Long-term capacity reservation and utilization with bonuses
in a multi-period setting with uncertain demand
Long-term capacity reservation and utilization with bonuses
in a multi-period setting with uncertain demand
Ilse Helder
Abstract
This paper investigates the long-term capacity reservation strategy for a retailer assuming uncertain demand, where for a certain minimum total reserved quantity, a bonus is awarded. The goal is to determine the optimal capacity reserved quantities and order quantities for all periods of the planning horizon. A quasi Newton method is used for determining the optimal quantities. If a planning horizon of one period is considered, it can be optimal to reserve more than the order quantity. Assuming a planning horizon of more than one period, the optimal reservation quantity is equal to the optimal order quantity for the first period. Considering a time horizon of two periods, for each of the parameters indicated in the paper, a threshold is found at which the retailer is indifferent between two strategies: reserving quantities such that the bonus is obtained and reserving quantities not taking the bonus system into account.
1
Introduction
Most companies can reduce their cost associated with stock acquisition by focusing on inven-tory control (Axs¨ater(2015)). Therefore, companies should pay attention to inventory control in combination with the possible supply contracts they have. It is very common that a company has contractual agreements with the seller that obliges the seller to reserve capacity (Zhao et al.
(2007)). With those reservations, the seller can forecast his production more precisely. In order to stimulate its buyers to purchase products, the seller can decide to award bonuses for large reservation quantities.
One of the companies facing the capacity reservation problem and being offered a bonus for large reservations, is PON Automotive. PON Automotive is a company in the Netherlands that, among other things, imports passenger cars for several brands. They import cars from large manufacturers, who produce for multiple companies. We consider one (unnamed) manufacturer in particular. For the long-term planning and sometimes possibly even expansion, the manufacturer wants its customers to reserve capacity, specified per month, one year in advance. A bonus is awarded when at least a certain quantity is reserved in a year.
This paper investigates the long-term capacity reservation strategy for the buying company (retailer) assuming uncertain demand, where for a certain minimum total reserved quantity, a bonus is awarded. The goal is to determine the optimal capacity reserved quantities and order quantities for all periods (e.g. months) of the planning horizon (e.g. one year). If the total reserved amount (over all periods) is at or over some threshold, the retailer receives a bonus. The retailer is not obliged to order (at least) the reserved amounts. If the order quantity for some period differs from the reserved amount, then the retailer has to pay penalty costs. An important reason for ordering more (less) than reserved is that the demand in the previous periods has been above (under) expectations. If the demand for a period turns out to be larger than the number of products in inventory, products can be backordered, which comes with backorder costs. This means that there are no sales lost when the number of products in inventory is not sufficient. If there are more products in inventory than there is demand, the retailer has to pay holding costs (e.g. storage costs).
penalty costs are small. Furthermore, if the expected demand is large and demand variation small, it is expected that the retailer reserves many products. It is assumed that the products that are in inventory at the end of the considered time horizon cannot be sold. Therefore, it is expected that it is optimal to reserve more products for the first periods than for the last periods. However, having products in inventory comes with holding costs. Therefore, there will be a crossover point at which the retailer reserves equal quantities for the first and last periods. Our main focus is on finding that point.
The rest of the paper is organized as follows: In the next section, a review on relevant existing literature is given. In Section 3 the described problem is modelled, which includes the costs functions that will be optimized and definitions of the used parameters. Then, in Section4, the cost functions are investigated in detail and the optimization algorithm is discussed. In Section
5, the results for several parameter settings and the effects of the parameters are investigated. Finally, in Section6, the most remarkable results, shortcomings of the research and future research directions are discussed.
2
Literature review
Much is written on capacity reservation with stochastic demand in a single-period setting.Ritchken and Tapiero (1986) consider the procurement problem and derive conditions for using capacity reservation as a function both of managerial attitudes toward risk and of the correlation between price and demand.Hazra and Mahadevan(2009) consider the procurement problem where a buyer reserves capacity from one or more suppliers. They derive the reservation price, which is a function of the supplier’s capacity, amount of capacity reserved by the buyer and other parameters. Other papers combine capacity reservation with spot markets. For example, Wu et al. (2002) model capacity reservation when the output is produced in the high-tech industry. Seifert et al. (2004) develop and solve mathematical models that determine the optimal order quantity to purchase via reservation contracts and spot markets.
Besides papers on capacity reservation with stochastic demand in a single-period setting, there are contributions that consider the problem in a multi-period setting. Some papers investigate this problem with an uncertain production capacity. Ciarallo et al.(1994) show that in a single-period setting with single-periodic review, the optimal policy is not affected by the capacity uncertainty. However, in a multi-period setting, order-up-to policies that are dependent on the distribution of the capacity are optimal. Iida (2002) obtains upper and lower bounds of the optimal order-up-to levels for a periodic review inventory control policy. The paper ofCosta and Silver (1996) discusses the procurement problem where the demand and the production capacity are assumed to be discrete random variables with known probability distributions. The reservations for each period have to be made at the start of the first period. Boulaksil et al.(2017) consider a manufacturer who outsources the production activities to a contract manufacturer. After the reservation is made, the contract manufacturer determines the actual reserved capacity, based on allocation rules that are unknown to the original manufacturer. They indicated that an increase in the capacity uncertainty increases the order quantity. However, the effect of an increase in the capacity uncertainty decreases substantially when the demand uncertainty increases.
Another research stream discusses capacity reservation in combination with the spot market in a multi-period setting. Serel et al. (2001) investigate outsourcing of a firm with a periodic review inventory control policy framework. The capacity reservation contract entails delivery of any desired portion of a reserved fixed capacity in exchange for a guaranteed payment by the buyer. Serel(2007) andInderfurth and Kelle(2011) consider the procurement model. Serel(2007) assumes uncertainty in the quantity available in the spot market, whileInderfurth and Kelle(2011) examine the joint effect of demand and spot market price uncertainty. The paper ofInderfurth et al. (2013) focuses on the cost-effective management of the combined use of two procurement options: a spot market with random price, and a multi-period capacity reservation contract with fixed purchase price and reservation level. They derive the structure of the optimal policy using stochastic dynamic programming.
uncertainty by simulation. Lee and Li(2013) and Akbalik et al. (2017) investigate the lot sizing problem with capacity reservation. Lee and Li(2013) study the general lot sizing problem with capacity reservation and the polynomial time solvable special cases of the problem and propose corresponding algorithms for them. Akbalik et al.(2017) investigate the case where a manufacturer is replenished by an external supplier with batch deliveries.
The following papers are not directly related to our research, but we discuss some seminal papers. Much is written on bonuses in the context of employer’s salaries. The effects of cash and stock bonus systems are investigated in the paper ofHan and Shen(2007). In their paper, the size of the bonus is based on the company’s profits, combined with pay grade, seniority, position and individual performance rating. Edmonds et al. (2013) investigate the effects of missing analysts’ revenue forecasts on the CEO bonus compensation. They consider a bonus system based on revenue.
Although many authors have considered the capacity reservation problem, to the best of our knowledge, none have considered capacity reservations for situations where bonuses can be achieved when the total reservation quantity across all periods is larger than a certain amount, which is studied in this paper.
3
Model
A company is considered that receives products from a manufacturer who supplies products to several companies. For the company, the contracted manufacturer is the only supplier of the products. The company reserves products for the multiple periods of a finite time-horizon. It is assumed that the products that are in inventory at the end of last period can not be sold any more. The notation of the parameters and variables can be found in Table1.
Table 1: Definition parameters Parameters
T Number of periods in the planning horizon p Purchase price per product
u Penalty costs per item that the order quantity is under the reserved quantity a Penalty costs per item that the order quantity is above the reserved quantity h Inventory holding costs per unit per period
b Backorder costs per unit per period
s Bonus amount
m Minimal total reserved quantity for receiving the bonus Random variables
Dt (Random) demand in period t
ft(dt) Probability density function of the demand in period t
dt Actual demand in period t
Decision variables
rt Reserved quantity determined before the first period for period t
qt Order quantity in period t
xt Inventory position at the start of period t before the order arrives
A planning horizon of T periods is considered. Before the first period of this horizon, reserva-tions are made for the T periods, specified per period. When the total reserved quantity is larger than bonus level m, the bonus s is awarded. The company determines the actual order quantities before the start of each period. Per item that the order quantity is under or above the reserved quantity, a penalty has to be paid, u or a respectively. We assume that these penalty parameters are non-negative. The purchase price is p per product. At the start of period t, the products are delivered and the purchase costs and possibly penalty costs for period t have to be paid. The purchase costs are given by qt· p and the penalty costs are given by u · (qt− rt)−+ a · (qt− rt)+.
The random demand, Dt, is assumed to be independently distributed in each period considered
in the time horizon T with known distribution function. At the end of period t, the demand dt
is the sum of the inventory position at the start of period t and the order quantity of period t minus the demand in period t. In mathematical notation, the inventory position can be written as in Equation3.1.
xt+1= xt+ qt− dt. (3.1)
In figure1, a time line of the process is shown.
Figure 1: Timeline
There are two types of decisions that have to be made in this problem:
1. Before the start of the first period, the reservations (r1, ..., rT) have to be determined.
2. Before the start of each period t (t = 1, ..., T ), the order quantity (qt) has to be determined.
3.1
Cost functions
There are several moments in the time horizon at which decisions have to be made. At the start of period t, the order quantity qt has to be determined. At that moment, the reserved quantity,
rt is known. Furthermore, the order quantity for the previous periods, q1, ..., qt−1, the demands
d1, ..., dt−1and the inventory positions x1, ..., xt−1are known. Dtis the random demand in period
t with density function ft(dt). The cost function for period t consists of purchase costs, penalty
ct(qt; rt, xt, Dt) = qtp |{z} purchase costs + (qt− rt) + a + (qt− rt)−u | {z } penalty costs + Ht(xt, qt, Dt) | {z }
holding and backorder costs
, (3.2) where Ht(xt, qt, Dt) = h Z xt+qt 0 (xt+ qt− dt)ft(dt)ddt | {z } holding costs + b Z ∞ xt+qt (dt− xt− qt)ft(dt)ddt | {z } backorder costs . (3.3)
At the start of the first period, the reservation quantities for all periods and the order quantity have to be determined. The cost function for the first decision moment is the sum of direct costs and expected costs of the upcoming periods and is given by Equation3.4. The direct costs consists of possibly the bonus and the cost for period 1, which includes purchase costs, penalty costs, holding and backorder costs. The costs for the upcoming periods, for which the optimal order quantities are determined at the start of the corresponding period, is not fixed at the start of the first period. However, it is assumed that the demand distributions for all periods is known. Therefore, an expectation of the costs is included in the costs for the first period.
c(r1, ..., rT, q1; D1, ..., DT) = q1p |{z} purchase costs − δs |{z} bonus + (q1− r1) + a + (q1− r1) − u | {z } penalty costs + H1(0, q1, D1) | {z }
holding and backorder costs
+ T X t=2 E [{ct(qt∗; rt, xt, Dt)}] | {z } expected costs , (3.4) where δ = ( 1 if PT t=1rt≥ m 0 otherwise , (3.5) and q∗t = arg min qt≥0 ( ct(qt; rt, xt, Dt) + T X t=t+1 E [ct(qt; rt, xt, Dt)] ) . (3.6)
4
Optimization
First, note that if we are considering a time horizon of at least two periods, it is optimal to set the reservation quantity for the first period equal to the order quantity. If one would select r1 > q1,
there is another strategy with r01 = q1 and r02 = r2 + (r1− q1), which is more flexible. This
alternative strategy is always at least as good as the original strategy, because the total number of reservations does not change and the penalties for reserving are at least as small as they were. A proof for the statement that r∗1= q∗1 can be found in AppendixA.
Now, we will show how we can optimize the cost function for period T . Since we consider the last period, there are no expected costs for other periods. Therefore, the optimal qT can be
found by minimizing cT(.). First, there is a linear part in the cost function, which is obviously
convex. Then there are penalty costs that are also convex. Furthermore, HT(.) is a convex function
(Porteus(2002)). Consequently, since the sum of convex functions is convex, the cost function for period T , is convex. Taking the constraint qT ≥ 0 into account, does not change the fact that
we are minimizing a convex function. The constrained minimization problem can be presented as given in expression4.1.
min cT(qT; rT, xT, DT)
s.t. qT ≥ 0
(4.1) Since this is an optimization problem with an inequality constraint, we will use Karush-Kuhn-Tucker conditions (Karush(1939),Kuhn and Tucker(1951)). The Lagrangian is given by Equation
4.2.
The corresponding Kuhn-Tucker conditions are given by Equations4.3,4.4and4.5.
c0T(qT; rT, xT, DT) − µ = 0 ⇐⇒ µ = c0T(qT; rT, xT, DT) (4.3)
µ · qT = 0 (4.4)
µ ≥ 0 (4.5)
Substituting the expression for µ from Equation4.3into Equation4.4gives result4.6.
c0T(qT; rT, xT, DT) · qT = 0 ⇐⇒ c0T(qT; rT, xT, DT) = 0 or qT = 0 (4.6)
Due to the penalty costs, the cost function is not differentiable at qT = rT. Since the
mini-mum can not be found in an exact way, we will use a numerical optimization method (See Sec-tion4.1). Note that qT∗ depends on rT and xT, where rT is determined in the first period and
xT = xT −1+ qT −1− dT −1 is determined in the previous period. The optimal qt for period t 6= T
can be found by minimizing the sum of ct(.) and the expected costs in upcoming periods. The cost
function of the first period is not continuous at the point wherePT
t=1rt= m. Furthermore, the
expectation of the costs in the upcoming periods are included in the cost for each period. That part of the cost function on its turn depends on the reservation quantities. Due to the complexity of this cost function, we will use a numeric algorithm to find the optimum. In Section 4.1, the algorithm is introduced.
4.1
Algorithm
Since we have to deal with a nonlinear optimization problem and we have to deal with a discon-tinuous function, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is used. It is proven that the BFGS algorithm performs well for non-smooth functions (Lewis and Overton(2013)). In Algorithm1, the pseudo-code for the BFGS algorithm can be found. The BFGS algorithm is a quasi-Newton method, which uses hill-climbing optimization techniques to search for a stationary point of a function. The Hessian is not computed, but approximated by using updates specified by (approximate) gradient evaluations. For the optimization, the algorithm is used with starting points in each of the segments that exist since we have to deal with a discontinuous cost function. Algorithm 1 BFGS algorithm
Require: ε > 0 (tolerance), K (maximum number of iterations)
1: x0∈ Rn, H0> 0 (for example In) 2: for k = 0 to K do 3: if ||gk|| then 4: break 5: end if 6: dk = −Hk−1∇f (xk)
7: Compute optimal step size ρk
5
Results
The results are obtained by implementing the cost function in R. For a time horizon of both one and two periods, the results are discussed in this section. If a time horizon of more than two periods is considered, the cost function has more than three dimensions and can not be presented in a levelplot. Therefore, we will focus on the one and two period setting. The demand is assumed to be Gamma distributed, because the Gamma distribution is adequate for representing the demand of both slow and fast-moving products (Burgin (1975)). Furthermore, it is defined only for non-negative values, which is realistic for demand. The shape parameter of the Gamma distribution, k, is assumed to be a non-negative integer. The initial parameter settings of the model are given in Table2.
Table 2: Initial parameter settings Symbol Value Definition
p 2 Purchase price per product
u 0.5 Penalty costs per item that the order quantity is under the reserved quantity a 0.5 Penalty costs per item that the order quantity is above the reserved quantity h 0.1 Inventory holding costs per unit per period
b 4 Backorder costs per unit per period
s 50 Bonus amount
m 100 Minimal total reserved quantity for receiving the bonus
kt 8 Shape parameter of the Gamma distribution for the demand in period t
θt 5 Scale parameter of the Gamma distribution for the demand in period t
ε 1.5e−8 Tolerance parameter for the BFGS algorithm
K 100 Maximum number of iterations for the BFGS algorithm
The backorder costs are twice the costs of purchasing an item, which is because it is assumed that the costs of unsatisfied demand are high. The holding costs are smaller than the backorder costs, which means that is it assumed that it is more expensive to backorder an item than to stock an item. The penalty costs per item that the order quantity is under the reserved quantity are the same as the costs per item above the reserved quantity. This implies that the manufacturer depreciates an order quantity that is larger than the reserved quantity as much as an order quantity that is smaller than the reserved quantity. Note that the parameters for the Gamma distributed demand are assumed to be equal across periods, with mean demand kt· θt= 8 · 5 = 40 and variance
kt· θt2= 8 · 52= 200. The bonus of 50 is obtained if the total reserved quantity is more than 100.
5.1
One period
Since, to the best of our knowledge, the reservation problem combined with bonuses is not con-sidered in the literature before, we will start with considering only one period. The cost function boils down to Equation5.1.
c(r1, q1; D1) = q1p − δs + (q1− r1) + a + (q1− r1) − u + h Z q1 0 (q1− d1)f1(d1)dd1+ b Z ∞ q1 (d1− q1)f1(d1)dd1, (5.1) where δ = ( 1 if r1≥ m 0 otherwise (5.2)
In Figure 2, a level plot can be found of the cost function using the initial parameter settings. At r1 = 100, the function is not continuous. For r1 < 100, the bonus will not be obtained, for
r1 ≥ 100, it will be obtained. In general, there are three possibilities for an optimum. The first
one is that the reserved quantity, r∗1, is equal to the bonus level, m, and larger than the order
quantity, q∗
1. This is optimal if the penalty costs for ordering less than reserved are much smaller
local optimum is that the reserved quantity is equal to the order quantity. It is optimal to select r1∗= q1∗ if expression5.3holds.
c(m, q∗1; D1) < c(q∗∗1 , q∗∗1 ; D1), (5.3)
where q∗∗1 is the optimal order quantity not taking a bonus into account. The third possible
optimum is that the total optimal order quantities are already larger than the bonus level. The reservation quantities are then set equal to the order quantities and the bonus is obtained. From the plot it can be seen that it is optimal to select q1 6= r1 = 100. Which implies that the bonus
will be obtained, while the order quantity is smaller than the reserved quantity.
Figure 2: Levelplot of the cost function for one period using the initial parameter settings
5.1.1 The effect of parameter values
First of all, the effect of the bonus amount, s, is investigated. A plot of the optimal reservation and order quantity can be found in Figure 3. It can be seen that if the bonus amount is smaller than 30, the optimal number of reservations is equal to the optimal order quantity. If the bonus amount is larger than or equal to 30, the optimal reserved quantity is equal to the bonus level, which is larger than the optimal order quantity. Note that the optimal order quantity is smaller if the bonus amount is smaller than 30 than it is for other values of the bonus amount. This can be explained by the penalty costs for ordering less than reserved.
In Figure4, the results for different values of the penalty costs for selecting an order quantity that deviates from the reserved quantity can be found. Note that only the cost per item ordered less than reserved, u, has influence. If only one period is considered, it is never optimal to order more than the reserved quantity, because the decision moments take place at the same time and reserving less than the order quantity will not help reaching the bonus level. For penalty costs per item more than 0.8, it is not optimal to reserve a quantity such that the bonus is obtained. This is in line with expectations. If the penalty costs are small, it can be optimal to reserve a lot more than the optimal order quantity. For large penalty costs, this is not the case. Another result is that for the values of the penalty costs for which the bonus is obtained the following holds: the larger the penalty costs, the larger the optimal order quantity. If the costs per item ordered less than reserved increase, selecting the order quantity closer to the reserved quantity, which probably comes with holding costs and unsold items, is less expensive than keeping the order quantity at the same level.
For values of the bonus level up to and including 140, the reserved quantity is selected such that the bonus is obtained. For larger values of the bonus level, the reserved quantity is equal to the order quantity. We can conclude that for some value of the bonus level between 140 and 145, the retailer is indifferent between reserving more that the optimal order quantity in order to obtain the bonus and reserving the optimal order quantity.
Figure 3: The effect of the bonus amount Figure 4: The effect of penalty costs
Figure 5: The effect of the bonus level
The other parameters do not have a direct influence on whether the retailer goes for the bonus, or not. However, they do influence the optimal order quantity. If the optimal order quantity becomes much smaller than the bonus level, it could be that it is not optimal any more to go for the bonus. Increasing the backorder costs lead to an increase of the optimal order quantity, which increases the total costs. The retailer orders more products, such that the probability of getting out of stock, which comes with backorder costs, reduces. The larger the value of the parameter for backorder costs, the smaller the increase in the order quantity and consequently the total costs. The holding costs have a similar but opposite effect on the order quantity and total costs. If the holding costs increase, the retailer reduces the probability of having products in stock at the end of the period. Note that if the holding costs are large, it optimal is to reserve the order quantity.
As mentioned before, the demand is Gamma distributed. One of the parameters of the Gamma distribution determines the scale of the distribution and is denoted by θ. The larger θ, the larger the mean demand and variance. If θ increases, the optimal order quantity increases, which increases the total costs. The larger θ, the larger the increase in optimal order size and consequently, the larger the total costs. For small values of the scale parameter, it is optimal to set the reserved quantity equal to the order quantity. The parameter of the Gamma distribution that determines the shape, is k. The larger k, the fatter the tails of the distribution and the larger the mean. An increase of k leads to an increase of the optimal order quantity, q∗
5.2
Two periods
Now, we consider a time horizon of two periods, which means that there are two decision moments. At the start of the first period, the order quantity for the first period and the reservation quantities for both periods are determined. At the start of the second period, only the order quantity of the second period is determined. For determining q2, the cost function c(q2; r2, q1, d1, D2) is minimized.
At the start of the first period, q1= r1and r2 are determined by minimizing the cost function for
that period. The cost for the first period as a function of q1 and r2 are given in a level-plot in
Figure6.
Figure 6: Levelplot of the cost function for two periods using the initial parameter settings
As can be seen from the cost function and the plot, the function is discontinuous at q1+r2= m.
The minimum can be found at about a reservation quantity of 70 for the first period and 30 for the second period.
5.2.1 Bonus
Keeping the other parameters constant, we vary the minimal total reserved quantity for receiving the bonus, which we will refer to as bonus level, m. In Figure 7, the results are presented for different values of m, the bonus level. For m = 75 and smaller values for m, the results are the same. The optimal strategy not taking a bonus into account has an r1∗and r∗2 such that the bonus is already obtained. For m = 190 and larger values of m, the sum of r∗1 and r∗2 is smaller than
the bonus level m. Therefore, the bonus will not be obtained. For values of m larger than 75 and smaller than 190, it is optimal to reserve amounts such that r∗1+ r∗2≥ m. The reservations for the
first period keep approximately constant for different values of the bonus level. An explanation of this result could be that the number of reservations for the first period is already enough to satisfy demand. Reserving more for the first period would yield holding costs, which do not have to be paid when the reservations are done for the second period.
Figure 7: Results for different bonus levels Figure 8: Results for different bonus amounts
5.2.2 Purchase Price
The purchase price is paid only for the items that are actually ordered. If a product is reserved, but not ordered, no purchase price has to be paid. In Figure9, the results for different values of the purchase price can be found. If the purchase is zero, the total reserved quantity is larger than the bonus level. This can be explained by the initial parameter settings, with the backorder costs being much larger than the holding costs. Therefore, ordering more products for the second period is cheaper than possible backorders. The larger optimal order quantity, the larger the optimal reserved quantity, which can be seen in the graph. The larger the purchase price, the smaller the order quantity and reservation quantity for the first period, because backordering is relatively cheaper. In order to reach the bonus level, which is optimal for all values of the purchase price we investigated, the reserved quantity for the second period increases with the purchase price.
Figure 9: Results for different values of the purchase price
5.2.3 Holding and backorder costs
Figure 10: Results for different values of the backorder costs per item
Figure 11: Results for different values of the holding costs
5.2.4 Penalty costs
The effects of penalty costs for ordering a different quantity than reserved are investigated. In Figure 12, the results for different values of the penalty costs for ordering more than reserved, a, are presented in a graph. First, note that it is always optimal to select the order quantity of the first period equal to the reserved quantity. Consequently, the penalty for ordering more than reserved does not have influence on the reservations for the first period. The optimal reserved quantity for the second period does not differ among different values for a. If there is no penalty for ordering more than reserved and the bonus system is not taken into account, it would be optimal to reserve no products for the second period. However, since the bonus amount is large and/or the bonus level is small, it is optimal to reserve products such that the bonus is obtained. Consequently, the reservation quantity for the second period is selected such that the bonus is obtained. Considering the demand distribution and the reserved quantity for the second period, the probability that the order quantity would be larger than the reserved quantity is close to zero. Therefore, the probability that penalty costs for ordering more than reserved have to paid is close to zero. Hence, using the initial parameter settings, the reservation quantities do not depend on a.
Figure 12: Results for different values of the penalty costs for ordering more than reserved
ordering less than reserved is costless, while ordering more than reserved is not. In case there is a bonus system, reservations for the second period are selected such that bonus level is reached. One would expect that at some point it is too expensive to select the reservation quantities such that the bonus level is reached. However, the bonus level is too small and/or the bonus amount is too large to make that happen. In case there is no bonus system, the larger the penalty u, the smaller the optimal reserved quantity for the second period. This can be explained by the larger costs if one orders less than reserved. The optimal order quantity of the first period is proven to be equal to the optimal reserved quantity. Therefore, the optimal reserved quantity for the first period only slightly increases when the penalty for ordering less than reserved increases.
Figure 13: Results for different values of the penalty costs for ordering less than reserved
5.2.5 Demand
The demand distribution has much influence on the reservation and order strategy. As mentioned before, a Gamma distribution is assumed for demand. With the initial parameter settings, the distribution is assumed to be the same for both the first and second period. One of the parameters of the Gamma distribution is the scale, it determines in what extend the distribution is spread out. The larger the scale parameter, the more spread out the distribution. In Figure14, the results for some values of the scale can be found. We can conclude that the larger the scale parameter, the larger the reserved quantity for the first period. This is a result of the increasing mean demand, while the decrease in variance has less influence. Larger mean demand for the first periods leads to a larger optimal order size, which is equal to the optimal reservation quantity. The optimal reserved quantity for the second period is selected such that the bonus level is reached.
Next to the scale parameter, we investigate the effects of the shape parameter of the Gamma distributed demand. The larger the shape parameter, the larger the mean and the variance. A Gamma distribution with k = 0 is an Exponential distribution and for large k, the Gamma distribution converges to a Normal distribution. In Figure15, the results for several values of the shape parameter are presented. From this graph it can be concluded that the higher the value of the shape parameter, the larger the optimal reserved quantity for the first period. This is a consequence of the increase in the mean and variance in demand. For values of the shape parameter equal to 10 or smaller, the optimal reserved quantity for the second period is such that the bonus level is reached. For larger values of the shape parameter, the reserved quantity for the second period slightly increases with the shape parameter. This is caused by the increase in the mean and the variance of demand.
Figure 14: Results for different values of the scale parameter of the Gamma distributed de-mand
Figure 15: Results for different values of the shape parameter of the Gamma distributed de-mand
period is investigated. The demand of the first period is Gamma distributed with the initial pa-rameter settings. The shape papa-rameter and scale papa-rameter of the second period change, while the mean demand is kept constant. In Figure16, the results can be found for different values of the variance in demand. The larger the variance of demand in the second period, the larger the optimal reserved quantity of the first period. Compared with the variance in the second period, the variance in demand for the first periods decreases. This means that one is more certain about the demand in the first period, which increases the order quantity (and the reservation quantity).
Figure 16: The effect of the variance in demand
6
Conclusion
In this paper, the long-term capacity reservation strategy for the retailer assuming uncertain demand, where for a certain minimum tot reserved quantity, a bonus is awarded, is investigated. A numerical study is conducted in order to reveal insights in the effects of the parameters on the reservation strategy. First, a time horizon of one period is considered, after which a time horizon of two periods is considered.
reser-vation and order quantity for the first period. Again, the reserreser-vation quantities for the second period are selected such that the bonus is obtained. Conversely, increasing holding costs decrease the optimal reservation and order quantity for the first period. With the parameter settings used in this research, the penalty costs for ordering a different amount than reserved do not have an effect on the reservation quantities. Only if there are no costs for ordering less than reserved, the optimal reservation quantity for the second period is larger than for other values. Increasing the scale parameter of the Gamma distributed demand leads to an increase in the optimal reservation quantity for the first period. For small values, an increasing scale parameter decreases the reser-vations for the second period, while for larger values, it increases the reserreser-vations for the second period. The same effects are observed for the shape of the Gamma distributed demand. If the variance of the demand for the second period increases, the reservation and order quantity for the first period increases. The reservations for the second period decrease just enough for the bonus to be obtained.
In this research, assumptions are made in order to create a model that can be dealt with. For example, it is assumed that there is a zero lead-time. The orders arrive directly when ordered, which is not realistic. Furthermore, there is no discount factor. The costs for the second period are not discounted to the present value. The reservations are done at the start of the first period, while in practice, it would take place earlier. Finally, the penalty costs for ordering a different amount than reserved are assumed to be linear. In practice, it is more realistic to have larger penalty costs for larger deviations. Small deviations are probably not very problematic for the manufacturer, while large deviations can be. Besides relaxing these assumptions, future research could consider a time horizon with more than two periods. For example, a time horizon of 12 periods can be considered, where each period is a month and reservations can be done one year ahead.
A
Proof reservations equal the order quantity for the first
period
Consider a time horizon of two or more periods. First of all, for the first period it would never be optimal to reserve less than the order quantity. It leads to penalty costs due to ordering more than reserved, and reserving less does not help reaching the total reserved amount needed for obtaining the bonus. Reserving more than the order quantity for the first period could be helpful for reaching the bonus reservation level. In order to find out whether it can be optimal to reserve more than the order quantity for the first period under some conditions, we explore the cost function in more detail. We have to focus on the penalty costs for order quantities that differ from the reserved quantities, and the bonus s, which is rewarded if the total reserved quantity is at least bonus level m. Only these parts of the cost function depend on the reserved quantities. In EquationA.1 an expression for the costs of interest can be found.
cres= T X t=1 (qt− rt)+a + (qt− rt)−u − δs, (A.1) where δ = ( 1 if PT t=1rt≥ m 0 otherwise . (A.2)
Suppose the parameters settings are such that it is not profitable to select the reserved quantities in a way that the bonus is obtained. Then the reserved quantities are selected such that the penalty costs are minimized. Consequently, since the reserved quantity and order quantity for the first period are determined at the same moment in time, r∗
1 = q1∗ is optimal. There are also
situations in which the bonus system has influence on the reserved quantities. In those cases, we havePT
t=1rt= m. Suppose r1 > q1∗ is selected for obtaining the bonus. Then, moving (r1− q1∗)
to r2, which will then be denoted by r∗2 and r∗1= q∗1, has the following three possible consequences
1. The reserved quantity for the second period is still smaller than or is now equal to the order quantity, r2∗≤ q2. Then the cost for the second period decrease with (r1− q1∗)a and the cost
2. The reserved quantity for the second period is larger than the order quantity, while it was larger than or equal to the order quantity, i.e. r2∗> q2∗, while r2≥ q∗2. Then the cost for the
second period increase with (r1− q1∗)u, which is exactly the amount at which the cost of the
first period decrease.
3. The reserved quantity for the second period is larger than the order quantity, while it was smaller than the order quantity, i.e. r2∗ > q2∗, while r2 < q2∗. The cost for the first period
decrease with (r1− q∗1)u. The cost for the second period decrease with (q2∗− r2)a and increase
with (r∗2− q∗
2)u. The total cost decrease is given by EquationA.3.
(r1− q1∗)u + (q ∗ 2− r2)a − (r2∗− q ∗ 2)u = (r1− q1∗)u + (q ∗ 2− r2)a − (r2+ (r1− q∗1) − q ∗ 2)u = (r1− q1∗)u + (q2∗− r2)a − (r2− q2∗)u − (r1− q∗1)u = (q∗2− r2)(a + u) (A.3) Since in the described case we have that r2< q∗2, and we assume that a ≥ 0 and u ≥ 0, the
decrease is positive.
In the first and third case, selecting r1 = q1∗ instead of r1 > q∗1 leads to a cost decrease. In the
second case, the cost in unchanged. Hence, selecting r1= q∗1 is always at least as good as selecting
r1> q∗1.
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