering cash overhead costs and non-cash overhead costs (Freeman et al., 2012). We consider a small size farmer with average potential demand of 60 0 0 0 0 lb that is varying normally with a standard deviation of 10 0 0 lb. The demand is considered to be linear in which $1 increase in price reduces the potential demand by 16% (Russo, Green, & Howitt, 2008). According to an increasing trend in almond consumption during the last decade (Russo et al., 2008), we assume that the average potential demand of the farmer increases 5% each year. 7.1. Effect of dynamic pricing In the study of de Vericourt and Lobo (2009) it was demonstrated that the initial asset of the firm has a reverse effect on the effectiveness of dynamic pricing. Here, we investigate the same question where the initial asset is determined by the acreage of the land that the farmer owns in the beginning of the first period. We compare two policies: a dynamic pricing policy in which the farmer can change the price in each period and a static pricing policy in which the farmer finds a single optimal price for all the periods and is not allowed to change it. Fig. 1 demonstrates the value of dynamic pricing in terms of the expected profit. In this figure, the y-axis indicates the percentage difference in expected value of profit between dynamic pricing and static pricing cases. In contrast to the finding of de Vericourt and Lobo (2009), the difference between the dynamic and static pricing is not monotone. The major reason for this difference is the shape of the demand and the nature of the production. In our analysis we assumed that demand is increasing; however, de Vericourt and Lobo (2009) considered no variation in the average of demand. Also in our study the production capacity is determined by the farmland. If the farmer wants to increase the production

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Fig. 2. Profit increase with respect to renting rate.

capacity she has to endure the acquisition cost. In the study of de Vericourt and Lobo (2009) the asset is money which increases when the firm makes profit. When the demand is increasing, in a static pricing strategy, the farmer faces a dilemma. Low prices may be suitable for the beginning periods (since the demand is low), but as the time goes on, the farmer can take the advantage of high demand in the last periods by charging high prices. When the initial acreage is low the farmer buys a large amount of farm in the first few periods to be able to satisfy the demand in the latter periods. In this way, the farmer can match the supply and demand from the second period but the first period remains a problem (the size of the farm is too small to satisfy the demand). In dynamic pricing the farmer sets a high price for the first period to match the production and demand. However in static pricing the price is almost low in the first period because it can not be changed through time and the farmer needs to compromise (high price is not optimum for the second and third periods). The difference between profits in the first period is the major reason for the high difference between the dynamic and static pricing strategies. As the size of the farm increases the variation of the price decreases for the dynamic case and it gets closer to the static case. In both cases the cost of acquisition prevents the farmer from buying more farm to satisfy the demand completely and the farmer relies more on high prices. When z1 is high, again the variation of the price goes up for the dynamic case. The farmer wants to take advantage of low demand in the beginning period and high demand in final periods when she has enough capacity for production. However in the static case, the price should stay constant and the farmer has to compromise between low demand in the beginning and high demand at final stages. We believe this is the main reason for the higher difference between profits for high values of z1 . Despite the difference between the profits, still the percentage of the difference stays below 3.5% which needs to be considered by the farmer in implementation of dynamic pricing as it may be costly. 7.2. Effect of renting option In this part we investigate the role of the renting option. Fig. 2 shows the gain for the farmer when there is a renting option with respect to the renting rate. As the renting rate goes up, the farmer’s profit increases. This is intuitive as the share of renting revenue in the farmer’s profit increases as the renting rate goes up. Fig. 3 shows the behaviour of the average optimal prices with respect to the renting rate. When the renting rate is low, the farmer prefers to allocate all the available land to production. As a result a slight increase in the renting rate does not change the decision of the production

Fig. 3. Behaviour of average optimal prices with respect to renting rate.

Fig. 4. Behaviour of production land with respect to standard deviation of supply.

quantity as well as the price. However when r increases, renting out the farm becomes more appealing. So the farmer assigns less farmland to production and increases the price to match the production and demand. This phenomena is interesting since the growth of renting fees not only affects the price of producers which use renting farm lands (by increasing the cost of production), it also affects the price of the farmers which are using their own land for production, even when their cost of production is constant. 7.3. Effect of uncertainty of supply Here we investigate the behaviour of the optimal policies as the uncertainty of yield increases. We assume that the average yield stays constant (2200 lb/acre) while the standard deviation increases. Fig. 4 shows the behaviour of the average amount of land that is assigned to production with respect to the uncertainty of yield. Since the farmer enjoys the benefit of the salvage value, the cost of underproduction is more than overproduction. As a result when the uncertainty of yield increases the farmer tends to increase q to alleviate the risk of underproduction. In the same manner, the farmer tends to acquire more land during different periods to increase the production capacity. This behaviour is demonstrated in Fig. 5. Another policy that decreases the chance of underproduction is increasing the price so that the demand declines. Fig. 6 shows the increasing trend of price as a mitigation strategy for the increase of yield uncertainty. As we increase the uncertainty of yield the expected profit decreases. This behaviour is demonstrated in Fig. 7. According to Lemma 1 part (a) the highest profit happens when the yield is certain. In this situation the farmer’s risk of underproduction is minimized.

A. Golmohammadi and E. Hassini / European Journal of Operational Research 275 (2019) 1037–1049

Fig. 5. Behaviour of average land change with respect to standard deviation of supply.

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Fig. 8. Behaviour of average production land with respect to measure of risk averseness.

Fig. 9. Behaviour of average price with respect to measure of risk averseness. Fig. 6. Behaviour of average price with respect to standard deviation of supply.

more sensitive to risk of underproduction and consequently, decreases the prices. 8. Conclusion

Fig. 7. Behaviour of expected profit with respect to standard deviation of supply.

7.4. Effect of risk averseness We numerically analyze how the attitude of the decision maker toward risk affects the optimal policies. To do so, we focus on the case where the utility function is exponential, B(x ) = 1 − e−κ x ∀κ = 0, where κ is the Arrow-Pratt measure of absolute risk aversion. Figs. 8 and 9 show the behaviour of average optimal production quantity and the average optimal price with respect to the level of risk averseness of the decision maker, respectively. As indicated in Section 3.5, the risk averse decision maker is sensitive to the risk of overproduction. Hence, as the level of risk averseness increases the decision maker tends to reduce the production quantity in order to avoid any overproduction. On the other hand, as the level of risk averseness increases, the sudden drop of optimal production quantity results in higher optimal prices. In this situation, the decision maker tries to balance the risk of underproduction with a mild increase of optimal prices. However, for high level of risk averseness, the decision maker becomes

In this article, we looked at the problem of production planning, pricing and capacity planning of a farmer in a single and multi-period cases. We introduced the concept of EDFR elasticity as an extension to LSR elasticity developed by Kocabiyikoglu and Popescu (2011). This elasticity facilitates the analysis of the newsvendor problem with pricing under uncertain supply and demand where the application of LSR elasticity seems to be cumbersome. In the single and multi-period cases we provided the condition where the optimal policy of the farmer is one-sided production and pricing. This is an interesting policy because it provides a structural form for pricing, despite the fact that some of the previous researchers such as Li and Zheng (2006) and Feng (2010) demonstrated that under uncertainty of supply, pricing decision usually does not show a well formed behaviour. In the case that there is a fixed cost for production, the farmer switches to a two-sided production and pricing policy. In this situation the farmer has to make sure that the benefit from production surpasses the fixed cost. Similar as the one-sided policy, optimal prices have a proper structural form in this situation too. We looked at the conditions where it is not optimal for the farmer to start production and interestingly we find that this decision is not affected by the salvage value. This result is not intuitive since an increase in the salvage value, decreases the risk of overproduction which seems to be a good motivation for starting the production. In contrast, a penalty cost has a direct affect on this decision. Increasing the penalty cost motivates the farmer to start production in order to reduce the unmet demand. Removing the uncertainty from the model can result in overestimation or

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underestimation of the profit. This can be influential on the production decision. The farmer may start or stop production while it is optimal to do the opposite action. We investigated the case in which the decision maker is risk averse. We showed the objective function is individually concave with respect to price and production quantity. In addition, we compared the optimal production quantity and price under the risk neutral and risk averse cases. We also numerically looked at the impact of the decision maker’s risk averseness on production and pricing decisions. We examined the case where the pricing decision can be postponed until after realization of yield. We provided the conditions that guarantee the uniqueness of the solution and also characterized the optimal production, pricing and capacity planning policy. We also looked at the case where the farmer has to wait for maturity of the trees to start production. We showed that the one-sided production, pricing and capacity planning policy is optimal in this situation as well. In our numerical analysis we found that the importance of dynamic pricing does not necessarily diminish as the initial asset goes up which is in contrast to the finding of de Vericourt and Lobo (2009). When increasing the production capacity is costly and the demand is increasing, in some cases the role of dynamic pricing becomes more important for the firm that has a larger initial assets. We also show that in the case where the cost of underproduction is relatively higher than that of overproduction, the farmer may choose to increase the production quantity as well as the price in order to alleviate the risks associated with a volatile yield. Our work can be extended in several ways. First, the farmer may consider to invest in technology to improve its crop yield. Therefore, adding an investment decision variable to the model would help in studying optimal investments policies for yield improvements. Second, in the agribusiness it is common to find farmers that are cash-strapped. In such situations the farmers often partner with handlers to invest in improving yield. An interesting extension would then be to investigate the investment behaviours of the handler and farmer. The authors are currently considering this line of research in the context of a Stackelberg game. Finally, our work can be extended to the case of multiple products. The can include complexities such as products substitutability and demand dependency. In addition, in an agricultural environment using the same farmland for different crops would usually imply that their yields are dependent and may require the rotation of the crops thorough consecutive periods. Acknowledgement The authors acknowledge support through a Natural Sciences and Engineering Research Council (NSERC) Discovery Grant (RGPIN/283135-2009 and RGPIN-2014-04827). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.ejor.2018.12.027. References Angelus, A., & Porteus, E. L. (2002). Simultaneous capacity and production management of short-life-cycle, produce-to-stock goods under stochastic demand. Management Science, 48(3), 399–413. Boersma, B. (2015). How do small-scale, local farmers adjust prices for inflation? https://goo.gl/z7UGHT. Online, accessed 2015-10-27. Braunscheidel, M. J., & Suresh, N. C. (2009). The organizational antecedents of a firms supply chain agility for risk mitigation and response. Journal of Operations Management, 27(2), 119–140. doi:10.1016/j.jom.20 08.09.0 06.

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