PRICING AND INVENTORY DECISIONS WITH

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Asia-Pacific Journal of Operational Research Vol. 30, No. 6 (2013) 1350030 (25 pages) c World Scientific Publishing Co. & Operational Research Society of Singapore DOI: 10.1142/S0217595913500309

PRICING AND INVENTORY DECISIONS WITH UNCERTAIN SUPPLY AND STOCHASTIC DEMAND

CHIRAG SURTI∗ Faculty of Business and IT University of Ontario Institute of Technology (UOIT ) 2000 Simcoe Street North, Oshawa, Ontario, Canada L1H 7K4 [email protected] ELKAFI HASSINI† and PRAKASH ABAD‡ DeGroote School of Business, McMaster University 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L8 †[email protected] ‡[email protected] Received 7 December 2010 Revised 10 March 2011 Accepted 10 July 2011 Published 21 October 2013 We consider a retailer, facing uncertain supply and price-sensitive stochastic demand, who has to make stocking and pricing decisions for a given selling period. We also consider the case when the demand is price-sensitive deterministic and provide a unified framework for the model with additive errors. For both scenarios, we look at the case when the price is set before receiving the supply, called simultaneous pricing and the case when the price is set after receiving it, which is called postponed pricing. We develop a procedure for finding the optimal policy for the retailer with general distributions for the supply and the demand. To study the effect of supply uncertainty on expected profit, we conduct sensitivity analysis and develop results for both pricing scenarios and give insights. The results have important implications for a retailer in the supply chain, where a portion of the inventory may be lost due to variety of factors including mishandling and failure to meet quality standards. The findings shed light on the nature and role of prices and their relationship to supply and demand. Keywords: Pricing; inventory; random yield; demand uncertainty.

1. Introduction Retailers are exposed to a variety of risks, one of which is having too much or too little stock to sell. This risk is usually due to uncertainty in consumer demand, which is amplified when the supply is uncertain. The uncertainty in supply can be ∗ Corresponding

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C. Surti, E. Hassini & P. Abad

due to long lead times, loss, damage or theft in transit. If the demand outstrips the supply then a mid-season recourse may not be possible, resulting in costly shortages. On the other hand, overstocking or using buffer stock is not always desirable as it can be costly. Supply may be uncertain in many different environments, such as electronic fabrication, chemical manufacturing, discrete parts manufacturing, food and agricultural products, pharmaceutical products and medical supplies. Jones et al. (2001), Kazaz (2004), Bakal and Akcali (2006), Tang and Yin (2007) and Hsieh and Wu (2008) are the most recent industry specific studies that have looked at the issue of yield uncertainty. For a risk neutral retailer, an order quantity and a price that maximize revenue and minimize salvage and shortage losses need to be determined, in order to balance these risks. We assume the yield to be stochastically proportional, i.e., the yield does not depend on the order size. In our study, we develop models that support uncertain supply with demand either being stochastic or deterministic, to yield results that explain the optimal price and order quantity and their relationship to the uncertain supply. The added uncertainty in supply makes the analytical determination of the optimal price and quantity complex. The existing body of knowledge on this issue does not always include pricing as one of the levers to reduce these risks. There are two distinct cases of pricing in relation to the realization of demand. In the first instance, the price set before the supply is received. We refer to this as the simultaneous pricing and ordering model. Such situations arise when the retailer needs to engage in advertising of the product in advance of the selling season and before receipt of stock, or when a sales contract has been signed before the supply is received. In the second case, we consider a retailer who may be able to delay the pricing decision until the receipt of the supply, but before the realization of demand. We refer to this as the postponed pricing model. In the postponed pricing case the decision process is two-staged. In stage one, the retailer places an order, optimizing for the anticipated unreliability of the supply as well as that of the demand. Once the supply is received, the retailer prices and sells such that the revenue is maximized. Some specific situations in the literature where postponed pricing is applicable include agribusinesses (Jones et al., 2001; Kazaz, 2004), re-manufactured automotive parts (Bakal and Akcali, 2006) and retailing (Tang and Yin, 2007). We assume the demand to be linear in price and consider both additive and multiplicative demand error that can assume any general distribution. In Sec. 2, we provide a brief review of the existing relevant literature on uncertain supply and demand. In Sec. 3, we develop the proposed model and analyze the problem with stochastic and deterministic demand. Within each demand case, we further analyze the two cases of simultaneous and postponed pricing. For stochastic demand case, we consider the case of additive as well as multiplicative demand error. In Sec. 4, we conduct a sensitivity analysis to study the effects of yield distribution parameters on the expected profit, the order quantity and the price. Finally we provide our conclusions as well as identify avenues for future research in Sec. 5. 1350030-2

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Pricing and Inventory Decisions with Uncertain Supply and Stochastic Demand

2. Literature Review The earliest model of random supply and demand was explored by Karlin (1958), where the amount received is independent of the order quantity. Silver (1976) explored the exact relationship between the optimal order quantity and the parameters of the yield distribution. He found that the optimal order quantity is a multiple of the EOQ and this multiple depends on the choice of mean and variance of the yield distribution. Analysis of supply uncertainty in a newsvendor context is provided by Rekik et al. (2007). The most comprehensive review of the literature on random yields, in a manufacturing context, is by Yano and Lee (1995), who broadly classify yields into two classes: random process yields and variable capacity yields. Examples for random process yields include a discrete manufacturing system, producing one unit at a time, with a quality control process that may accept or reject the components. For a batch or a continuous type system, a stochastically proportional yield is used, where only a fraction of the whole batch is fit for consumption. For random capacity, where the actual process capacity is unknown, examples include land under agriculture cultivation or a scarce manufacturing resource such as a highly specialized machine that is not always available either due to breakdown, routine maintenance or because it is processing another job. Henig and Gerchak (1990) provide a comprehensive analysis of a periodic review inventory system with random yields. All the studies mentioned do not incorporate price as a decision variable, since they are focused on production environments. Li and Zheng (2006) incorporate the simultaneous pricing decision in this model with full backlogging of unmet demand. They show, for a general stochastic demand function, that the objective function is jointly concave in price and order quantity. Any unmet demand at the end of the selling cycle is fully met, with the help of a special order, in which all units of good quality and surplus inventory are carried forward. Although such assumptions are normal in production/manufacturing type settings, they are not realistic in a retail setting where the unmet demand at the end of the selling season is lost and surplus is usually salvaged. Van Mieghem and Dada (1999), and later Gerchak (2012), analyze the benefits of a price-postponement strategy under demand uncertainty. Although product postponement is well studied and understood as a means to reduce demand uncertainty, they propose a price-postponement strategy as a cost effective tool to reduce uncertainty in the supply chain when only demand is uncertain. Other work involving postponed pricing with uncertain yield and demand includes Jones et al. (2001), Kazaz (2004) and Bakal and Akcali (2006), where they take the problem and separate it into two stages. Stage one deals with the realization of random yield and stage two with a recourse action in terms of either placing a second order with no yield uncertainty (Kazaz, 2004) or placing a second order with yield uncertainty (Jones et al., 2001) or setting a postponed price (Bakal and Akcali, 2006). Wang (2006) considers the problem of simultaneous pricing and ordering as well as postponed 1350030-3


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pricing in a decentralized supply chain setting with n manufacturers and a single retailer with certain yield for complementary products. In our case we consider a retailer facing an uncertain supply and a price-sensitive demand and no recourse in terms of placing a second order. More recently, Rekik et al. (2007) consider random demand with additive and multiplicative errors for the case when yield and demand are either uniform or normal, obtain closed form expressions and provide some statistical insights for a choice of uniform distribution. Tang and Yin (2007) develop a model with uncertain supply yield for the case when demand is deterministic and price-sensitive. They also develop a model for simultaneous pricing and postponed pricing. Our model extends that of Tang and Yin (2007) in two ways: (1) it assumes demand to be stochastic and similar to existing stochastic inventory models, includes shortage and surplus cost; (2) it assumes yield to be stochastically proportional to the lot size and represents it by a general distribution. Hsieh and Wu (2008) consider a decentralized supply chain, consisting of a manufacturer, distributor and an original equipment manufacturer (OEM). The source of randomness in supply comes from capacity at the manufacturing end and random demand at the distributors end. The random demand faced by the distributor is iso-elastic. They find that coordination, depending upon the scenario, may be beneficial to the manufacturer but not to the OEM. The distributor makes ordering and pricing decisions simultaneously. Our paper assumes uncertain supply as well as stochastic demand allowing for both the additive and multiplicative error. Given stochastic demand, we incorporate shortage and salvage values in our analysis. We consider both simultaneous and postponed pricing and show analytically that postponed pricing leads to higher profits and lower prices. We develop the analytical results and conditions that guarantee optimality, as well as provide simple yet effective solution procedures. The analytical results are sometimes intractable and closed form solutions are not always possible, especially when the yield and demand error distributions are complex. We provide numerical examples as well as conduct sensitivity analysis for the models using the Beta distribution for the yield. 3. Model 3.1. Notations and assumptions We define the notation used throughout this section in Table 1. In addition to the requirement that the unit price cannot be lower than the unit cost, i.e., p ≥ c, we also assume that the retailer only pays for the units that are good (salable). Such an assumption is consistent in a pure inventory setting such as ours, where costs are related to the salable quantity received by the retailer (Yano and Lee, 1995). Furthermore, in order to ensure that the demand is nonnegative, we put an upper bound on the price p ≤ a+A b . This ensures that the demand D(p, ε) ≥ 0. The stochastically proportional yield, r should be nonnegative and independent of Q (Grosfeld-Nir and Gerchak, 1990). 1350030-4

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Pricing and Inventory Decisions with Uncertain Supply and Stochastic Demand Table 1. Definition of terms. Symbol c p Q h s r

u D(p, ε)

ε

v D(p, v)

Definitions unit cost, c ≥ 0. price per unit charged, a decision variable such that p ≥ c. lot size, a decision variable such that Q > 0. salvage price such that 0 ≤ h ≤ c. shortage cost s ≥ c. random variable representing the supply yield defined over the range [0, 1], with mean µr and standard deviation σr . Its probability density and cumulative functions are denoted by f and F , respectively. realized value of yield r. is the demand function such that D(p, ε) = y(p) + ε, where y(p) = a − bp and a > 0 and b > 0 are known constants. a is the market size parameter and b is the price elasticity. random variable representing the uncertainty in demand defined over range [A, B], such that A > −a, with mean µε and standard deviation σε . Its probability density and cumulative functions are denoted by g and G, respectively. realized value of the demand error ε. is the realized demand D(p, v) = y(p) + v.

3.2. Model with stochastic demand 3.2.1. Simultaneous pricing and ordering In this case, the retailer determines the order size Q and price p at the same time. The retailer has no recourse and thus must mitigate the risk of shortages and risk of being left with excess inventory at the end of the selling period. The profit function is Π(p, Q) = p min{D(p, ε), rQ} − crQ + h(rQ − D(p, ε))+ − s(D(p, ε) − rQ)+ ,

(1)

where x+ = max{0, x}. Upon substituting D(p, v) = y(p) + v in Π and simplifying, the expected profit function is given as E[Π(p, Q)] = (p − h)[y(p) + µε ] − (c − h)µr Q 1 B + (p + s − h) [uQ − y(p) − v]g(v)dvf (u)du. 0

(2)

uQ−y(p)

The problem of maximizing the expected profit is thus formulated as max E[Π(p, Q)]. p,Q

(P1)

The optimization problem (P 1) has been stated in the unconstrained form, since we deal with the simple bounds on the price in the solution procedure itself. Analysis In the newsvendor model for a retailer facing only demand uncertainty, it is possible to use a change of variable z = Q − y(p) in order to simplify the analysis of the problem (without guaranteeing joint concavity) (e.g., Petruzzi and Dada (1999)). 1350030-5


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Unfortunately, when supply uncertainty is combined with that of demand uncertainly, it is no longer possible to make use of this change of variable idea. Due to the existence of the random yield variable, a change of variable does not lead to an expected profit that is simpler than what we currently have and, in particular, it does not lead to an expression that recasts the expected profit as a separable sum of expected leftovers and shortages. It will also not allow us to distinguish between the riskless profit and the loss associated with uncertain supply and demand. Li and Zheng (2006) and Kazaz (2004) were able to obtain analytically tractable objective functions by allowing for any unmet demand to be satisfied by placing a special order with no supply yield uncertainty at the end of the season. Such a recourse may be plausible in a manufacturing or production environment. However in retail operations, lost sales are the norm and it is not reasonable to assume that we can place a special order at the end of the season to satisfy unmet demand without supply uncertainty. The expression for the Hessian is very complicated and proving joint concavity analytically, even under some conditions on the problem parameters, leads to very complex and meaningless expressions. The lack of joint concavity properties is in general true for all price setting newsvendor problems and requires complete enumeration (Petruzzi and Dada, 1999). In such circumstances one would seek to at least show concavity for separate decision variables (e.g., Agrawal and Nahmias (1997) and Karakul and Chan (2008)). Likewise, we show that the expected profit function is concave in each decision variable in Proposition 1. Proposition 1. For any given price p, E[Π(p, Q)] is concave in Q. For any given order quantity Q, E[Π(p, Q)] is concave in p. Proof. First we consider a fixed price p. From (1) we see that the terms rQ and −rQ are concave in Q, since they are linear in Q, and so min{D(p, ε), rQ}, (rQ−D(p, ε))+ and (D(p, ε) − rQ)+ are also concave in Q, because concavity is preserved under maximization and minimization. Finally, taking expectation preserves concavity, hence E[Π(p, Q)] is concave in Q, when p is fixed. A similar argument can be used for the case when Q is fixed. Thus, we can calculate p∗ (Q) by solving the first-order conditions given as ∂E(Π(p, Q)) = (y(p) + µε ) − (p − h)b ∂p 1 B [(uQ − y(p) − v) + b(p + s − h)] + 0

uQ−y(p)

× g(v)f (u)dvdu = 0.

(3)

Solution Procedure To find an optimal order quantity Q∗ , price p∗ and expected profit E[Π(p∗ , Q∗ )] we propose a simple search procedure. From Proposition 1 and Eq. (3), we note that, for 1350030-6


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a given order quantity Q, we have a unique p∗ (Q). To ensure that the demand takes a meaningful form, we require that the price be such that D(p, ε) ≥ 0 ⇒ p ≤ a+A b . The lower bound on the order quantity is the natural limit: Q ≥ 1. Observation 1 establishes an upper bound on the optimal quantity. Observation 1. Let Ql = c ≤ p ≤ (a + A)/b, Q ≥ Ql .

(a+µ −bh)2 4b(c−h)µr .

Then E(Π(p, Q)) ≤ 0 for all p, Q such that

Proof. The last term of (2) is negative for all p and Q. Hence, E(Π(p, Q)) < (p − h)[y(p) + µ ] − (c − h)µr Q

(4)

for all p, Q such that c ≤ p ≤ (a + A)/b, Q > 0. Since max{(p − h)[y(p) + µ ]} = p

(a + µ − bh)2 4b

it follows that (p − h)[y(p) + µ ] − (c − h)µr Ql ≤ 0

(5)

for all p, Q such that c ≤ p ≤ (a + A)/b, Q > Ql . Combining (4) and (5), we get the desired result. With the above bounds on the decision variables, we propose the following solution procedure: (1) Initialize i = 0 and Qi = 1. (2) Find pi , s.t.

∂E[Π(b pi | Qi )] ∂p

a+A

= 0 and set pi =

b

p bi , c,

,

if pi ≥

a+A b , pi ≤ a+A b ,

if c ≤ if pi ≤ c.

(3) Calculate E[Π(pi , Qi )]. Store pi , Qi and E[Π(pi , Qi )] in a table. (4) Let i = i + 1 and Qi = Qi−1 + ∆ where ∆ > 0. If Qi ≥ Ql then end else goto Step 2. The global maximum (or maxima) is identified by searching the table developed in Step 3. 3.2.2. Postponed pricing In this case, the retailer first determines the optimal order quantity Q in Stage I and after observing the value of supply yield, determines the optimal selling price p in Stage II. Thus the problem is formulated in stages, with Stage II solved first where the retailer knows the value uQ ≥ 0. The profit function for Stage II is ΠII (p | uQ) = p min(D(p, ε), uQ) − cuQ + h(uQ − D(p, ε))+ − s(D(p, ε) − uQ)+ . Note that in the above expression the value of the yield is already known. 1350030-7


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The problem of maximizing the expected profit in Stage II is max E[ΠII (p | uQ)].

(P2)

Let p∗ (uQ) be the optimal price that maximizes the expected profit in Stage II for a realized value of supply yield u and order quantity Q. The expected profit function for Stage I is given as 1 E(ΠII (p∗ (uQ) | uQ))f (u)du. (6) E[ΠI (Q)] = 0

The problem of maximizing the expected profit in Stage I is max E[ΠI (Q)].

(P3)

The optimization problems (P 2) and (P 3) have been stated in the unconstrained form since we handle the simple bounds on the price and order quantity explicitly in the solution procedure proposed in Sec. 3.2.2. Using the same logic in Proposition 1 we can show that E(ΠII (p | uQ)) is a concave function in p, for a given uQ value. Hence, there is a unique p, given as p∗ (uQ), that satisfies the first-order condition ∂E(ΠII (p | uQ)) = (y(p) + µε ) − (p − h)b ∂p B [(uQ − y(p) − v) + b(p + s − h)]g(v)dv = 0. +

(7)

uQ−y(p)

Thus the optimal price p∗ (uQ) can be determined using the first-order condition in Eq. (7). Solution Procedure To find the order quantity Q that maximizes E(ΠI (Q)), we propose the following solution procedure. To ensure that the demand takes a meaningful form, we require that the price be such that D(p, ε) ≥ 0 ⇒ p ≤ a+A b . A result similar to Observation 1 can be shown for E(ΠII (p | uQ)) and thus the bounds for Q are 1 and Ql . The procedure is described in the following two steps: (b pi |uQ)] = 0 values for a set of (uQ) values, (1) We obtain a set of pi (uQ), s.t. ∂E[ΠII∂p 0 ≤ uQ ≤ Ql . Since in Stage II, the values of u and Q are numerically known, i.e., yield r takes a value u for a given order size Q, the problem simply reduces to using the corresponding price p∗ (uQ) that maximizes the expected profit in (P 2). This price is given as a + A a+A  , if pi (uQ) ≥ ,    b  b p∗ (uQ) = p (uQ), if c ≤ p (uQ) ≤ a + A , i i   b    c, if pi (uQ) ≤ c.

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(2) Next, we vary Q ∈ [1, Ql ] by increment ∆ > 0 to find the global optimum Q. For the problem in Stage II, p(uQ) can take values depending upon the value of Q and the realized value u. We calculate the expected profit for the value of Q using Eq. (6) as follows 1 E(ΠI (Q)) = E(ΠII (p, Q))|p=p(uQ) f (u)du, 0

≈

E(ΠII (p, Q))|p=p(uj Q) f (uj )∆u.

uj

We store the values of Q and the corresponding E[ΠI (Q)] in a table. The global maximum (or maxima) are then identified by searching the table. The average price, which we define as p = uj p(uj Q)|Q=Q∗ f (uj )∆u, can be calculated at this stage. 3.3. Extension to multiplicative demand error We consider the case where demand is multiplicative, such that D(p, ) = y(p), where the random error range is [A, B] and A > 0, without loss of generality. To emphasize the similarity to the additive demand case we use the same equation numbers with letter ‘m’, for multiplicative, attached to them. 3.3.1. Simultaneous pricing and ordering The profit function is Π(p, Q) = p min(D(p, ε), rQ) − crQ + h(rQ − D(p, ε))+ − s(D(p, ε) − rQ)+ , and the expected profit is E[Π(p, Q)] = (p − h)[y(p)µε ] − (c − h)µr Q + (p + s − h)

1

0

B

[uQ − y(p)v]g(v)dvf (u)du.

(1m)

uQ/y(p)

The problem of maximizing the expected profit is formulated as max E[Π(p, Q)]. p,Q

As in Proposition 1 we can show that expected profit function (1m) is concave in p for any given Q. Thus, we can calculate the optimal p∗ (Q) by solving the following first-order condition ∂E(Π(p, Q)) = (y(p)µε ) − (p − h)bµε ∂p 1 B + [(uQ − y(p)v) + bv(p + s − h)]g(v)f (u)dvdu = 0. 0

uQ/y(p)

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The solution procedure is similar to the additive demand case, except for steps 2 and 4, where we are required to change some values and limits for the decision variables: (1) Initialize i = 0 and Qi = 1. (2) Find pi , s.t.

∂E[Π(b pi | Qi )] ∂p

a

= 0 and set pi =

b, p bi , c,

if pi ≥ a b, if c ≤ pi ≤ if pi ≤ c.

a b,

(3) Calculate E[Π(pi , Qi )]. Store pi , Qi and E[Π(pi , Qi )] in a table. 2 (a−bh) (4) Let i = i + 1 and Qi = Qi−1 + ∆ where ∆ > 0. If Qi ≥ µ4b(c−h)µ then end else r goto Step 2. The global maximum (or maxima) is identified by searching the table developed in Step 3. 3.3.2. Postponed pricing The profit function for Stage II is ΠII (p | uQ) = p min(D(p, ε), uQ) − cuQ + h(uQ − D(p, ε))+ − s(D(p, ε) − uQ)+ , and the expected profit function for Stage II is E(ΠII (p | uQ)) = (p − h)[y(p)µε ] + (h − c)uQ B + (p + s − h) [uQ − y(p)v]g(v)dv. uQ/y(p)

The problem of maximizing the expected profit in Stage II is max E[ΠII (p | uQ)]. Let p∗ (uQ) be the optimal price that maximizes the expected profit in Stage II for a realized value of supply yield u and order quantity Q. The expected profit function for Stage I is given as 1 E(ΠII (p∗ (uQ) | uQ))f (u)du. E[ΠI (Q)] = 0

The problem of maximizing the expected profit in Stage I is max E[ΠI (Q)]. In a similar way to Sec. 3.2.2 we can show that E(ΠII (p | uQ)) is a concave function in p, for a given uQ value. Hence, there is a unique p, given as p∗ (uQ), that satisfies the first-order condition ∂E(ΠII (p | uQ)) = (y(p)µε ) − (p − h)bµε ∂p B + [(uQ − y(p)v) + bv(p + s − h)]g(v)dv = 0. (2m) uQ/y(p)

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Thus the optimal price p∗ (uQ) can be determined using the first-order condition in Eq. (2m) and the same procedure used in Sec. 3.2.2 can be used here with some modifications to the decision variables limits and values: (b pi |uQ)] = 0 values for a set of (uQ) values, (1) We obtain a set of pi (uQ), s.t. ∂E[ΠII∂p ∗ 0 ≤ uQ ≤ Ql . The price p (uQ) that maximizes expected profit is given as  a a   , if pi (uQ) ≥ ,   b b p∗ (uQ) = p (uQ), if c ≤ p (uQ) ≤ a , i   i b    c, if pi (uQ) ≤ c.

(2) Next, we vary Q ∈ [1, Ql ] by increment ∆ > 0 to find the global optimum Q. We calculate the expected profit for the value of Q using Eq. (3.3.2) as follows 1 E(ΠII (p, Q))|p=p(uQ) f (u)du, E(ΠI (Q)) = 0

≈

E(ΠII (p, Q))|p=p(uj Q) f (uj )∆u.

uj

We store the values of Q and the corresponding E[ΠI (Q)] in a table. The global maximum (or maxima) are then identified by searching the table. The average price, which we define as p = uj p(uj Q)|Q=Q∗ f (uj )∆u, can be calculated at this stage. 3.4. Model with deterministic demand We now analyze the case where the demand is deterministic (i.e., D(p) = y(p) = a − bp) and the supply is unreliable. We first develop and analyze the case when the price p and the order quantity Q are simultaneously determined and then develop and analyze the case when the price is set after realizing the supply yield r, for a given order quantity Q. 3.4.1. Simultaneous pricing and ordering The profit function is Π(p, Q) = p min(D(p), rQ) − crQ + h(rQ − D(p))+ − s(D(p) − rQ)+ . Upon simplification the expected profit is given as E[Π(p, Q)] = (p − h)y(p) − (c − h)µr Q + (p + s − h) y(p) Q × [uQ − y(p)]f (u)du. 0

The problem of maximizing the expected profit can now be stated as max E[Π(p, Q)]. p,Q

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As in Proposition 1, we can show that for a given price p, E[Π(p, Q)], as defined in (8), is concave in Q. For any given order quantity Q, E[Π(p, Q)] is concave in p. The first-order conditions for the expected profit are given as y(p)/Q ∂E(Π(p, Q)) = −(c − h)µr + (p + s − h) uf (u)du, (9) ∂Q 0 ∂E(Π(p, Q)) = y(p) − (p − h)b ∂p y(p)/Q + [(uQ − y(p)) + (p + s − h)b]f (u)du.

(10)

0

By solving Eq. (9) for Q, we get the following implicit expression for Q∗ as a function of p:

y(p)/Q c−h uf (u)du = (11) µr . p+s−h 0 It is interesting to note that condition (11) is similar to the fractile form in Whitin (1955). The optimal price p∗ (Q) can be determined using the first-order condition given by Eq. (10). The solution procedure for the simultaneous pricing problem with deterministic demand is similar to the solution procedure in Sec. 3.2.1, thus we do not repeat it. 3.4.2. Postponed pricing In this section, we consider the scenario where the price will be set after the order quantity, Q, has been received and the outcome of the supply yield, r, is known. Unlike the model in the previous sections, in the case of postponed pricing with deterministic demand, there is no further uncertainty left for the retailer (no demand uncertainty). The retailer, based on the available stock, must decide on the optimal price at which his revenue is maximized. Note that the retailer will not face shortage loss in this case, i.e., D(p) ≤ uQ. However it is still possible that the retailer may have supply that exceeds the revenue maximizing quantity in which case the retailer will salvage the remaining units. Optimal Price Given an order quantity Q and an observed value of supply yield u, the retailer can sell, at most, uQ units at price p. The postponed pricing problem can be formulated as max{(p − h)(a − bp) + huQ | a − bp ≤ uQ, h ≤ p ≤ a/b}. p

(P4)

Note that the lower bound on p is set to h as the retailer can price below cost as long as the price is higher than the salvage value. Depending on the state of the first constraint in (P 4), the optimal price can take two values as per Proposition 2. 1350030-12


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Proposition 2. The optimal postponed price when demand is deterministic is  a + bh  , if uQ > D(p0 ), p0 = 2b ∗ p (u, Q) =  p1 = a − uQ , otherwise. b Furthermore, p0 ≤ p1 . Proof. It is easy to show that a+bh = arg maxp {(p − h)(a − bp) + huQ}, i.e., 2b the unconstrained solution for problem (P4). We will show that p∗ (u, Q) satisifes the constraints. We start with the bounds on the prices and show that h ≤ p0 ≤ p1 ≤ a/b: p0 − h = p1 − p0 =

a − bh ≥ 0 ∵ D(h) ≥ 0; 2b (a − bh) − 2uQ ≥ 0 ∵ D(p0 ) ≥ uQ; 2b

uQ a − p1 = ≥ 0. b b Next we consider the demand (no shortage) constraint. If uQ > D(p0 ) then p0 , the optimal price to the unconstrained version of (P4), also solves (P4). In this case we 0 have ample supply and should charge a+bh 2b and salvage the remaining [uQ − D(p )] 0 at a salvage price h. If uQ ≤ D(p ) then the first constraint in (P4) will be binding at optimality, i.e., the price should be p1 such that D(p1 ) = a − bp1 = uQ, or p1 = a−uQ b . In this case the retailer does not have enough supply to maximize profit and should charge a price that will clear the supply with no quantity left for salvage. Optimal Order Quantity Given the above discussion and noting when uQ ≤ D(p0 ), we can have either Q ≥ D(p0 ) or Q < D(p0 ). The profit, as a function of Q, is defined as p0 D(p0 ) − cuQ + h[uQ − D(p0 )] uQ ≥ D(p0 ), Π(Q) = (12) uQ < D(p0 ). (p1 − c)uQ In order to calculate the expected profit we consider two cases. Case A: Q < D(p0 ). Since Q < D(p0 ) it follows that uQ < D(p0 ) for all 0 < u ≤ 1. From (12), the profit for Stage I in this case is ΠA (Q) = (p1 − c)uQ and the expected profit is 1 µr Q Q2 2 − [σ + µ2r ]. E(ΠIA (Q)) = (p1 − c)uQf (u)du = [a − bc] (13) b b r 0 1350030-13


WSPC/S0217-5959

APJOR

1350030.tex


The problem of maximizing the expected profit for Stage I can be formulated as max

0