Dynamic demand allocation in a multi-channel company
M.H. Jacobs
Master thesis Operations Research
Supervisors Rijksuniversiteit Groningen: Prof. Dr. K.J. Roodbergen and A.S. Dijkstra Co-assessor Rijksuniversiteit Groningen: Prof. Dr. R.H. Teunter
Dynamic demand allocation in a multi-channel company
Maartje Jacobs
Abstract
Multi-channel companies often handle their online and offline supply chain separately. This thesis con-tributes to the research on integrating both supply chains. The goal is to minimize the total costs, which consists of handling, transportation, holding and lost sales costs. We developed an Integer Linear Pro-gramming Allocation Heuristic (ILPAH), which dynamically decides on the allocation location for each incoming demand, taking the current inventory level and the multiple future demand scenarios into ac-count. We compare the performances of the ILPAH with the total costs of the lower bound and with four greedy allocation heuristics we constructed. Experiments show that the ILPAH performs better than all four greedy heuristics, and that the percentage of the costs above the lower bound is small for large initial inventory levels. Moreover, the integrated supply chain performs better than the separated supply chain, and a cost reductions up to 33% can be obtained if the online demand is 66% of the total demand. In addition, a case study on a multi-channel company is performed, where the total costs of the integrated supply chain are equal to the lower bound.
Contents
1 Introduction 4
1.1 Situation . . . 4
1.2 Goal, scope and research framework . . . 5
1.3 Outline . . . 7
2 Literature 8 3 Problem description 11 3.1 Problem description . . . 11
3.2 Mathematical model: allocation with perfect information . . . 13
3.3 Lower bound . . . 15
4 Heuristics 16 4.1 Demand arrival . . . 16
4.2 ILPAH: Integer Linear Programming Allocation Heuristic . . . 17
4.3 Greedy heuristics . . . 19
5 Model evaluation 22 5.1 Situations . . . 22
5.2 Randomness . . . 22
5.2.1 Number of cycles and scenarios . . . 23
5.3 Experiments and Results . . . 26
5.3.1 Sensitivity analysis on cost parameters . . . 29
5.3.2 Other demand parameters . . . 31
6 Case study 36 6.1 Experiments and Results . . . 37
1
Introduction
1.1
Situation
In the mid 1990s, the CD Ten Summoners Tales by Sting was sold via the Web retailer NetMarket. It was the first secure retail transaction on the Internet (Cnet.com, 2004). Since then, e-commerce has grown enormously. For example, the revenue of online shopping in the Netherlands rose from 2.8 billion Euros in 2005 to 10.6 billion Euros in 2013 (Thuiswinkel.org, 2013).
With the growth of e-commerce, more and more multi-channel companies arise. Multi-channel companies are serving offline customers in normal shops as well as online customers in their web shop. The rise of these companies comes from pure e-tailers, who are opening physical shops, and traditional retailers, who develop web shops to serve their customers. There are multiple advantages when the two channels are combined. For example, customers can do research on the Internet before buying the products in a physical shop. In addition, when a product is not available in the shop, customers can order the product online. Another ad-vantage of multi-channel companies is that they can offer the possibility of picking up the order in a physical shop, instead of home delivery.
Multi-channel companies have different characteristics than single-channel companies. First, a multi-channel company has different types of customers. Some prefer to order their products offline, while other prefer to order online. Second, there is a difference in the inventory locations. While a traditional retailer has a distribution center (DC) from where the shops are supplied, a multi-channel company also has a dedicated location for online orders (e-DC) where the products are picked and packed and then delivered to the cus-tomers. The supply chain of the multi-channel company is therefore different than the supply chain of a single-channel company.
The relevant part of the supply chain of a classical multi-channel company is given in Figure 2. There are two ways where the two channels are handled completely separate. The customer can either go to a shop and buy his product there (green arrow in Figure 2), or order online and receive his product home delivered from the e-DC (purple arrow in Figure 2). There is also the possibility that the customer picks up his online ordered product in the shop. In the supply chain of the current practice, the product is then shipped from the e-DC to the shop, and the customer comes there to pick his product (the red arrows in Figure 2). From the customers view, the two channels are combined because the shops are involved in the process of online shopping. However, from the supply chain view, the process is still separated because the shop inventory is only used for offline customers and the e-DC inventory is only used for online customers. A disadvantage of the current practice is that an online customer cannot be served if the e-DC is out of stock, although some of the shops might still have the product in their inventory. When the inventory of the shops can be used to fulfill the demand of online customers, the customer can still be served and will not be disappointed. In this integrated supply chain, there are two new ways for the online customer to receive his product, apart from the ways already described in Figure 2. The first is that an online customer can pick up his product at a shop, and that the inventory of the shop is used (blue arrow in Figure 3). The second is that the customer can receive his product at home, and the inventory of a shop is used (yellow arrow in Figure 3). In this way, also from the supply chain perspective the two channels are integrated, because the inventory of the shops can be used for both offline and online customers. This thesis contributes to the study of the integrated supply chain of a multi-channel company.
Figure 2: Relevant part of the separated supply chain of a classical multi-channel company. Shop inventory is only used for offline customers, and e-DC inventory is used for online customers.
has to decide which demand is allocated to which location.
Companies want to serve their customers. If a customer wants to order a product which is not available anymore, the customer cannot be served. We say that the customers is lost. There are different costs in-volved with the supply chain of a multi-channel company. If there are a lot of products stored, the chances of stocking out are low, and almost all customers can be served. However, holding costs must be paid over the products in the inventory. Therefore, companies do not want to keep very high levels of inventory. However, with lower inventory levels, the risks of stocking out rise. A method to prevent lost customers is to transport a product from a location which has inventory to another location which is out of stock. Of course, trans-portation costs must be paid if that happens. If a customer orders online, serving this customer means that the product has to be picked from the shelves and packed into a box to be ready for delivery. This takes time from an employee of the company, and the costs which are involved are called handling costs.
1.2
Goal, scope and research framework
The goal of this research is to develop a model which minimizes the cost of the allocation of offline and online demand in a multi-channel company with multiple inventory locations.
Figure 3: Relevant part of the integrated supply chain of a multi-channel company. Shop inventory can be used for both offline and online customers, and e-DC inventory is only used for online customers.
online order is placed, the model decides which storage location is used to fulfill this demand. The relevant part of the supply chain of this system is shown in Figure 3.
Within most of the research done on supply chains, the type of the customer that orders the product (offline, online with home delivery or online with pick up in a shop) determines the location where the demand is served. This is called a static order allocation policy. One could imagine that this is not necessarily the best policy. For example, if the static allocation policy allocates a customer to a location without inventory, the customer is lost. If one could decide on the allocation location at the moment the demand occurs, one could allocate the demand to a location which still has inventory, and thereby prevent the lost customer. To use such a dynamic allocation policy, it is important that the inventory levels at the locations are known at all times. The dynamic allocation policy might reduce the costs compared with the static allocation policy. Hence, studying such a dynamic allocation policy is interesting, and the following research questions are formulated.
Research questions:
• Which inventory location should be used for an incoming order? • What is the influence of the different costs?
• How does the future influence the allocation decision?
• How do the current inventory levels influence the allocation decision?
1.3
Outline
2
Literature
The supply chain of retailers has received a lot of attention in the literature. One of the first models on the supply chain of a retailer is from Eppen and Schrage (1981). In their model, they have a central depot which supplies multiple shops, and they assume that the demand at the shops is normally distributed. After fixed time intervals, the inventory levels at the shops are checked and the inventory will be replenished. The check of the inventory levels at fixed times is called period review, and since an order can only be placed at these fixed times, they ordering method is called periodic ordering. The amount that is ordered from the central depot is the difference between a certain level and the current inventory level. The certain level is called the order-up-to level, and this ordering policy is named a base stock policy. The model derives approximately optimal order-up-to policies. Many articles on supply chains build on the work of Eppen and Schrage (1981). One of the major differences between the model of Eppen and Schrage (1981) and the model in this thesis is that the inventory of different locations can be shared. This is called pooling. If one shop has a low inventory level, another shop could send some products in order to prevent the first of going out of stock. This delivery of products is called a lateral transshipment. In a classical company, there are no lateral trans-shipments. Every shop serves his own customers with his own inventory. In our model, there are also no lateral transshipments between different shops. However, (virtual) transshipments between a shop and the e-DC are possible. These types of transshipments are handled later in this section.
Paterson et al. (2011) give an overview of the research on inventory models with lateral transshipments. They devide the types of lateral transshipments into two categories: proactive and reactive lateral trans-shipments. Where proactive lateral transshipments are transshipments in order to redistibute the inventory at predetermined moments in time, reactive lateral transshipments respond to situations where one of the inventory locations faces a stock out. The last type is also called emergency transshipments.
E-commerce brings new difficulties to the traditional supply chain. Where there were only ”normal” cus-tomers in a classical company, there now are different types of cuscus-tomers: offline and online. Agatz et al. (2008) write a review on e-fulfillment and multi-channel distribution. They devide the literature into two classes. The first class is about sales and service design, which includes transportation planning and deliv-ery service planning. The second class, about supply management, discusses distribution network design, warehouse design, and inventory and capacity management.
Another overview is given by Swaminathan and Tayur (2003). They look at models for supply chains in e-business. Among other things, they discuss the difficulties of the distribution of goods in these supply chains and the advantages of the real-time acces to information. These topics are important in this thesis. Continu-ous review is real-time information on the inventory levels. In this thesis, continuContinu-ous review is used to make a decision on the allocation location of the demand of a customer. The distribution of goods is difficult in two ways. We use periodic ordering, and hence the replenishments from the DC are after fixed periods of time. We call the period between two replenishments a (replenishment) cycle. One of the difficulties is that, at the start of the replenishment cycle, one should decide on the distribution of the products among the inventory locations: how many products are distributed from the DC, and what fraction of those products is put in the e-DC and what fraction in the shops. This decision is beyond the scope of this thesis. However, within a replenishment cycle, the allocation of an online customers demand affects the distribution of products from the e-DC to a shop or the home of the customer. This type of distribution is part of the research of this thesis. In the separated supply chain of a multi-channel company, the e-DC handles all the online customers, and the shops serve the offline customers. The integrated supply chain in this thesis merges these channels; the shops can also handle online customers demand. When this happens, it can be seen as a virtual lateral trans-shipment. For example in case of a home delivery, the product from the inventory of the shop is virtually send to the e-DC, and then send to the home of the customer. Of course, the product can be home delivered straight from the shop, but from the view of the inventory, it can be seen as virtual lateral transshipments.
their own area. Lateral transshipments take place if shops serve online customers coming from another area. His model makes use of periodic review, as the model of Eppen and Schrage (1981). When a customer cannot be served because the shop is out of stock, a backorder is placed: the customer receives his product as soon as the product is on stock again after a replenishment. This comes with higher delivery costs.
Another model on the supply chain of a multi-channel company is given by Bendoly et al. (2007). They develop two models with periodic review for minimizing the costs while maintaining a service level. The first is a two-system model where the shops and the online depot operate as independent subsystems. This is the separated supply chain previously described. The second is an so called equal lead time model, where both the online depot and the shops are replenished from the central warehouse. The lead time is the time between the placement of an order by a shop or the e-DC and the time of the replenishment, i.e. the time it takes to process the order and the transportation time. In the equal lead time model, the lead time from the central warehouse to all inventory locations is the same. Bendoly et al. (2007) conclude that there exist threshold levels of online demand, which determine whether the inventory for online demand is either com-pletely decentralized at the shops or comcom-pletely centralized at the online depot.
Lateral transshipments are also used by Seifert et al. (2006a). They compare a separated supply chain, in which lateral transshipments are not allowed, with a integrated supply chain, where excess inventory at the shops can be used to fill those online demands that the online depot cannot meet from stock. They perform a case study on HP, which shows that both the service level and the cost savings benefit from the integration of the supply chain. However, the excess inventory of the shops can only be used at the end of the period. If the e-DC has no inventory anymore and the shops have low inventory levels, it is not known until the end of the period whether the demand of the online customer will be fulfilled or not. In our study, we want to notify the customers directly after the order is placed whether or not they will receive their product. The model of Bretthauer et al. (2010) also uses an integrated supply chain. They determine how many and which of a firm’s capacitated locations should handle online demand to minimize the total costs. Case studies show that the fraction of online orders compared with the total number of orders determines how many locations should serve online customers. The number of such locations does not necessarily increase with the fraction of online orders. As a service to the customer, we want every shop to have the possibility of serving online customers. Therefore, we do not investigate which and how many locations should serve online customers in our research.
Despite the amount of literature on the supply chain of multi-channel companies, there are only few pa-pers on the possibility of picking up a product in a shop after the order is placed online. We are interested in these papers, since this is one of the main issues of this thesis. Gallino and Moreno (2014) study the impact of buying online and picking up in the store. They show that the online sales decrease and the in store sales increase when the pick up in store is possible. An explanation could be that the customers do research online using the available information on the products and the stores, and then buy their items offline. Another explanation is that customers buy additional products in the stores when they are picking up their online bought product. However, they do not investigate which shops should serve which online customers.
review and periodic review lost sales models, with fixed and variable order sizes. If continuous review is used as method, a replenishment order is placed as soon as the inventory level drops below a certain limit.
One of the papers on lost demand is the previously discussed paper from Seifert et al. (2006a). With their case study on HP, they conclude that the integration results in high cost savings if the lost sales costs are high. Moreover, high cost savings are achieved if the number of retail stores is large, transshipment costs are relatively low, and inventory holding costs, demand variability, and lead time are high. The supply chain of Seifert et al. (2006a) comes close to the method we are using. However, they use a static allocation method, where we want to dynamically assign the incoming demand, making use of the knowledge of the current inventory levels.
The dynamic assignment of online orders is investigated by Mahar et al. (2009). They construct a static assignment algorithm, and two dynamic assignment algorithms for minimizing the total costs. The dynamic assignment algorithms allocate an online demand to the e-fulfillment location that will incur the least ex-pected holding, backorder, and transportation costs between the time the demand is allocated and the end of the period. The second dynamic assignment algorithm uses, besides the current inventory levels, the in-formation on inventories that have been allocated to the retailers, but remain in transit between the central warehouse and the shops. They show that the two dynamic assignment algorithms perform better than the static algorithm, and the second dynamic algorithm (with more information) performs better than the first. They also show that the results depend on the percentage of online demand. The problem investigated by Mahar et al. (2009) is closely related to the problem we consider. However, we assume that the lead time is equal to zero. However, the online demand of Mahar et al. (2009) is only of one type, namely home delivery. In our problem, we consider online customers of different types: home delivery and pick up in a shop. Dynamic assignment of demand and integration of the supply chain by lateral transshipments are ways to reduce the costs compared with the classical supply chain. Bhatnagar and Syam (2014) and Mahar and Wright (2009) introduce other methods to reduce the supply chain costs of multi-channel companies. Bhat-nagar and Syam (2014) investigate the possibility of deviding the products into two categories: fast and slow moving items. Fast moving items are available both online and offline, while slow moving items are only available online. Since the inventory holding costs at the e-DC are usually less than at the stores, withdrawing the slow moving items from the offline channel reduces the inventory costs. However, we are only investigating products which are available both offline and online, since for these products.
Another way of saving cost is by postponing the decision of which location is fulfilling an online order. Mahar and Wright (2009) are investigating this possibility by accumulating the online orders. They also make use of the information on the inventory position when assigning an order instead of assigning it to the closest fulfillment location. They develop a quasi-dynamic allocation policy that assigns accumulated online orders to fulfillment locations based on expected inventory, shipping and customer wait costs. Because the allocation is postponed, there is more information available on the demand afterwards, and a better allocation choice can be made. However, we want to allocate the demand directly. In this way, the customer knows directly if he can be served or not, and if the product has to be shipped from one place to another, the transportation can start immediately.
3
Problem description
In this section, we define the problem and give the mathematical model for the allocation of demand when we have perfect information about the future demand. This ILP can serve as a lower bound on the total costs, which is explained in Section 3.3.
3.1
Problem description
Within this research, we investigate the supply chain of a multi-channel retailer with continuous review and periodic ordering. The retailer has multiple shops and one e-DC, that all hold inventory to serve the offline and online demand of the customers.
Hence, we have the following sets:
I = {0, 1, . . . , l} inventory locations, which are the e-DC and the shops
J = Joff∪ Jpick∪ Jhome demand types. The demand types and their possible allocation locations are:
• Joff= {1, . . . , l}. Offline demand for a certain shop (per shop). There is only
one possible location which can serve the demand. This is the shop where the customer arrives.
• Jpick= {l + 1, . . . , 2l}. Online demand which is picked up in a certain shop
by the customer (per shop). There are two possible locations which can serve this demand, namely the shop where the product is picked up and the e-DC. • Jhome= {2l + 1 = d}. Online demand which will be home delivered. Every
location is a possible location to fulfill this demand.
The shops and the e-DC are replenished by the DC (see Figure 3), and we assume that the inventory of the DC is sufficient for the replenishments. We also assume that the replenishments all occur at the same time at all locations, and that the replenishments have zero lead time. This assumption is also made by Seifert et al. (2006a) and Schneider and Klabjan (2013). In practice, this will not be the case since the transportation of the products from one place to another takes time. However, this time will only be a small fraction of the total time. Moreover, when the previous two assumptions are made, the period between two replenishments is independent of what happens before and after these replenishments. The period between two replenishments is called a cycle, which has length R. We assume that the replenished amount is the same as the number of products that left the location during the cycle, i.e. an order-up-to level is used.
We assume that for each demand type j ∈ J , the probability of a given number of customers within a cycle is given by the Poisson distribution with mean µj. In addition, we assume that the time between two
customers, the interarrival time, is memoryless. That is, the time left until the next customer arrives does not depend on the time that has passed since the last customer arrived. The exponential distribution is the only continuous probability distribution which has the memoryless property. Hence, the customers arrive according to a Poisson process, which is also assumed by Schneider and Klabjan (2013). We assume that each customer has a demand for a single product of a certain type. This assumption is not always true in practice, but we assume that the fraction of customers with demand for multiple products is small, and hence these customers do not have a large influence on the supply chain. We assume that each demand for a single product has to be allocated to a location (shop or e-DC) at the moment the demand occurs, and it will be confirmed to the customer whether his product is available or not. If the demand cannot be fulfilled, the demand is lost and we assume that a penalty cost for lost sales must be paid. If it is possible to fulfill the demand, we assume this must be done.
We have no influence on the time and type of the demand arrivals. Hence, the only decision variable is the allocation location of the arrivals:
in∈ I allocation location of customer n ∈ N.
We use the following parameters:
N random total number of customers in the cycle,
In∈ I set of possible allocation locations of customer n of type j,
t time,
en unit vector in Rm, with m = |I|, representing the allocation location of customer n, with 1 at in and
0 elsewhere,
Q(t) random inventory level vector in Rm, with m = |I|, at the time t, R length of the replenishment cycle,
gij handling cost of demand type j at location i. This cost per product is for picking and packing a
product. The height of this cost can be different for different locations. This cost is only nonzero when the demand is online,
tij transportation cost for demand type j when allocated to location i. This cost per product is for the
transportation of a product between the e-DC and a shop (if the product is picked up in a shop), or between the e-DC/shop and the home of the customer (if the product is home delivered). The height of this cost can be different for different shops. This cost is only nonzero when the demand is online, hi inventory holding cost per unit at the end of the cycle at location i. This cost is per product that is
left in the inventory of a location at the end of the cycle. The height of the cost can be different for different locations,
sj lost sales cost per unit of demand type j. This cost is a penalty cost which occurs when a customers
demand cannot be satisfied. The cost can be different for the various demand types.
At each time in the cycle, it is unknown how many customers will arrive between the current time and the end of the cycle. Furthermore, the type of future customers (offline in shops, online with home delivery or online with pick up in a shop) is unknown. We assume that we do know the inventory levels at the current time, q(t) = Q(t), and we know the time left in the cycle (R − t). The goal of this research is to allocate the demand of all customers during a cycle so as to minimize the expected total costs within a cycle. Since we cannot change the demand allocations of the previous demand arrivals and the accompanied costs, the challenge is to allocate the current incoming demand in such a way that as much future customers as possible can also be served, and that the costs are as low as possible. We define zin,j as the cost induced by the
incoming customer n of type j when allocated to location in:
zin,j =
gin,j+ tin,j if qin(t) > 0
sj if qin(t) ≤ 0,
and E[f (t, q(t) − en)] as the expected future costs at time t and with inventory levels q(t) − en. Then, the
allocation location is determined by: arg min
in
In addition to the previous discussed parameters, we use the following parameters: Xn random demand type of customer n,
An random interarrival time of customer n,
Tn random arrival time of customer n,
∆t time to allocate the incoming demand, µj mean demand per cycle of demand type j.
The arrivals of customers in a cycle is a stochastic process. Both the interarrival times and the type of the customers are random variables, where An is the interarrival time of the nth customer, and Tn is the arrival
time of the nth customer. Let Xn be the random demand type of the nth customer. Then, P(Xn = j0) = µj0
P
jµj. The state of the supply chain is represented by the time t and the inventory level vector q(t). At
t = 0, we have the initial inventory levels q(0). When a customer arrives, the demand has to be allocated. Within a cycle, the inventory of the locations only changes when a demand arrives. The allocation of the demand is done within time ∆t (where ∆t → 0 since the allocation is done immediately), hence,
Q(Tn) = Q(Tn−1+ ∆t),
and
Q(Tn+ ∆t) = Q(Tn) − en.
The timeline of a cycle with the arrivals of customers is given in Figure 4.
Figure 4: Timeline of a cycle with demand arrivals. An, Tn and Xn are the random interarrival time, the
time of arrival and the demand type of customer n, respectively.
The difficulty of the problem is that we do not know the expected future costs. The future costs are the total handling, transportation, lost sales and holding costs, which all depend on the allocation of the (unknown) future demand. Even though we assume we know the expected future demand, this does not give us the expected future costs, since the costs do not linearly depend on the number of future customers. Instead, a difference of one future customer does not only influences the costs of that customer, but could also influence the optimal allocation of the other customers. Hence, we can only determine the optimal allocation and accompanied expected future costs if we know the future demand. In Section 3.2, we assume that we have perfect information on the future demand, and we can calculate the optimal allocation location with the ILP we will formulate. Since we do not have perfect information, we construct heuristics which will solve the problem for multiple demand scenarios. In this way, the expected future costs are approximated. The construction of the heuristics are discussed in Section 4.
3.2
Mathematical model: allocation with perfect information
Within a cycle, customers arrive one after each other. Every time a customer arrives, the decision on the allocation location has to be made. The allocation location of customer n at time tn will be the location
which incurs the least total costs, arg minin(zin,j+ E[f (tn, q(tn) − en)]), as explained in the Section 3.1.
Hence, for every possible allocation location of the incoming demand, the current and expected future costs are calculated. If there is a tie in the total minimum costs, the allocation location will be the location with the lowest number amoung the locations in the tie. The e-DC has the highest location ”number”, and hence will never be the allocation location when such a tie occurs.
When we have perfect information on the future demand, this demand has to be allocated to the locations with the lowest costs possible. This can be done with an integer linear program (ILP). The mathematical ILP model of the allocation with perfect information uses the sets and parameters discussed above. Everything that happens previously in the cycle cannot be changed. Hence, the only decision variable of the ILP is the allocation of future demand:
nij number of future demand of type j allocated to location i.
The parameters of the ILP are:
pj number of future demand of type j until the next replenishment,
mij number of future demand of type j allocated to location i which can be served,
kij number of future demand of type j allocated to location i and assigned as lost sale,
q(tn, in) inventory level vector at time tn when the incoming demand n is allocated at location in,
cin total costs of the future demand when the incoming demand is allocated to location in,
ain future allocation costs when the incoming demand is allocated to location in,
bin future inventory holding costs when the incoming demand is allocated to location in,
uin future lost sales costs when the incoming demand is allocated to location in.
The objective function shows the minimization of the total costs when the incoming demand is allocated to location in. Constraint 1 states that the total future costs are equal to the sum of the future allocation costs,
the future inventory holding costs and the future lost sales costs. Constraint 2 ensures that the inventory level of the allocation location of the incoming demand is reduced by 1. In Constraint 3, the future alloca-tion costs are equal to the unit allocaalloca-tion cost (unit handling cost plus unit transportaalloca-tion cost) times the number of customers served at that location. The number of customers of a given type served by a location is less than or equal to the number of customers of that type allocated to that location, which is explained in Constraint 4. Constraint 5 ensures that the total number of future customers served by a location is less than or equal to the number of products in the inventory of that location. Constraint 6 states that demand allocated to a location must either be lost or fulfilled. In Constraint 7, the future inventory costs are calcu-lated. These are equal to the unit inventory holding cost times the inventory level at the end of the period. Constraint 8 and 9 handle the lost sales costs. Constraint 8 states that the future lost sales costs are equal to unit lost sales cost times the number of lost sales at the end of the period, and Constraint 9 states that the number of lost sales at a location is the surplus of the number of allocated demand over the inventory at that location. Constraint 10 ensures that all demand is allocated to some location. Constraint 11 ensures that the cost variables are nonnegative, and Constraint 12 ensures that the demand variables are natural numbers. The future costs for every possible allocation location in are calculated with the ILP. The incoming
de-mand also induces costs when allocated to in, zin,jn, which is explained before. Hence, if we have perfect
information on the future demand, the definite allocation location of the incoming demand is determined by arg min
in
(zin,jn+ min cin) .
3.3
Lower bound
At the end of a cycle, just before the replenishment, we know all the handling, transportation and lost sales costs of the customers during the cycle. For the inventory that is on hand at the end of the cycle, we can calculate the holding costs. These four costs together are the total costs of the cycle. The total costs alone do not tell whether the allocation is done well or not; this depends on the randomness of the demand. If there is a lot of demand, there might be a lot of transportation and handling necessary. In addition, if the inventory is not sufficient, there might be a lot of lost sales and thus high total costs. Hence, if the model does allocate the demand in the best possible way, it performs well even though the costs are high. Therefore, we want to compare the total costs to some value that says something about the randomness of the cycle. We can use the previously described ILP to construct a lower bound on the total costs of a cycle, and compare the obtained total costs with this lower bound.
The ILP allocates the known future demand to the locations, taking all inventory levels into account. At the end of the cycle, we have perfect information on all demand that occured within the cycle. Hence, we can use the ILP to allocate all demand optimally within the cycle. Constraint 2 is not applicable when we use the ILP for the allocation of all demand in the cycle with perfect information. Instead, the initial inventory levels are used, hence Constraint 2 is replaced by q(tn, in) = q(0). Moreover, cin is replaced by c,
4
Heuristics
In this section, the heuristics to solve the allocation problem are described. We call the newly introduced heuristic the Integer Linear Programming Allocation Heuristic (ILPAH). Besides this main heuristic, we construct four greedy heuristics. These greedy heuristics make use of simple allocation rules and are developed to measure the quality of the ILPAH. All heuristics are developed in the mathematical optimization platform AIMMS (Aimms.com, 1989). The ILPAH is described in Section 4.2, and the greedy heuristics are discussed in Section 4.3. First, the arrival of demand is explained, since this is necessary for all heuristics.
4.1
Demand arrival
As described in Section 3.1, we assume that customers arrive according to a Poisson process. Every time a customer arrives, the decision has to be made which inventory will be used to serve the customer. All heuristics developed in this research make this allocation decision. They do so as long as the arrival of the demand occurs before the next replenishment. Therefore, every time a new demand arrives, a check is performed whether the time is still within the replenishment period R. After the allocation decision is made, the time continues and a new demand arrives again after an exponentially distributed time, see Figure 5. This process continues untill the time exceeds the length of the replenishment period.
New demand arrival
Time ≤ R? Allocate incoming
demand Start cycle
End cycle
Yes
No
Figure 5: Arrival of new demand. If the replenishment period is not over yet, the demand has to be allocated.
When a new demand arrival is simulated, two parameters have to be determined: the interarrival time Anand the demand type Xn of the new arrival (customer n). The interarrival time is the time between two
successive arrivals. We assumed that the demand arrivals of all types are according to a Poisson process. Since the sum of Poisson processes is still a Poisson process, we obtain the mean total number of demand arrivals in a replenishment cycle by adding the means of all demand types in a cycle. The interarrival time can be drawn from the exponential distribution with a mean as just described: A ∼ ExpP
jµj. Hence, after
an exponentially distributed time with mean P
jµj, a new demand arrives.
The type of demand Xn of arriving customer n is constructed as follows. We assumed that we know
µj, the mean number of demands per cycle for each demand type. Therefore, we know which fraction of all
fraction between 0 and 1, where demand type j0 is assigned to the customer n if yn≥ j=j0−1 X j=0 µj X j µj and yn≤ j=j0 X j=0 µj X j µj , where we define µ0= 0.
For instance, if we have three demand types and 30% of all demands is of the first type, 10% of the second type and 60% of the third type, we assign every random uniform variable between 0 and 0.3 to the first demand type, every random uniform variable between 0.3 and 0.3 + 0.1 = 0.4 to the second demand type, and every random uniform variable between 0.4 and 0.4 + 0.6 = 1 to the third demand type. This method of assigning a demand type to an incoming demand can also be used for more or less than three demand types.
4.2
ILPAH: Integer Linear Programming Allocation Heuristic
In Section 3, we calculated the best allocation location with the ILP. However, we can not implement this straight into the model, because the ILP assumes that we know exactly what the demand will be in the future. Since we do not have perfect information about what happens in the future, we have to come up with a solution for this. In this section, we explain the main heuristic of this thesis, which is called the Integer Linear Programming Allocation Heuristic (ILPAH). This heuristic uses the information on the inventory levels and the mean demand within a cycle to decide which location will be the allocation location. Hence, the ILPAH is a dynamic allocation heuristic.
In Figure 6, the flow chart of the ILPAH is given. At time tn, customer n arrives, and his demand type jn
is checked first. If the demand is offline (jn ∈ Joff), there is no choice in the allocation location: only the
inventory of the shop where the customer arrives can be used. Hence, if for example jn= 1, the only possible
location is in= 1. The demand jnis allocated to the corresponding shop in and qin(tn+ ∆t) = qin(tn) − 1. If
qin(tn+ ∆t) ≥ 0, the demand can be satisfied. Since jn ∈ Joff, the allocation costs gin,jn= 0 and tin,jn= 0,
as assumed. If qin(tn+ ∆t) ≤ 0, the demand cannot be satisfied, and lost sales costs sjn have to be paid.
The lost sales costs are stored for the calculation of the total costs at the end of the cycle.
If the demand type of customer n is online (jn ∈ Jpick∪ Jhome), there is a choice in the allocation
loca-tion in. We want to allocate this demand to the location in which incurs the least expected total costs:
arg minin(zin,j+ E[f (tn, q(tn) − en)]), as explained in Section 3. At the end of the cycle, all demand is
allocated and the total costs are calculated. Hence, we want to allocate the incoming demand jn to the
location in ∈ In which adds the least costs at the end of the cycle. However, the height of the total costs
is among others determined by the future demand pj, and we do not know yet what the demand will be
in the future. For instance, if p1 (future offline demand in shop 1) is large, it would not be a good idea to
allocate the current demand to shop 1, since then it is more likely that future demand is lost. To find the best location, the ILPAH tries to predict the future using multiple scenarios in which the future demand is set. This is done as follows.
We assumed that the demand arrivals from the different types come in according to a Poisson distribution and we know the mean demand µj per replenishment cycle for each demand type. The mean future demand
of type j at time tn, νj(tn), is linear to the time left in the replenishment cycle. Hence, νj(tn) = (R−tRn)µj.
Once we know the mean demand until the next replenishment, we can draw demand scenarios P from the Poisson distributions for each demand type. Hence, a scenario is the number of future demand for each demand type within a cycle, with Pj ∼ Pois(νj(tn)) ∀j.
If a future demand scenario is known (pj ∀j), the current demand can be allocated to a location in ∈ In.
That is, the inventory level of the allocation location in is reduced by one (all other inventory levels stay the
same): q(tn, in) = q(tn) − en. In addition, the allocation costs or lost sales costs of the current demand are
calculated, which is
zin,j =
gin,j+ tin,j if qin(tn) > 0
jn ϵ Joff? Draw future demand
Pj ~ Pois(νj(tn)) j
Allocate jn to in ϵ In and
set q(tn,in) = q(tn) - en
Allocate pj using the
ILP with inventory levels q(tn,in)
Calculate total costs for scenario k Zi (k) + min ci (k) jn allocated to all in ϵ In? q(tn,in) = q(tn) k < K in: E[ f(tn, q(tn) - en) ] = Σk (zi (k) + min ci (k)) / k Allocate jn to argmini E[f(tn, q(tn) - en)] End allocation Start allocation No Yes A n n Yes No A n n n No k = k + 1 Yes
The drawn scenario pj ∀j is used as input for the ILP, as if we know what the future will be. Since we
now ”know” the future, the ILP can indeed be used, as described in Section 3. Moreover, if we allocate the incoming demand to location in, the ILP uses the reduced inventory level q(tn, in). Since the ILP allocates
the future demand in the best possible way, but not necessarily feasible (since it does not take the order of arrivals into account), it calculates a lower bound on the future costs in the current scenario. The total costs of this demand scenario pj and this allocation location in are stored. This procedure is repeated for
all possible allocation locations of the current demand. Of course, when a new possible allocation location is chosen, the inventory levels are reset to their levels when the demand came in.
Now we know the lower bound on the total costs of allocating the current demand to all possible allo-cation loallo-cations in the scenario that is drawn. However, this scenario is not necessarily the outcome of the future. There are way more scenarios that could be the future demand. Therefore, we repeat the procedure of drawing the future demand from the Poisson distributions with mean νj(tn). In this way, we represent
the future better than with only one scenario. For each of these scenarios, we calculate the expected future and current costs for each possible allocation location. The number of scenarios we draw is K, and when all scenarios are handled, we have a lower bound on the costs for each possible allocation location and each scenario. To decide which location will be the allocation location, the mean of the lower bounds of all scenar-ios is calculated for each possible allocation location of the incoming demand. Since every scenario is drawn with the same probability, the mean total costs are the expected lower bound costs over all scenarios. The location with the least expected lower bound costs will be the allocation location.
We assumed that after time ∆t after the arrival time, the allocation is definite. Then, the inventory level of the definite allocation location in is reduced by one: q(tn+ ∆t) = q(tn) − en. Again, if qin(tn+ ∆t) ≥ 0,
the demand can be satisfied. In this case, since jn ∈ Jpick∪ Jhome, there are allocation costs which have to
be paid. When qin(tn+ ∆t) < 0, the demand cannot be satisfied, and lost sales costs have to be paid. The
allocation costs or lost sales costs are calculated and stored, since we need them when we are calculating the total costs at the end of the cycle.
Within the cycle, there are multiple customers arriving, and each demand is allocated with the above described method. As explained in Section 4.1, the next interarrival time is drawn from the exponential distribution. After this time, a new demand arrives, and the type of demand is again determined. Of course, every time a new demand arrives, some time is elapsed and the mean νj(t) decreases. Hence, future demand
scenarios have on average less demand than before. At the end of the cycle, the total costs of the cycle are calculated by adding all allocation costs, holding costs and lost sales costs.
When a demand comes in, the ILPAH solves the ILP for the allocation of the future demand. This is done with the mathematical optimization software CPLEX, version 12.6 (IBM.com, 1988).
4.3
Greedy heuristics
Besides the ILPAH, there are also four greedy heuristics constructed. Greedy heuristics apply decision rules which are intuitive, but often do not come with an optimal solution (Wolsey, 1998). The greedy heuristics are developed to test whether the main heuristic performs well or not.
The first greedy heuristic is called Greedy Allocation Cost Heuristic (GACH). This heuristic allocates the incoming demand to the location which incurs the least direct allocation costs. Here, the direct allocation costs are the sum of the handling costs and the transportation costs of that product. Hence, for incoming demand jn, the allocation location in is determined by arg minin(gin,jn+ tin,jn). This heuristic is a static
allocation heuristic, since the demand type of a customer determines the allocation location in. The GACH
does not take the future demand nor the inventory levels into account. When the allocation is done, the inventory level of the location is reduced by one (q(tn+ ∆t) = q(tn) − en), and if the incoming demand incurs
allocation costs or lost sales costs (zin,jn), these are calculated and stored.
Ǝ in ϵ In | qi (tn) > 0? in ϵ In | qi (tn) > 0: zi = gi j + ti j In = In \ { in } Allocate jn to Argmini (gi j + ti j ) End allocation Start allocation qi (tn) > 0 for in: min zi ? List zi in ϵ In with increasing costs No Allocate jn to in n A n n n n n A n n n n n n n Yes Yes No n
Figure 7: Flow chart of the Adapted Greedy Allocation Cost Heuristic.
allocation costs. However, when there is no inventory available at that location, the second best location will be used. In this way, direct high lost sales costs are avoided. If there is also no inventory left at the second best location, the next best location will be used, etcetera. If there are no possible locations with inventory, the demand is allocated to the location with the least direct allocation costs (and hence the demand is lost). Hence, if there is a possible location with inventory, the allocation location in is determined by
arg min
in
(gin,j+ tin,j|qin(tn) > 0),
and if there is no possible location with inventory, the allocation location in is determined by
arg min
in
(gin,j+ tin,j).
Since the allocation location is only determined once the demand comes in, the AGACH is a dynamic allo-cation heuristic. The inventory level of the alloallo-cation loallo-cation is reduced by one: q(tn+ ∆t) = q(tn) − en. In
addition, the allocation or lost sales costs zin,jn are calculated and stored. The flow chart of the AGACH is
Calculate ν1(tn), …, νl(tn). Let ν0(tn) = 0 and w(tn) = ν(tn). q’(tn) = q(tn) - w(tn) Allocate jn to Argmini q’i (tn) End allocation Start allocation n n
Figure 8: Flow chart of the Adapted Greedy Inventory Level Heuristic.
The third greedy heuristic is the Greedy Inventory Level Heuristic (GILH). This heuristic allocates the demand to a possible location with the highest inventory level. The allocation location in is determined by
arg maxinqin(tn). The GILH is a dynamic allocation heuristic, since the allocation location is determined
once the demand arrives. This heuristic does not take the allocation costs into account, nor the future demand. Again, the inventory level of the allocation location is reduced by one, and the allocation or lost sales costs are calculated and stored, which is done with q(tn+∆t) = q(tn)−enand zin,jnas described before.
The fourth and last greedy heuristic is the Adapted Greedy Inventory Level Heuristic (AGILH). This heuristic chooses the allocation location based on the current inventory level, like the GILH. However, this heuristic does take the expected future demand into account. To do so, the expected number of offline customers for the remaining part of the cycle is calculated. For these customers, there is no choice for the inventory location. Let w(tn) ∈ Rm (with m = |I|) be the vector with the expected number of offline customers
for the locations. Since i = 0 represents the e-DC, there are no offline customers at that location. Hence, w0(tn) = 0, w1(tn) = ν1(tn), w2(tn) = ν2(tn), . . . , wl(tn) = νl(tn). A new virtual inventory level q0(tn) is
calculated by subtracting w(tn) from the current inventory level of that shop:
q0(tn) = q(tn) − w(tn).
The demand will be allocated to the possible location in with the highest virtual inventory level:
arg max
in
qi0n(tn).
5
Model evaluation
The ILPAH allocates the incoming demand. The heuristic does so taking the expected demand and the current inventory levels at all locations into account. There are multiple ways of evaluating the performance of the ILPAH. To evaluate the heuristic, we compare the total costs of a cycle of the ILPAH with another allocation method. One way is comparing ILPAH with the lower bound. Another way is to compare ILPAH with one of the greedy heuristics. The construction of the lower bound and the greedy heuristics are discussed in previous chapters. In Section 5.1, we discuss another method to evaluate the ILPAH. In Section 5.2, we explain how to cope with the randomness of the heuristics. After that, we explain in Section 5.3 which experiments are performed and show the results of these experiments.
5.1
Situations
In this section, an alternative method to review the ILPAH is presented. To do so, we first summerize the main characteristics of the ILPAH.
When a demand arrives, the allocation is done via analysis of multiple future demand scenarios. The al-location of the demand is determined by evaluating the lower bound on the future costs and the current costs for the different locations. The allocation location is the location which has the lowest expected total costs over all future demand scenarios. However, since the future demand is unknown, we are not sure whether this choice of the allocation location would in the end indeed result in the best allocation location. We are interested in determining whether the customer is allocated to the right location, or if another location is more profitable.
A Markov decision process calculates the costs of every allocation possibility and stores the costs made. In this way, a kind of tree diagram is made, with the probability of each branch given. However, the number of possibilities of arrival times and types of customers in a cycle is infinite, and hence Markov processes cannot be used to calculate the best allocation location. However, we could use the idea of the Markov decision process. That is, exploring all allocation options and keeping track of the costs. Since we cannot investigate the allocation choice of all demand arrivals, we will only look at the allocation choice of the first arrival. Hence, instead of exploring all branches of the tree diagram, only all branches from the root node are explored. We only follow the branch with the least expected costs from the rest of the nodes.
For example, if the first demand is an online demand which is picked up in shop 1, there are two allocation possibilities: shop 1 and the e-DC. If the expected total costs of shop 1 are lower than the expected total costs of the e-DC, we would normally allocate the demand to shop 1 and then continue the run. However, besides only keeping track of the costs of the model of allocation to shop 1 (the best expected allocation location), we now also keep track of the costs of the other allocation possibilities (in this case the e-DC). The model is thus simultaniously keeping track of different situations, with a different first allocation locations. The demand in each situation is the same, but the inventory levels at the different locations and the total costs at the end of the cycle may be different.
The number of situations we keep track of depends on the type of demand of the first arrival. If this demand is an offline demand, there is only one situation, since this demand can only be allocated to one shop. If the first demand is an online demand which is picked up in a shop, there are two situations. If the demand is an online demand with home delivery, there are as many situations as there are locations, i.e. the shops and e-DC. A flow chart of the evaluation method with situations is given in Figure 9.
5.2
Randomness
t < R? Calculate E[ f(t1), q(t1) – e1 ] i1 ϵ I1 with ILPAH End cycle Start cycle
New demand arrival
n = 1? Yes
For i1 ϵ I1: calculate cost
with the ILPAH at tn
for jn and qi (tn) ci = E[ f(t1), q(t1) – e1 ] and qi (t n) = q(t1) – e1 i1 ϵ I1 ci = ci + zi j ILPAH allocations done i1 ϵ I1? Yes No No A A 1 1 A 1 1 n n No Yes
Figure 9: Flow chart of the evaluation of the ILPAH with multiple first allocation situations.
Suppose we have two shops and one e-DC, all with a current inventory level of two products, and that the unit allocation cost for online demand which is picked up in a shop is lower when the demand is allocated to the shop than with allocation to the e-DC. Suppose that demand x comes in, an online demand which is picked up in shop 1, and the future demand scenario predicts that there is only one offline demand for shop 1 and one online demand which will be picked up in shop 1. With this scenario, one could allocate demand x to shop 1, since the allocation costs are less than the allocation to the e-DC, and the future demand can be allocated in a way that no demand is lost. Hence, after allocating demand x, we have one product in shop 1 and two products at shop 2 and the e-DC. Suppose at the end of the cycle, it turns out that there were three demands coming in after the demand x: one online customer who picks up his product in shop 1, and two offline customers which come to shop 1. Since there are two offline customers for shop 1, and there is only one product left at shop 1, at least one of them is lost (depending on the choice of the allocation of the online demand).
If the two offline customers come first, the inventory level of shop 1 is reduced to zero, and the online customers after them will receive their products from the e-DC. In this way, there are no lost sales and only two products in the inventory at the end of the cycle. On the other hand, if the online customers come first, their allocation can be done to shop 1, since the unit allocation cost is the lowest. However, the two offline customers are then lost and at the end of the cycle, and four products are left in the inventory. Hence, the second order of arrival and allocation method incurs more costs.
This example illustrates the large effect of randomness. Since this effect is so large, all different models are compared with the same demand arrivals in a cycle, since then the comparison of the different models is as fair as possible.
5.2.1 Number of cycles and scenarios
of scenarios, we used the ILPAH to allocate the demand of 596 cycles, and calculated the average total costs. The cycles are the same for each number of scenarios, to make a fair comparison between the different number of scenarios. The average total costs and the standard deviation of the mean are shown in Figure 10.
Figure 10: Average costs over 600 cycles for different numbers of scenarios.
Figure 10 shows that the average total costs of the different number of scenarios are close to each other and do not differ significantly from each other. Especially for 10 scenarios and more, the average total costs differ only by 0.2. However, we do have to make a decision on the number of scenarios. Since 500 scenarios performs best with an average total costs of 71.528, we take this number of scenarios for the rest of the experiments.
To make a fair comparison between the different heuristics, the costs of all heuristics are calculated given the same demand arrivals in a cycle. However, comparing the different heuristics only for one cycle does not give a good idea of the performance. This is because some heuristics may perform good on certain cycles while they perform bad on other cycles, since there are different numbers and types of customers arriving in the different cycles. Therefore, the total costs are calculated for multiple cycles and the average costs of these cycles are compared. The number of cycles is determined as follows. We use the ILPAH to cal-culate the total costs per cycle. However, we only want to have one type of randomness when determining the number of cycles, namely the demand arrivals in the cycles. Hence, we do not draw multiple random future demand scenarios. Instead, we want to use a future demand that only depends on the time and the type of demand. Therefore, at time t, we take νj(t) = (R−t)R µj, and round up the number to the nearest
in-teger, since we cannot have partial demand. Hence, we have dνj(t)e as future demand for each demand type j.
A run of 1000 cycles is performed. For each cycle, the total costs are shown in Figure 11.
Figure 11: Cost per cycle with one scenario. The scenario we used is not randomly drawn from the Poisson distribution, but the average future demand, which is rounded up.
Figure 12: The average costs of the cycles.
Figure 12 shows that after about 500 cycles, the average costs over the first z cycles stays the same. There-fore, we conclude that we need to perform the experiments with 500 cycles.
Table 1: Values of the parameters in the base case.
Parameter Value
Initial inventory level
Shop 1 10
Shop 2 10
e-DC 10
Average demand per cycle
Offline, shop 1 7
Offline, shop 2 6
Online, pick up in shop 1 3
Online, pick up in shop 2 3
Online, home delivery 10
Transportation costs
From e-DC to a shop 3
From shop/e-DC to a customer 3
Handling costs Shop 1/2 2 e-DC 1 Holding costs Shop 1/2 1.5 e-DC 1
Lost sales costs
Offline shop 1/2 15
Online pick up in shop or home delivery 10
5.3
Experiments and Results
In this section, we discribe the experiments for the Integer Linear Programming Allocation Heuristic (ILPAH) and the greedy heuristics to test the performance of these heuristics. The results of the experiments are shown and interpreted. The values of the cost parameters and the parameters of the initial inventory levels and the average demand per type and per cycle are highly important for the outcomes of the experiments. Therefore, we use a base case which we analyse and afterwards, we will perform sensitivity analysis on different parameters.
The values of the parameters of the base case are shown in Table 1. The total average demand per cycle is 29, where the total inventory is 30. Hence, on average the inventory should be enough for the demand. In the base case, the fraction of online demand is high. The high fraction of online demand is chosen because the integrated supply chain has multiple allocation options for online demand, whereas there is only one option for offline demand. The higher the fraction of online demand, the more possibilities the integrated supply chain has above the separated supply chain. Since we want to show the possibilities of the integrated supply chain with the ILPAH, this is a fair choice.
The transportation costs from the e-DC to a shop as well as the transportation costs for home delivery are set to 3. The handling costs in the shops are twice as high as the handling costs in the e-DC. This is because the e-DC is fully designed for fast picking and packing of products. In a shop, this is not the case, and the picking process is even more difficult as there are also customers around. The holding costs in the shops are higher than the holding costs in the e-DC, since the e-DC is often built at a cheap location, whereas the shops are located in the busy areas, which yields higher holding costs. The offline lost sales costs are choosen to be higher than the online lost sales costs. This is the case because when customers go to a shop and find out that their product is not available anymore, we expect them to be more disappointed than when they cannot order online.
Inventory Level Heuristic (GILH) and the Adapted Greedy Inventory Level Heuristic (AGILH). As can be seen, the ILPAH outperforms all greedy heuristics. The adapted versions of the GACH and the GILH out-perform their basic heuristics. This is what we expect since the adapted versions have smarter decision rules than the basic heuristics; the AGACH only allocates a demand to the location with the lowest allocation costs if there is still inventory left, to prevent lost sales costs. The AGILH also tries to reduce the lost sales costs, by reducing the virtual inventory level with the expected future offline demand, since these customers only have one location which can be their allocation location.
Figure 13: The average total costs of the ILPAH and the four greedy heuristics over 500 cycles in the base case.
Figure 14 displays the average total costs over 500 cycles of the ILPAH and the four greedy heuristics. This is not only done for the base case where the initial inventory level is set to 10 for all locations, but also for the cases where the initial inventory levels for all locations is set to 8, 12 and 15. All other parameters have the same values as in the base case. As can be seen from this figure, as the initial inventory level increases, the average total costs of all heuristics first decrease and then increase. Since the average number of customers is 29 in the base case, little initial inventory means a lot of lost demand, which induce high costs. As the initial inventory grows, these costs are reduced.
We are most interested in the main heuristic ILPAH. Figure 14 shows that with the initial inventory levels at 15, the costs of the ILPAH are higher than with these levels at 12. This can be explained by the fact that, in the case with initially 15 products at each location, there are on average 3 × 15 − 29 = 16 products left in the inventories at the end of the cycle, which all induce holding costs. In the case with 12 products at each location, there are on average only 3 × 12 − 29 = 7 products left.
Figure 14: The average total costs of the ILPAH and the four greedy heuristics over 500 cycles, shown for four different initial inventory levels.
not allocate in the more expensive way, the total costs are lower than the total costs of the AGACH. However, we assumed that the customers have to be served if possible, so the GACH produces infeasible solutions due to the lost sales which could be served, whereas the AGACH remains feasible due to the adapted allocation rule. Figure 15 shows the average percentage of the total costs of the ILPAH above the lower bound. This is done with the initial inventory levels at 8, 10, 12 and 15, and all other parameters are the same as in the base case. With low initial inventory levels, the average percentage above the lower bound is high, and this reduces as the initial inventory levels are raised. At initial inventory levels of 8, the total costs are on average 18.8% above the lower bound, whereas this value is only 1.4% with initial inventory levels of 15. One could say that the ILPAH only performs good when the initial inventory level is high. However, the lower bound does not necessarily give a feasible solution, whereas the ILPAH is restricted to feasibility. In the case of low initial inventory levels, there is lost demand in most of the cycles. Since there is a difference in lost sales costs for different demand types, the lower bound first labels the demand with low lost sales costs as lost sales, and only if there are still products short, the expensive products are labeled as lost sales. Hence, the ILP does not take the arrival times of the customers into account, and therefore the ILP can produce infeasible solutions. This cannot be done by the ILPAH, since the demand must be served in order of arrival. There is almost always enough inventory for all customers when the initial inventory levels are 15. Therefore, it is only logical that the ILP finds a better solution with low initial inventory levels, and that the difference between the lower bound and the total costs becomes smaller when the initial inventory level raises. We expect that the costs of the separated supply chain are higher than the costs of the integrated sup-ply chain modeled by the ILPAH, since the ILPAH has more options to allocate the demand, and takes the current inventory levels and the expected future demand into account. This is indeed what we see in Figure 16, where the total costs of the separated supply chain are shown with the total costs of the integrated supply chain. The costs of the separated supply chain are much higher than the costs of the integrated supply chain. For the base case (with initial inventory levels of 10), the costs are 114.01 ± 1.80 and 81.02 ± 1.73, respectively. The costs of the supply chain are reduced by 28.9% when the supply chain is integrated. When the initial inventory levels are set to 15, the costs are 103.15 ± 1.41 and 75.38 ± 0.57, respectively. Hence, even with 5 products more at each location, the separated supply chain still performs worse than the integrated supply chain.
Figure 15: The average percentage of the total costs of the ILPAH above the lower bound, shown with different initial inventory levels.
small for initial inventory levels of 8, and as the initial inventory levels increase, the difference first increases and then decreases. The integrated supply chain modeled by the ILPAH has more options to allocate the online demand. However, the added options are more expensive, since the handling costs and transportation costs are higher compared to the basic allocation options. Therefore, these options are only profitable if lost demand is prevented. Since initial inventory levels of 8 lead relatively often to empty inventories at all locations, the extra (more expensive) options do not add much gain. When the inventory levels rise, there is more to win since some locations might still have inventory at the end of the cycle while other have not. When the initial inventory levels are high compared to the average demand that comes in, there are almost no customers lost, and the extra expensive options do not have to be used, which leads to a lower difference in the costs of the separated and integrated supply chain.
5.3.1 Sensitivity analysis on cost parameters
We want to perform sensitivity analysis on the cost parameters. In Table 2, the values of the relevant cost parameters are given. The values in bold face are the values of the base case. For each experiment, we will change the value of one cost parameter, and let all other parameters remain the same as in the base case.
Figure 16: The average total costs over 500 cycles of the separated and the integrated supply chain, shown for different initial inventory levels.
of the sensitivity analysis are presented.
In Figure 17, the average costs are shown for different values of the transportation costs from the e-DC to the shops, both for the separated and the integrated supply chain. The average total costs of the separated supply chain do not change for different values of the unit transportation cost. This is what we expect, since it is not possible to transport products from the e-DC to a shop in the separated supply chain, hence, changing the costs for this transportation does not influence the costs of the supply chain. The average total costs of the integrated supply chain do increase as the transportation costs increase. When increasing the value of the transportation costs from 1 to 5, the average total costs increases with 8.4% from 77.08±1.68 to 84.15±1.81. The average total costs for the separated and the integrated supply chain are shown in Figure 18, for different values of the handling costs at the shops. As with the different values for the transportation costs from the e-DC to the shops, the separated supply chain is insensitive to the handling costs at the shops. This is because there is no picking and packing of products in the shops for online customers. As can be seen,
Table 2: Values of the cost parameters for the experiments with sensitivity analysis. The values in bold face are the values of the base case.
Parameter Value
Transportation costs
From e-DC to a shop 1 3 5
Handling costs
Shop 1/2 1 2 3
Holding costs
Shop 1/2 1 1.5 2
Lost sales costs
Figure 17: The average total costs over 500 cycles of the separated and the integrated supply chain, shown for different values of the unit transportation cost from the e-DC to the shops.
the average total costs of the integrated supply chain changes with the height of the handling costs, as we expect. The average total costs of the integrated supply chain is 73.84 ± 1.50 with a unit handling cost at the shops of 1 and 85.70 ± 1.71 with a unit handling cost of 3, which is an increase of 13.8%.
In Figure 19, the total average costs over 500 cycles are displayed against the different values of the holding costs at the shops, both for the separated and the integrated supply chain. As can be seen, the integrated supply chain seems insensitive for the change in the holding costs at the shops, whereas the average total costs separated supply chain increases as this unit holding cost increases. Since all parameters but the unit holding cost have the values as in the base case, on average, there are 29 customers in a cycle, and the total initial inventory has 30 products. The customers in the integrated supply chain can be devided evenly among the locations in most cases, so there are almost no products at the end of the cycle. Therefore, a change in the holding costs does not lead to a significant change in the average total costs. For the separated supply chain, this does not hold. The shops only serve the offline customers, and on average there are 7 products left at the end of the cycle, for which holding costs have to be paid.
The last sensitivity analysis of the cost parameters is on the lost sales costs of offline customers. In Figure 20, the average total costs of the separated and integrated supply chain are plotted for offline lost sales costs of 10, 15 and 20. There is no significant change in the average total costs of the separated supply chain. As mentioned earlier, in most of the cases, all offline customers can be served in the separated supply chain. This is because there are 3 or 4 products more than the average number of offline customers at the shops. In the integrated supply chain, online customers can be allocated to the shops. If an online customer arrives before an offline customer and empties the inventory, the demand of the offline customer will be lost. The total average costs for the integrated supply chain with offline lost sales costs of 10, 15 and 20 are 76.83 ± 1.44, 81.02 ± 1.73 and 84.81 ± 2.05 respectively.
5.3.2 Other demand parameters
Figure 18: The average total costs over 500 cycles of the separated and the integrated supply chain, shown for different values of the unit handling cost at the shops.
Figure 19: The average total costs over 500 cycles of the separated and the integrated supply chain, shown for different values of the unit holding cost at the shops.
