EL PROYECTO DE TECNÓPOLIS EN LA UNIVERSIDAD

Within the following sections we present the formulas for stochastic and price-dependent demand. Demand at the retail level is now affected by the retail price r as well. In other words, an altered retail price leads to a change in demand for the products. We therefore modify our initial model by a distribution of the demand parameterized by the retail price r. Two groups of stochastic and price-dependent models can be found in literature: the additive demand model, D(r, x) = y(r) + x (Mills (1959) [37]), and the multiplicative demand model, D(r, x) = y(r)x (Karlin and Carr (1962) [30]) where y(r) is decreasing in the retail price r and x is the random component with mean 1. In what follows, we utilize the multiplicative model that is formulated in Emmons and Gilbert (1998) [16] and make necessary extensions in order to include customer returns. Following its wide use throughout the literature, expected demand is assumed to be of the form D(r) = b × (r − k) with b < 0 and k > 0. Because demand can’t be negative, we require k ≥ r. Since D(r) is only defined on [c, r_up], D(r) = 0 ∀ r ≥ r_up. The expected demand quantity D(r) is assumed to be decreasing in the retail price, to be continuous, nonnegative and also twice differentiable. Consequently, the actual demand, y, can be modeled as a product of the expected demand D(r) and the positive random variable x. Hence, the density function for demand can be expressed as follows:

g(x, r) = D(r)⁻¹f ( x

D(r)) with y ≥ 0, (3.14)

where f (·) is the density distribution function of x and F (·) is the corresponding cumulative distribution function. F (·) is assumed to be invertible and f (·) shall have a continuous derivative f⁰(·). Both players, manufacturer and retailer, have knowledge of the respective demand distribution and the retailer can, up to a certain extent, control demand with the setting of the retail price r. As a consequence the

retailer finds his optimal order quantity now to be dependent of D(r) and profits of the players are dependent of D(r) as well.

In the next sections we present the formulas for the profits of both, manufacturer and retailer as well as the optimal order quantities. We start out with the centralized supply chain and then go over to the decentralized case. The model that we are considering in the following is an extension of the classical newsvendor problem (see Emmons and Gilbert (1998) [16]). Additions to include consumer returns occurring at the retail level are studied by Ruiz Benitez and Muriel (2007) [42]. For the decen-tralized systems the terms include the buy-back option. Again, by simply setting s to v the wholesale contract can be modeled.

3.3.1 Centralized System Ignoring Returns

Total profit for the supply chain is:

Π^IR(Q, r) = (r − v)(D(r) − Z _∞

Q/D(r)

(D(r)x − Q)f (x)dx) − Q(c − v) (3.15)

By differentiating the latter expression with respect to Q we find the optimal order quantity:

Q^IR∗(r) = D(r)F⁻¹³ r − c r − v

´

(3.16) For a fixed retail price r, the expected optimal profits are:

Π^IR(r) = (r − v)D(r)

Z _F⁻¹_(ξ1)

0

xf (x)dx (3.17)

where ξ₁ = (r − c)/(r − v).

However, the inverse cumulative distribution function makes it difficult to analyze and obtain a closed form expression for the optimal retail price. For the purpose of gaining more insight into the problem, we can simplify the expression by assuming

f(x) to be a uniform distribution on the interval [0,2]. With this assumption the retailer’s profit reduces to:

Π^IR(r) = (r − c)² r − v D(r)

As we assume a linear demand model with the form D(r) = b × (r − k), where b < 0 and k > 0, D(r) is strictly decreasing in r. Thus, an explicit expression for the retail price r can be easily obtained by solving the equation resulting when taking the derivative of (3.18) with respect to r and equal it to zero. The retailer’s profit maximizing price is

r^IR∗ = 3v + k +p

(k + 8c − 9v)(k − v) 4

Having r^IR∗ the optimal order amount Q^IR∗ is easy to obtain:

Q^IR∗ = 2D(r)(r^∗− c) r^∗ − v Considering Returns

The supply chain expected profit is:

Π^CR(Q, r) = ((1 − α)r − α(l − vr) − v)(D(r) − Z _∞

Q/D(r)

(D(r)x − Q)f (x)dx) − Q(c − v) (3.18) The optimal order quantity Q^∗ is found by differentiating Π^CR(Q, r) with respect to Q and simplifying:

Q^CR∗(r) = D(r)F⁻¹³ (1 − α)r − α(l − vr) − c (1 − α)r − α(l − v_r) − v

´

(3.19)

Then, for a fixed retail price r, the expected optimal profits follow

Π^CR(r) = ((1 − α)r − α(l − vr) − v)D(r)

Z _F⁻¹_(ξ2)

0

xf (x)dx (3.20)

where ξ2 = ((1 − α)r − α(l − vr) − c)/((1 − α)r − α(l − vr) − v).

As shown by Ruiz Benitez and Muriel (2007) [42] under deterministic and price dependent demand the optimal retail price increases and thus the optimal ordering quantity decreases when consumer returns are considered. This is valid if D(r)×(r−c) is unimodal.

By making the assumption that f (x) is a uniform distribution defined on the interval [0,2] it is possible to find an explicit expression for r. Taking the specified uniform distribution, the expected optimal profit for the centralized supply chain becomes:

Π^CR(r) = ((1 − α)r − α(l − v_r) − c)² (1 − α)r − α(l − v_r) − v D(r)

Taking derivative of the latter expression with respect to r, equal it to zero and simplifying gives:

And thus, the optimal order quantity is:

Q^CR∗ =³ (1 − α)r^∗− α(l − v_r) − c (1 − α)r^∗− α(l − v_r) − v

´ 2D(r)

r^CR∗ = 3α(l − vr) + 3v + (1 − α)k 4(1 − α)

+

p((1 − α)k + 8c − 9v − α(l − v_r))((1 − α)k − v − α(l − v_r)) 4(1 − α)

3.3.2 Decentralized System

In the decentralized system players act rational. In other words, they act individ-ually and seek to maximize their own profits instead of the total supply chain profits.

This section presents the respective formulas for the expected profits and order quan-tities of the manufacturer and retailer in a decentralizes setting. We first present the expressions in case of consumer returns are ignored in the player’s individual

optimization process. Secondly, we consider the setting when consumer returns are included in the decision process. The latter formulas are presented in Ruiz Benitez and Muriel (2007) [42], whereas Emmons and Gilbert (1998) [16] consider the former case.

Players Ignoring Returns

Depending on the values w and s, which are set by the manufacturer, the retailer faces profits according to

For a specific r, the problem can be reduced to the traditional newsboy problem.

Thus, the optimal order quantity in dependency of r is:

Q^IR∗(r) = D(r)F⁻¹³r − w r − s

´

(3.22)

Hence, the retailer’s profit function for the optimal order quantity and with given r is:

As under stochastic demand, the manufacturer finds her profits, given that the retailer acts optimally by choosing (Q^∗, r^∗), according to the following formula:

Π^IR_M(w, s; r^∗, Q^∗) = (w − c)Q^∗− (s − v)

Z _Q^∗_/D(r^∗₎

0

(Q^∗ − xD(r^∗))f (x)dx (3.24)

Identical to the centralized case, the inverse cumulative distribution function makes it difficult to analyze the formulas. Again, the assumption of a uniform

distri-bution on the interval [0, 2] and D(r) = b(r − k), the optimal order quantity and the retailer’s profit function, given a retail price r when ignoring returns, are, respectively:

Q^IR∗(r) =³r − w the item at a profit, since D(r) is negative for any r > w ≥ k. Taking the derivative of 3.23 and simplifying results in the retailer’s optimal resale price:

r^IR∗_R = 3s + k +p

(k + 8w − 9s)(k − s) 4

Consequently, the manufacturer’s optimal profits are

Π^IR_M(w, s; r_R^∗, Q^∗_R) = (w − c(1 − η₂))Q^∗_R− (s − v)(Q^∗_R)² 4D(r_R^∗) where η₂ = (r^∗_R− w)/(r^∗_R− s).

Players Considering Returns

When considering returns in the optimization process, the retailer expects profits, depending on w and s, to be as follows:

Π^CR_R (Q, r) = ((1 − α)r − α(l₂ − w) − s)(

For a given retail price r the latter expression can be reduced and the optimal order quantity is:

Q^CR∗(r) = D(r)F⁻¹³(1 − α)r − α(l₂− w) − w (1 − α)r − α(l₂− w) − s

´

(3.26)

The retailer’s profit function for the optimal order quantity and with given r is

The manufacturer finds her profits including the costs of returns according to the following formula. The retailer is expected to act rational again.

Π^CR_M (w, s; r^∗, Q^∗) = (w − c)Q^∗− (s − v)

Again, assuming f(x) to be a uniform distribution defined on [0, 2] and Demand to be of the type D(r) = b(r − k), we find the optimal order quantity and the profit function of the retailer:

Of course, the optimization formulae include the costs of returns. The manufac-turer’s share w + l1− vr, whereas the retailer faces r − w + l2 of the total return costs r + l − v_r. The following expressions for the manufacturer’s profit function and the retailer’s optimal price are obtained by Ruiz Benitez and Muriel (2007) [42]:

r_R^CR∗ = 3α(l₂− w) + 3s + (1 − α)k

Π^CR_M (w, s; r_R^CR∗, Q^CR∗_R ) = (w − c − α((w + l₁− v_r) + v)(1 − η₂))Q^CR∗_R

− (s + (α(w + l1− vr) − v))(Q^CR∗_R )² 4D(r_R^CR∗)

where η2 = ((1 − α)r_R^CR∗− α(l2− w) − w)/((1 − α)r^CR∗_R − α(l2− w) − s).