Master’s Thesis Econometrics, Operations Research and Actuarial Studies
Multi-location spare parts inventory system with lateral transshipments, time based service and stocking decisions
Pam van Benthem s2453754
Supervisor: Prof. Dr. R.H. Teunter Second Assessor: Dr. X. Zhu
Public Version
January 11, 2018
Multi-location spare parts inventory system with lateral transshipments, time based service and stocking decisions
P. van Benthem, s2453754 Abstract
In this paper we consider a single-echelon inventory system of multiple stock-keeping locations. A multi-item setting is applied, with an item classification. A stocking rule determines whether an item will be stocked at a specific warehouse due to historic sales.
For all stocked items, warehouses apply batch ordering policies (R, Q) to reach a target fill rate. Lateral transshipments are allowed between warehouses to move stock in reaction to a stock-out. We develop a mathematical model to minimize total costs consisting of transportation and inventory costs, constraint to reaching the fill rate, and furthermore, maximizing service response time. This implies that we will model time based service levels. To solve the problem and find the stocking decisions and corresponding policies, we apply a Hybrid Genetic Algorithm. A case study is performed for a technical wholesaler using the proposed model and solution technique, giving advice on the stocking policies to consider in different scenarios.
Keywords: inventory model, multi-location, multi-item, batch ordering, stocking decisions, lat- eral transshipments, service response time, genetic algorithm, case study
Contents
1 Introduction 2
2 Literature 4
3 Problem formulation 7
3.1 Problem description . . . . 7
3.2 Deriving total demand distributions, costs and service . . . . 10
4 Solution method 17 4.1 Genetic algorithm . . . . 17
4.1.1 Search space . . . . 17
4.1.2 Solution representation . . . . 18
4.1.3 Evaluation of individuals . . . . 18
4.1.4 Crossover and mutation . . . . 19
4.1.5 Population management . . . . 20
4.1.6 Stopping criteria . . . . 21
4.1.7 Parameter tuning . . . . 21
5 Case study 22 5.1 Case description . . . . 22
5.1.1 Warehouses . . . . 22
5.1.2 Suppliers . . . . 23
5.1.3 Customers and service . . . . 23
5.1.4 Products . . . . 23
5.1.5 Costs . . . . 25
5.2 Data preparation . . . . 26
5.3 Model validation . . . . 28
5.4 Results . . . . 30
5.4.1 Effect of price on the stocking rule . . . . 34
5.4.2 Full pooling vs no pooling of inventory . . . . 35
5.4.3 Sensitivity analysis . . . . 36
6 Conclusion 40
7 References 41
List of Figures
1 Network with lateral transshipments . . . . 3
2 Illustration of the fill rates βij, αijj0 and Aij for j = 1 and j0∈ {2, 3} . . . . . 9
3 Service response times . . . . 9
4 Classification at Spare Parts Inc . . . . 24
5 Total expected monthly demand in quantity per warehouse . . . . 28
6 Costs per stocking rule . . . . 30
7 Time based service per stocking rule . . . . 31
8 Positioning the stocking rules in terms of costs and direct and indirect fill rate 32 9 Sensitivity analysis on total inventory costs with multipliers 0.5 and 2 . . . . 36
10 Sensitivity analysis on fixed inventory parameter with multipliers 0, 0.5 and 2 37 11 Sensitivity analysis on risk probability with multipliers 1.1, 1.2 and 1.3 . . . . 37
12 Sensitivity analysis on transport costs with multipliers 0.5 and 2 . . . . 38
13 Sensitivity analysis on logistic handling costs with multipliers 0, 0.5 and 2 . . 39
14 Sensitivity analysis on backorder handling costs with multipliers 0, 0.5 and 2 39 List of Tables 1 Notation . . . . 16
2 Population representation . . . . 18
3 Crossover example with parents P1 and P2, |C| = 2, |J | = 4. . . . 20
4 Mutation at the fourth element . . . . 20
5 Parameter values after tuning . . . . 22
6 Removed or adjusted data . . . . 26
7 Final demand data set . . . . 27
8 Item classes and their fill rates and number of SKU . . . . 27
9 Validation time based service with a universal stocking rule of 1 order line . . 29
10 Total costs per stocking rule . . . . 31
11 Constraints on time based service used in the genetic algorithm . . . . 32
12 Optimal solution returned by the algorithm . . . . 33
13 Optimal stocking rule versus rule of 1 and current stocking rule . . . . 33
14 Costs per category for AC items per stocking rule . . . . 34
15 Total costs for CC items per stocking rule . . . . 34
16 Optimal solution in case of no pooling . . . . 35
17 Full pooling vs no pooling in case of the optimal rule, rule of 1 and the current rule . . . . 35
1 Introduction
This research is motivated by a real-life case study. The company of interest, which will remain anonymous and referred to as ”Spare Parts Inc.” in the remainder of this paper, is a technical wholesaler active in 19 countries with 10 warehouses and approximately 50.000 customers. In many cases, they are able to deliver the next morning, if not the next day, due to the many items they have on stock. To date, their assortment consists of around 500.000 items, from which 200.000 are on stock in the central warehouse. Spare Parts Inc. wants to control their stock more efficiently, but still wants to deliver the service they currently do, or potentially even better service.
They are not the only company with a need for more effective stock control. Companies that operate internationally usually face a lot of competition. Therefore, it is very important to keep customers satisfied. This implies that the service has to be quick, which is only possible when parts are stored at the right points in the network with the right amount. This is even more so the case in the industry of spare parts, where availability of parts is of great importance. However, a very important objective of firms is that they want to minimize costs, next to maximizing customer service. Firms do not want to overstock and face extra holding costs or even obsolete stock costs. A balance has to be found between service in terms of response time and inventory and transportation costs. We will address this problem by determining the stocking policies for each item.
Most international networks are multi-echelon systems, which consist of multiple suppliers, who supply their products to multiple warehouses, from where multiple customers are served.
An echelon can be distinguished as a stage in a network. Hence, the suppliers form one echelon, as do the warehouses. Most practical situations can be fitted to this network. We will only consider the warehouses as a stocking point. Therefore, for an inventory analysis, this results in a single-echelon system.
As mentioned, controlling stock more efficiently is a goal for many firms, and one way of doing this is allowing for lateral transshipments. These are stock movements within the same echelon. Different types of lateral transshipments can be distinguished (Paterson et al.
(2011)). These include reactive and preventive lateral transshipments, but also partial or complete pooling. Reactive transshipments are performed when a location faces a stock-out, but can receive parts from an other location in the same echelon. Preventive, as already implied by the term, is used to put items on stock for future demand. The amount of pool- ing implies whether locations in an echelon are willing to share all their available inventory (complete pooling), or reserve an amount for own demand (partial pooling). However, partial pooling can also imply that not all locations are considered for sharing their inventory. In this paper we will consider reactive lateral transshipments. Also, we assume that warehouses are willing to share their complete inventory. However, not all warehouses are linked for lateral shipments. Hence, we consider a special case of partial pooling. In practice, this can be due to the distance between the warehouses, or import laws or taxes. We assume that each warehouse has a pre-specified sequence of back-up warehouses, where it can differ be- tween warehouses how much back-up locations they have assigned. In Figure 1, a network of suppliers, warehouses and retailers is depicted, with lateral transshipments allowed between all warehouses.
Suppliers Warehouses Retailers
Flow of goods Lateral transshipments
Figure 1 Network with lateral transshipments
As mentioned, we will focus on decision making in supply chain networks regarding the stocking of items. In supply chain management, decisions have to be made on several levels.
These include the strategic, tactical and operational level, which represent the long term, mid term and short term decisions to be made, respectively. Strategic decisions can be of the form of facility location problems, whereas tactical and operational decisions usually represent inventory policies, routing decisions or warehouse management. It is believed that decisions on all levels should be incorporated instead of deciding separately, in order to make optimal decisions instead of suboptimal ones. However, we will not focus on the strategic level, which implies that we will use the supply chain network as given. Based on this supply chain network and the fact that we allow complete pooling between some pre-specified warehouses, we will make tactical decisions on which items to stock where, and operational decisions on how much to stock of these items.
In this paper we will develop a mathematical model to represent the single-echelon network and the inventory stocking decisions for item classes and warehouses. The goal is to find the stocking rules and policies that minimize costs, maximize service response time, and also maintain a pre-specified service level per class. To achieve this, lateral transshipments are included in the tactical and operational planning.
The remainder of this paper is organized as follows. In the following section, we will give an overview of relevant literature. In Section 3, we will formally define the problem. Section 4 will present the solution method proposed.A case study is performed in Section 5. We will finalize this paper with a conclusion and further research directions in Section 6.
2 Literature
In this section, all relevant literature on the aspects of the problem we introduce will be described. We also identify the gap in literature and define the contribution of this paper.
First, lateral transshipments are discussed, followed by inventory problems in spare part supply chains. Furthermore, we will discuss some literature on item classification and the solution method. Lastly, we will consider genetic algorithms and discuss why this approach is suitable for the problem at hand.
Lateral transshipments
An overview of the literature on lateral transshipments and gaps in literature regarding this topic is given by Paterson et al. (2011). They indicate that lateral transshipments have been researched in many different settings, both in single and multi-echelon systems, reactive or preventive, and with partial pooling or complete pooling of inventory. Also the case of trans- shipment of demand is considered, where not the stock is moved between two locations, but demand is directly satisfied from an other location than usually. Furthermore, they indicate that in the spare parts environment it is common to apply a complete pooling strategy due to the cost structure as holding and backordering costs are typically large. A key conclusion is that it has been proven in literature that lateral transshipments are of great importance in reducing costs or increasing service. The authors also state future research ideas as in de- termining optimal transshipment policies, however, that will not be of consideration in this paper. We assume a pre-specified transshipment rule regarding the sharing locations and the amount of inventory allowed to share.
Most research on spare parts inventory models with lateral transshipments consider one- for-one replenishment policies, but when items are ordered at different suppliers, it is not logical to order items in batches of one immediately after demand. Other research on (R, Q) policies in settings with lateral transshipments include Xu et al. (2003), Evers (2001), Olsson (2009), Olsson (2010), Huo and Li (2007), Minner and Silver (2005), Minner et al. (2003), Axs¨ater (2003), Chiu and Huang (2003) and Ching et al. (2003), as depicted by Paterson et al. (2011). None of this research considers a multi-item setting. Wong et al. (2006) and Kranenburg and van Houtum (2009) do consider multiple items, but apply a base stock policy to control stock. They both consider mean waiting time constraints, which can be considered as a variant of the time based service levels we consider. Furthermore, only one period is considered.
In case of lateral transshipments, it can also be decided that some warehouses do not want to share their complete inventory. van Wijk et al. (2012) researches this option, however, their method is very general as it can also be used by setting the so-called hold-back levels to zero and assume complete pooling of inventory. They assume that each warehouse is assigned a pre-determined sequence of back-up warehouses. This will be also the case in this model.
However, they assume that demand is Poisson and they apply the basestock policy. In this paper, a continuous (R, Q) policy is applied.
We will consider multiple products, which is an area that has not been under great con- sideration yet. Also, most of the research is limited to small networks, whereas we will apply lateral transshipments to a more complex network with multiple locations. This implies that it is applicable to real life situations.
Spare parts inventory models
Research on inventory problems regarding spare part networks is anything but scarce. Sher- brooke (1968) was the first to develop an approximate technique to address the multi-echelon inventory model with Poisson demand and one-for-one replenishments. His technique is also the basis of many other papers published in this field. Lee (1987) considers pooling groups, as not all warehouses can share their inventory, unless they are in the same group, Axs¨ater (1990) allows non-identical locations, and Sherbrooke (1992) also account for expected back- orders. Alfredsson and Verrijdt (1999) also allow for emergency shipments, in addition to reactive lateral transshipments.
Following on their research, Botter and Fortuin (2000) point out the importance of the availability of spare parts, and answer crucial questions like which parts to stock and where, and how much to stock. They answer the proposed questions by means of a spreadsheet tool. They base the items that have to be stocked on the VED (vital, essential, desirable) approach, instead of based on historic sales. Furthermore, they do not include the supply chain as a whole and do not consider lateral transshipments.
The spare parts logistics network is also the topic of the paper written by Kutanoglu and Mahajan (2009). They consider a two-echelon network with the possibility of inventory sharing by means of lateral transshipments. Furthermore, they consider time-based service targets based on an earlier paper (Kutanoglu (2008)), as they argue that in the service parts industry customers do not care where the product comes from, but how quickly they can access them. This is also the reason for including these time-based service levels, or service response time, in this paper. They show that the sharing strategy is effective in reducing costs compared to no inventory sharing. Also interesting is the prioritizing of warehousing in fulfilling emergency transshipments, which will also be considered in this current paper.
Limiting in their paper is that they consider a one-item setting. A similar setting is consid- ered by Patriarca et al. (2016), based on the METRIC model (Sherbrooke (1968)), as they determine the stock levels of repairable items in a complex multi-echelon network. They do, however, consider a multi-item setting. In addition, they show the relevance by a case study of a European Airline.
Most papers discussing the multi-location inventory problem with lateral transshipment assume that demand is Poisson distributed. However, Porras and Dekker (2008) find that, against expectations, the normal distribution performs very well. Furthermore, for slow- moving items with not many observations, one can not make any clear statement on the demand distribution. Therefore, the normal distribution is used in this research. Also, all papers use an iterative procedure for determining the direct and indirect fill rates from the main warehouse and back-up warehouses. As we set a target for the direct fill rate, we can easily determine the reorderpoint without an iterative procedure.
We can conclude that research on spare part inventory problems is extensive, however, they all deal with determining only stocking levels, either in multi-item or single-item set- tings. We believe that it should first be determined which parts to put on stock at which locations, before deciding on the stock levels of the stocked parts. Also since the stock levels depend on these stocking decisions. It can be argued that for some items, due to their cost price, volume or holding costs, it may not be necessary to stock them at all locations, possible only when the item has enough sales. In case when it is decided that an item is not stocked, a customer can still be served by means of reactive lateral supply by an other warehouse, or backorder from the supplier.
Item classification
As mentioned by Teunter et al. (2010), when considering a multi-item setting, it is often useful to apply a classification. Especially spare part organizations often deal with a too large number of different SKU types, that some classification is necessary in order to make inventory decisions. The model in this paper will be designed for a setting where items are classified. Mostly, this classification is based on the demand value and demand volume of an item. There are single and multi-criteria classifications, however, Teunter et al. (2010) show that when aiming for a cost-optimal classification, a single criterion suffices. Nevertheless, this criterion can account for multiple parameters. We want to point out that the presented model can account for any classification method.
Solution method
With a spreadsheet tool, one can evaluate the model given the input of stocking rules. To find the best option for the stocking rules, it is possible to evaluate all possible combinations.
However, if there is no guideline on the stocking rules to use, this will be very time consuming.
Therefore, a genetic algorithm will be applied to the problem as literature has shown that these are effective in solving supply chain network problems and decision problems.
Nakandala et al. (2016) used the genetic algorithm for making sourcing decisions in the supply chain environment. By creating random initial solutions, applying hill climbing and gene slicing, they find that their heuristic is capable of finding very effective solutions. Ge- netic algorithms also seem to be very useful in solving multi-objective optimization problems, as among others, Altiparmak et al. (2006) and Liao et al. (2011) have proved. The first pre- sented a mixed-integer non-linear programming model for a supply chain problem and solved it by means of a genetic algorithm, and the latter integrated inventory control decisions in facility location models. Govindan (2016) shows that nowadays the supply chain network problems become more complex and therefore harder to solve, hence, the interest in evolu- tionary algorithms is growing as they are effective in solving mathematically complex and NP-hard problems. Also, they mention that these algorithms are greatly used in solving decision problems. Another overview is given by Jauhar and Pant (2016), who state that genetic algorithms are effective in solving real-life problems which are difficult to solve by traditional methods. Combining all off the above, we can state that a genetic algorithm will be a good framework for solving the decision problem of the stocking rules in the inventory model with lateral transshipments, which is applicable to a real-life situation.
To our knowledge, there is no literature yet on how specific, easy implementable decisions on which items to stock where in a single-echelon multi-location system have impact on the costs and service to the customer, while taking lateral transshipments into account.
We will address this problem, by not only analyzing the system as other papers do, but also adding the tactical decision of stocking locations of items. Furthermore, by setting a target fill rate for each item, the analysis of the model is very straightforward instead of iterative.
3 Problem formulation
In this section, we will define the problem and all its aspects. Furthermore, all model as- sumptions are given, as well as the mathematical problem.
3.1 Problem description Setting
We consider a single-echelon spare parts network with multiple locations (j ∈ J ). These locations provide items to the customers (k ∈ K). The network is single-echelon as only the inventories at the warehouses are considered. Multiple SKU’s (i ∈ I) are divided in several classes (c ∈ C). Especially in spare parts industries, the number of spare parts to consider is often too large to consider on the item level. Some classification has to be made in order to focus on the important items, and to be able to efficiently address spare parts inventory problems. Furthermore, using a classification ensures that the number of variables in the presented model is kept low and that it is still implementable in practice. The presented model can account for any type of classification, as well as any number of distinct classes. At every warehouse, every stocked item belonging to class c ∈ C has to reach a target fill rate Fc, individually. Hence, also included is the decision on whether item i should be stocked at location j. We will elaborate on this decision later in this section.
Lateral transshipments
A customer k ∈ K is linked to one of the warehouses, also called his main or default ware- house. The expected demand per time unit this customer has for item i ∈ I is ˜µik, with standard deviation ˜σik. We assume that all items have normally distributed demand in each period, according to these parameters. If the main warehouse is out of stock, one of the other warehouses can ship the items by means of a reactive lateral shipment. We assume that a customer accepts partial shipments, and hence, available inventory is immediately shipped.
Each warehouse has a standard set of back-up warehouses which will be considered for a lateral shipment in case of a stock-out. Also, the back-up warehouses will be prioritized to perform such reactive transshipments. It might be that not all warehouses are considered as back-ups. In practice, this can be dependent on import or tax rules, or simply distance between warehouses. The part of demand that is not fulfilled by either the main or any of the back-up warehouses, will be backordered at the main warehouse.
Including lateral transshipments in the planning
The total expected demand per time unit for item i ∈ I at warehouse j ∈ J , without con- sidering the lateral transshipments, is equal to µij with standard deviation σij. As multiple customers can be in the service region of warehouse j, µij is the sum of ˜µik for all customers k which are in this service region. The same holds for σij, where the variances of demand at the customer level add up. Next, we adjust the demand rates to incorporate lateral transship- ments. Given the target fill rate, it is known which part of demand can not be fulfilled and will be forwarded to the first back up warehouse. This is called overflow demand to the back up warehouse. In the same way, the overflow to the other back up warehouses can be calcu- lated, using the fill rate at the first back up warehouse. By adjusting the demand parameters in this way, one includes lateral transshipments in the planning. In this way, as we already expect that a warehouse will face extra requests by other warehouses, the reorderpoint is correctly aligned with the fill rate. The calculation will be outlined in the following section.
In practice, often firms do not do this, but do allow lateral shipment after the planning has been made. This results in wrongly calculated reorder points, and hence, the target fill rate will often not be reached. Furthermore, the costs are wrongly calculated.
Rules on what items to stock
Included in this model, contrary to other models, is the stocking decision of an item. In practice, this is often decided based on historic sales. If the sales were high enough, an item will be stocked such that future demand can be fulfilled from stock. Otherwise, the demand will be fulfilled completely by the back-up warehouses and backorders. Hence, specific rules will be created that will form the guidelines for stocking an item. These can be warehouse and class specific, and will be denoted by ljc, which represents the minimum number of order lines needed for an item belonging to class c at warehouse j in order to be stocked. Hence, an item will be stocked when the number of order lines in the previous year, dlij, is high enough, i.e. higher than the lower bound ljc if item i belongs to class c. The variable zij is a binary variable equal to one if item i will be stocked at warehouse j. I.e. we have that zij is equal to one if item i belongs to class c at warehouse j and dlij ≥ ljc. Otherwise, we have that zij = 0.
The number of order lines indicates how often an item was demanded. This business rule is chosen instead of using the sales quantity, as the number of events (number of order lines) might be more important than the size of the event (total quantity). These specific rules are chosen in this way as they are easy to understand, easy to implement, and often already used in practice. This results in employees relating to these rules.
Inventory policy
Whenever it is decided that an item i is stocked at warehouse j, the stock control will be by means of an (Rij, Qij) policy. This implies that whenever the inventory drops below Rij, an order of size Qij will be placed. Hence, Rij should cover the expected demand dur- ing the lead time, depending on the target service level, in this case a fill rate. Note that one should use the adjusted expected demand, including the overflow demand, for an ac- curate representation of the actual expected demand the warehouse faces. We assume that the order quantity Qij is predetermined by e.g. the economic order quantity (EOQ) and/or the minimum order quantity (MOQ), while the reorder point is treated as a decision variable.
Service
Service is measured in two ways in this paper. On the first hand, we measure part availability in terms of fill rates. Whenever a customer places an order, there are three ways in which it can be fulfilled. If the item is stocked, a fraction βij can be directly fulfilled by this warehouse, which is the direct fill rate. This holds for both the initial demand as the overflow of demand.
The reorderpoint is calculated based on this fill rate, such that it will be achieved in practice.
Another part is fulfilled trough lateral transshipments (LT). This part is denoted by Aij, and can be split up per back up warehouse. Then, αijj0 is the part that is fulfilled by back up warehouse j0, and Aij is the sum of αijj0 over all j0. In Figure 2, these fill rates are shown in an example with 3 warehouses, where warehouse 1 is the main warehouse, and warehouse 2 and 3 are the first and second back up warehouse, respectively. βi1 is the fraction fulfilled by the main warehouse and Ai1 is the total fraction fulfilled through LT. Specifically, αi12 is fulfilled by the first back up warehouse (warehouse 2), and αi13is fulfilled by the second back up warehouse (warehouse 3), and αi12+ αi13 = Ai1. The part that can not be fulfilled by the main warehouse or any of the back up warehouses is backordered at the main warehouse.
This fraction is denoted by θij, and hence, is equal to 1 − βij − Aij
Main warehouse Retailers 2
1 3
Back-up 1 Back-up 2
αi13
αi12
βi1 Ai1
Figure 2 Illustration of the fill rates βij, αijj0 and Aij for j = 1 and j0 ∈ {2, 3}
Next to the part availability expressed in fill rates, we are also interested in the time based service, or also called the service response time. It is important to note that customers often do not care from which warehouse their items are shipped, but how fast they receive the items. The time based service is measured as the fraction of demand that can be delivered to the customer within a specific time window, and hence, is related to the direct and indirect fill rates. Furthermore, it depends on the transportation time between the warehouses and the customers. An example is shown in Figure 3. It shows that it takes 12 hours to deliver items from the main warehouse (warehouse 1) to the customers. Note that a fraction βi1 is thus actually delivered in this time window. Second, it takes 12 hours and 24 hours to transport the items from the first and second back up warehouse to the main warehouse, respectively. Hence, if the customer receives the item through a LT from the first back up warehouse, it takes an extra 12 hours for the customer to receive the item. This holds for a fraction αi12 of demand. It is assumed that an LT always goes via the main warehouse, and thus, always takes extra time to deliver to the customers.
Main warehouse Retailers
2
1 3
Back-up 1 Back-up 2 24 hr
12 hr
12 hr
Figure 3 Service response times Costs
In the end, we want to minimize the costs of the system. Hence, we want to determine which stocking rules result in the lowest cost, given the demand parameters and target fill rates. These costs consist of the inventory costs, lateral shipment costs and the backorder costs. Additionally, we consider a risk cost depending on the stocking decisions. For example,
stocking an item after one sale in the previous year will result in a higher risk of not selling the item, and thus, obsolete stock.
In most research, it is assumed that holding costs are a percentage of the inventory value on stock. That will also be the case in this paper. Next to that, one can introduce a fixed cost for putting an item on stock. The lateral transshipment cost is defined as the extra cost one has to pay in comparison to a normal shipment. These can depend on the location of the customer and the warehouse performing the lateral shipment. The same holds for backorder costs. As all costs might differ among warehouses, this implies that we allow for non-identical locations.
3.2 Deriving total demand distributions, costs and service
In this section we will derive the total demand distribution by including the lateral trans- shipments in the demand parameters. These are used to evaluate the needed inventory to achieve the fill rates, the costs and the time based service. Then, the MIP to find the optimal set of stocking rules is formulated. The section finalizes with an overview of all notation used in this section.
As mentioned, the following evaluation is based on the stocking decisions. Hence, the first thing to do is to check which items will be stocked at which locations. Recall that this is done based on the stocking rules, as outlined in the previous section, and the stocking decision is denoted by the binary variable zij.
The stocking decisions are made based on historic demand of the previous year. However, for the remaining calculations, we need the expected demand per time unit. Also, from now on, everything will be based on quantities instead of on order lines, as inventory can only be based on quantities. In this paper, we assume that future demand per time unit follows a normal distribution. The mean and standard deviation can be based on historical data by using forecasting techniques. Note that the sum of normal distributed parameters is again normal. Therefore, both demand at the customers and warehouses are treated as normal.
Recall that ˜µik and ˜σik represent the mean demand and standard deviation of demand for item i by customer k. A binary variable ujk is introduced which is equal to one if warehouse j is the main warehouse for customer k, and thus, the demand of customer k is initially send to warehouse j. Then, the mean demand µij and its standard deviation σij on the warehouse level can be found by using the following equations.
µij = X
k∈K
˜
µik· ujk ∀i ∈ I, j ∈ J, (1)
σij = s
X
k∈K
˜
σ2ik· ujk ∀i ∈ I, j ∈ J. (2)
˜
µik and ˜σij can be estimated from historic demand. In this paper, it will be estimated using the simplest form of forecasting, namely, using the average periodic demand from historic sales and its standard deviation. Any stocked part should reach the target fill rate Fc subject to the class the item belongs to. Therefore, the combined target fill rate per item and warehouse βij can be determined by the stocking decision and the class the item belongs to.Hence, βij = Fc if item i belongs to class c at warehouse j, and the item is stocked. If the item is not stocked, we have that βij = 0.
We will base the reorder point on this combined target fill rate. Furthermore, this fill rate also determines the overflow demand to the other locations. Let φj(i) denote the ith
backup location of warehouse j. This implies that φj(0) = j. The number of elements in this sequence |φj|, and thus the number of back up warehouses of warehouse j, is such that 0 ≤ |φj| ≤ |J | − 1, with |J | the total number of warehouses. We assume that customers which are assigned to the same default warehouse, also have the same sequence of back-up warehouses. Initially, warehouse j faces a mean total demand of µij. A fraction βij will be directly fulfilled, and a part (1 − βij) will be forwarded to the first backup. Let eijj0 denote the overflow demand from warehouse j to warehouse j0. Then, we can recursively compute eijj0.
eijφj(l)= µij l−1
Y
h=0
(1 − βiφj(h)) ∀i ∈ I, j ∈ J, l = 1, . . . , |φj|. (3) In the same way, we can compute the variance of the overflow. We will denote this by vijj0.
vijφj(l)= σij2
l−1
Y
h=0
(1 − βiφj(h))2 ∀i ∈ I, j ∈ J, l = 1, . . . , |φj|. (4) It is important to note that the calculations are approximate. However, these are reason- able approximations. Furthermore, as the evaluation is approximate, the model is applicable to real-life situations. As we set a target for the fill rate, we can use this to immediately determine the adjusted demand, instead of iteratively as done by many papers (Kranenburg and van Houtum (2009), Reijnen et al. (2009), van Wijk et al. (2012), Kutanoglu (2008), among others). The adjusted mean demand and standard deviation can be calculated as the sum of initial demand and the overflow to the warehouse. This is then the total demand a warehouse can expect.
µadjij = µij +X
j0∈J
eij0j ∀i ∈ I, j ∈ J, (5)
σadjij = s
σij2 +X
j0∈J
vij0j ∀i ∈ I, j ∈ J. (6)
As eµijj0
ij represents the fraction of demand forwarded to backup warehouse j0, which has a fill rate of βij0, we can calculate the fractions of demand satisfied by lateral transshipments as
αijj0 = βij0eijj0
µij ∀i ∈ I, j ∈ J. (7)
As αijj0 represents the fraction of demand for item i at warehouse j which is satisfied by j0, the fraction of demand that is satisfied by all backup warehouse together follows from
Aij = X
j0∈J\{j}
αijj0 ∀i ∈ I, j ∈ J. (8)
Also, by definition we must have
βij + Aij+ θij = 1 ∀i ∈ I, j ∈ J, (9)
from which we can determine θij.
Using the adjusted demand and standard deviation, the lead time Lij, the order size Qij
and fill rate βij, we can calculate the reorder point Rij using the standard normal loss formula (Axs¨ater (2006)).
G Rij − µadjij · Lij σijadj· L1/2ij
!
= (1 − βij) Qij σijadj· L1/2ij
∀i ∈ I, j ∈ J. (10)
Note that Qij is predetermined by the minimum or economic order quantity, the latter being equal to
Qij = s
2 · Oij · µij
hij ∀i ∈ I, j ∈ J, (11)
with Oij the order costs for item i at warehouse j and hij the holding costs for item i at warehouse j.
In Excel, Goal Seek can find Rij based on eq. (10). However, there also exist approxima- tions to compute the corresponding safety factor with a fill rate, and use this to compute the reorder point (Silver et al. (1998)). They approximate the safety factor z by
z = a0+ a1k + a2k2+ a3k3
b0+ b1k + b2k2+ b3k3+ b4k4, (12) with
k = s
ln
25 G(z)2
, (13)
a0 = −5.3925569, a1 = 5.6211054, a2 = −3.8836830, a3 = 1.0897299, b0 = 1,
b1 = −0.72496485, b2 = 0.507326622, b3 = 0.0669136868, b4 = −0.00329129114.
G(z) in eq. (13) can be calculated using the right-hand side of eq. (10). Compared to a goal seek, this approximation can save significant calculation times, depending on the size of the problem. Straightforward, the reorder point is then calculated as Rij = µadjij · Lij + z · σadjij · L1/2ij .
Low service levels can lead to negative reorderpoints. This implies that one implicitly allows for backorders, before re-ordering. However, in this case we will set the reorderpoint to zero. Hence, we update Rij as Rij = max(Rij, 0). The average inventory on hand can be determined based on the reorderpoint, order quantity and the average demand during the lead time. When an item is not on stock, this is of course equal to zero. Therefore, we
include the stocking decision zij in the equation, and then the average inventory on hand can be calculated as
Iijavg = (Rij +Qij
2 − µadjij · Lij) · zij ∀i ∈ I, j ∈ J. (14) In this way, the calculation can be applied to all items and warehouses, also if the item is not stocked. To ensure that we do not have a negative average inventory, we update the average inventory the same as the re-orderpoint: Iijavg = max(Iijavg, 0).
Costs
Now we have calculated all values that are needed for determining the costs. Recall that these consist of holding costs, lateral transshipment costs and backorder costs. Additionally, it is possible to include extra risk costs for stocking slow moving items. This then includes the risk of these items becoming obsolete stock. The holding costs consist of a fixed cost per stocked item, fj, which can be seen as a minimum inventory cost to be paid, and a percentage that has to be paid per period over the average inventory value, hj. Hence, the inventory costs per period per unit hij are equal to hij = pij · hj. Hence, the total inventory costs to be paid are equal to
IC =X
i∈I
X
j∈J
fj· zij+ Iijavg· hij ∀j ∈ J. (15)
For the lateral transshipment costs, we introduce the variable CijkL . This is the extra costs if customer k is served by warehouse j, instead of the main warehouse for the customer.
Hence, if the main warehouse of customer k is warehouse j, then CijkL = 0. The same holds for all warehouse that are not a backup warehouse for customer k. Recall that in case of a lateral transshipment, the main warehouse (which is receiving the transshipment) is paying.
This is to keep the formulation easier, as in the end it has no effect which warehouse has to pay these costs as all costs are within the same system.
LC =X
i∈I
X
j∈J
X
j0∈J\{j}
X
k∈K
CijL0k· αijj0· ˜µik· ujk ∀i ∈ I, j ∈ J. (16)
Items that can not be delivered from the main or any of the backup warehouses, are backordered at the main warehouse. Note that we set target service levels to control the inventory, however, backorder costs are necessary for the analysis and show the preference of one solution over an other. I.e. two different solutions of stocking rules can result in different time based service levels. However, if there are no costs associated with backordering, one might choose the one with the lower service indicating that their costs will be lower. However, in reality, this might be misleading. Therefore, next to a target fill rate, also backorder costs are included in the mathematical model. The extra costs incurred by a backorder for item i are denoted by CikB, hence, they might be customer specific. Then, the total backorder costs are
BC =X
i∈I
X
j∈J
X
k∈K
CikB· θij· ˜µik· ujk ∀i ∈ I, j ∈ J. (17)
As mentioned, we add risk costs RC depending on the stocking rule decision. In all cases, there is a certain chance that an item will not sell and one has obsolete stock. The higher the sales in the past year, the lower this probability. Hence, the historic sales order lines will
determine the amount of risk. The value of the obsolete stock is then considered as a cost.
Hence, this expected risk cost is computed using the probability of not selling and the value of the stock. This results in a cost CijR per warehouse j and item i. It is expected that the total risk decreases when a higher stocking rule is applied, as the items with the highest risk are not stocked in this case. We have
RC =X
i∈I
X
j∈J
CijR· zij ∀i ∈ I, j ∈ J, (18)
which implies that you only have to pay these costs over the items that are stocked.
The total system costs can then easily be computed as
C = IC + LC + BC + RC. (19)
Time based service
Now we will use the fractions βij, αijj0 and θij to determine the time based service levels.
Based on Kutanoglu (2008) we first determine the time-based demand, which is defined as all demand coming from within a specified time window from a warehouse. Let ∆ijw be the expected demand per time period for item i at warehouse j which can be satisfied within time window w. Furthermore, Gwjk is a binary variable equal to 1 if customer k can be served within time window w from warehouse j. Then,
∆ijw =X
k∈K
˜
µik· ujk· Gwjk ∀t ∈ T, i ∈ I, j ∈ J, w ∈ W. (20)
However, this is now only defined for the main warehouse. Demand can also be fulfilled by a lateral transshipment. The same reasoning holds for this case, only now the time window depends on the backup warehouse j0 and the customer k. ∆ijj0w is then defined as total demand for item i at warehouse j which can be fulfilled by backup warehouse j0 within time window w.
∆ijj0w= X
k∈K
˜
µik· ujk · Gwj0k ∀i ∈ I, j ∈ J, w ∈ W. (21)
As we know which fractions are satisfied directly by the main warehouse and which fractions by the backup warehouses, we can also determine how much demand is satisfied within each time window. Recall that αijj0 denotes the fraction of demand of warehouse j that is satisfied by backup warehouse j0. Also, an assumption is made on the time window in which a backorder is fulfilled. This is modeled by introducing the variable bw, which is equal to one if it assumed that all backorders are fulfilled within the time window w. Hence, the total demand for item i at warehouse j which is fulfilled within time window w, represented by Λijw, is based on the fill rates and the time-based demand.
Λijw = βij · ∆ijw+ X
j0∈J\{j}
αijj0 · ∆ijj0w+ θij · µij· bw ∀i ∈ I, j ∈ J, w ∈ W. (22)
Then, the total fraction of demand satisfied within time window w can be computed by dividing all demand that is satisfied in time window w by total demand.
TSw = P
i∈I
P
j∈JΛijw P
i∈I
P
j∈Jµij ∀w ∈ W. (23)
It is straightforward that P
w∈W TSw = 1.
MIP
Now that we know how to evaluate the costs and time based service given a set of stocking rules, we can formulate the Mixed Integer Program. This will search for the solution of stocking rules that minimizes costs and satisfies all constraints.
A solution will be denoted by sn, and is of the form {ljc ∈ R, j ∈ J, c ∈ C}. Hence, a solution is characterized by its vector of stocking rules. The search space of solutions is denoted by S. Given a solution we know the stocking locations of all items and we can calculate the adjusted demand, the time based service and the costs as outlined in this section.
Now we want to find the solution sn such that the time based service is at a desirable level.
Therefore, the time based service is included as a constraint, where the time based service within time window w should be at least λw. We can formulate the MIP as follows.
sminn∈SC(sn) (24)
s.t. TSw(sn) ≥ λw ∀w ∈ W (25)
where C(sn) and TSw(sn) are the costs and time based service level corresponding to solution sn, as calculated using the mathematical analysis outlined in this section.
In Table 1 we have summarized all notation used in this section.
Table 1 Notation Indices
i Items, i ∈ I j Warehouses, j ∈ J k Customers, k ∈ K c Item classes, c ∈ C w Time window, w ∈ W Decision variables
ljc Lower bound on number of sales for items in class c at warehouse j in order to be stocked Rij Reorder point for item i at warehouse j
Parameters
dlij Total demand (in order lines) at warehouse j for item i in the previous year ujk Equal to one if warehouse j is the default for customer k
zij Equal to one if item i is on stock at warehouse j aijc Equal to one if item i belongs to class c at warehouse j Qij Order quantity for item i at warehouse j (EOQ or MOQ) Oij Order cost for item i at warehouse j
Lij Lead time in number of periods for item i at warehouse j
βij Fraction of demand for item i of warehouse j that is satisfied directly by the main warehouse Aij Fraction of demand for item i of warehouse j that is satisfied through LT
αijj0 Fraction of demand for item i of warehouse j that is satisfied through LT from warehouse j0 θij Fraction of demand for item i of warehouse j that is backordered
µij Mean of demand for item i at warehouse j
σij Standard deviation of demand for item i at warehouse j
˜
µik Mean of demand for item i by customer k
˜
σik Standard deviation of demand for item i by customer k µadjij Adjusted mean demand for item i at warehouse j after LT
σijadj Adjusted standard deviation of demand for item i at warehouse j after LT φj Sequence of backup warehouses for warehouse j
eijj0 Overflow demand of item i from warehouse j to warehouse j0
vijj0 Variance of overflow demand of item i from warehouse j to warehouse j0 Fc Target fill rate for items in class c
Iijavg Average inventory per period of item i at warehouse j pij Cost price in Euros of item i at warehouse j
hj Holding cost per Euro per period at warehouse j hij Holding cost per period for item i at warehouse j IC Total inventory costs
CijL0k LT cost per item i from warehouse j0 to customer k LC Total lateral transshipment costs
CikB Backorder cost per item i for customer k BC Total backorder costs
CijR Risk cost per item i at warehouse j RC Total risk costs
C Total costs of the system
Gwjk Equal to one if customer k can be served within time window w from warehouse j
∆ijw Demand for item i at warehouse j that can be satisfied within time window w
∆ijj0w Demand for item i at warehouse j that can be satisfied by j0 within time window w bw Equal to one if backorders are satisfied within time window w
Λijw Total demand for item i at warehouse j satisfied within time window w TSw Fraction of total demand satisfied within time window w
λw Target for TSw
