Because of the suboptimal stocking strategy, as discussed in the previous section, company X might stock too many parts unnecessarily or some parts not at all/insufficiently. In this section a new inventory control model will be proposed. First the model will be formulated, second the evaluation will be discussed, in which the performance of the system is calculated given certain base stock levels, reorder points and order quantities and finally a procedure is discussed that calculates feasible, near optimal stocking levels. The notation used in this section is displayed in table 4.1.
4.2.1 The model
Bi0b Number of outstanding backorders of SKU i ∈ I for warehouse b ∈ B at C Bibl Number of outstanding backorders of SKU i ∈ I for warehouse l ∈ L at B Iin Inventory on hand of SKU i ∈ I for warehouse n ∈ N
IPin Inventory Position of SKU i ∈ I for warehouse n ∈ N
Xil Number of outstanding orders for part i ∈ I at warehouse l ∈ L Xib Number of outstanding orders for part i ∈ I at warehouse b ∈ B Yin Demand during lead time for part i ∈ I at warehouse n ∈ N
chi Daily holding cost for SKU i
λin Average daily demand rate of SKU i at warehouse n Λi Total daily demand rate for part i at the central warehouse βnobj Target fill rate for warehouse n ∈ B ∪ C
βin Part fill rate of SKU i for warehouse n ∈ N βn Aggregate fill rate for warehouse n ∈ N
Ki Fixed ordering cost of part i for the central warehouse Qi Order quantity for part i at the central warehouse Tio Lead time for part i ∈ I from the outside
supplier/inbetween hubs/local warehouses to the central warehouse Tib Transportation time from central warehouse to inbetween hub Til Transportation time from inbetween hub to local warehouse
Ri Reorder point for part i at the central warehouse Sib Base stock level of SKU i for warehouse b ∈ B Sil Base stock level of SKU i for warehouse l ∈ L
Network
We consider a three-echelon spare part network for both repairable and non-repairable parts. As was discussed in section 1.4, the company operates through central, local and regional warehouses. Two of the local warehouses in the lowest echelon level are supplied directly by local warehouses and one by a regional warehouse. To avoid confusion, we will use a different name for the groups of warehouses than that is currently used by company X (see figure 4.1). The figure shows how the two central warehouses and the demand streams have been merged. The figure on the right doesn’t display the part stream, because this remains as is. Company X’s parts originate from European suppliers and from North American suppliers. Parts originating from European suppliers are always first delivered to the European central warehouse and from there on distributed to other warehouses. Likewise, parts originating from North American suppliers are always first supplied to the North American central warehouse and from there on distributed to other warehouses. Because of this structure, the two central warehouses will be modelled as one central warehouse and denoted by C = {0}. The local and the regional warehouses in the second echelon level will be named inbetween hubs, B, and the local warehouses in the third echelon remain local warehouses, L.
Figure 4.1: Three-echelon spare part inventory system
The demand of the inbetween hubs that have child locations exists of the external demand for that inbetween hub and the demand of the child location. The same holds for the central warehouses: the demand of the central warehouses is the sum of internal and external demand streams. More explanation on the demand streams is given below, under the Demand heading.
Reorder policy
The central warehouse operates according to the (Q,R) policy with reorder point Ri and order quantity Qifor each i ∈ I. This assumption is made because outside suppliers prescribe minimum order quantities.
Since a large number of parts has a low demand rate, a continuous review base stock (S-1,S) policy is assumed in the inbetween hubs and the local warehouses, with base stock levels Sin for each i ∈ I and n ∈ B ∪ L.
Demand
Customers are served through an installed base, consisting of technical systems. Technical systems consist of SKUs I, numbered , . . . , | I |. Each SKU is considered a critical component that fails independently.
Failures occur according to a Poisson process and are assumed to be Poisson with rate λin, for all n ∈ N per day. Poisson demand is common in literature (Caglar et al, 2004; Sherbrooke, 1968; Graves, 1985;
Topan et al, 2017) and has been justified in practice for the company in an earlier thesis (Thijssen, 2007) and in other literature(Hopp et al, 1999). If demand occurs at a warehouse and the part is in stock, this part is immediately sent to the customer. Otherwise, a backorder is created. Since the local warehouses operate through base stock policies and demand is assumed to follow a Poisson process, the total demand at the inbetween hubs equals λib plus the demand of the child location, i.e. local warehouse Australia is replenished by inbetween hub Singapore and Singapore is replenished by the central warehouse, therefore λi,singapore is the summation of the demand for that part in Singapore and in Australia. Following the same reasoning, total demand at the central warehouse also constitutes a Poisson process with rate Λi.
Λi=
N
X
i=0
λin (4.1)
All orders are handled First Come First Serve.
Although the company does perform lateral transshipments between the warehouses in case of stock-out, the model will not include it. The warehouses are located in different countries and for some parts it
is relatively difficult to send items cross borders because of infrastructure and compliance rules. This reduces the benefits of lateral resupply.
Lead times
If stock is available, orders arrive from the central warehouse at the inbetween hubs after a constant lead time Tib and orders arrive from the inbetween hubs at the local warehouses after a constant lead time Til. Constant lead times are realistic for the company, since the activities for dispatching orders can be controlled well. For the replenishment lead times a distinction is made between non-repairable and repairable parts. Replenishment lead times are constant with rate Ti. Ti for repairables is defined as the summation of repair lead time and transportation lead time from the warehouses in the second and third echelon level to the repair facility and for non-repairables as the supplier lead time. This is in line with how the company currently determines the replenishment lead times. However, because it still might happen that the lead times fluctuate, sensitivity analysis will be performed on this parameter.
The model includes both repairable and non-repairable parts. Note that since we assume no condemna-tion for repairable parts, the only difference in the treatment of repairable and non-repairable parts is the replenishment lead time. Consider the following: If demand for a repairable SKU occurs the failed item is replaced by another item at the customer. In case of no inventory on hand, the SKU is back ordered. The same happens when demand for a non-repairable part occurs. The only difference is that in case of demand for a non-repairable part, the old part is scrapped instead of sent back for repair. The only difference with respect to the treatment in the model is the replenishment lead time. For repairable SKUs the replenishment lead time is the transportation time from the customer to the central warehouse plus the repair time and for non-repairable SKUs the replenishment lead time is the supplier lead time.
Taking this into account, the models and techniques for non-repairable parts can also be applied to repairable parts (Axs¨ater, 2015)
Costs
Holding costs are denoted by chi for each SKU i ∈ I per day. In addition to holding costs, fixed ordering costs Ki for each SKU i ∈ I are applicable in the central warehouse.
Service measure
The company uses fill rate as service level, where fill rate is defined as the number of times all parts of an order line can be delivered within 24 hours and is measured per warehouse. The definition of fill rate as is going to be used in te model will differ from the definition as used by the company, namely the number of times a part can be delivered from stock in case of demand, βn = P
i∈I
SKUs are almost always delivered within 24 hours if stock is available.
Only a small percentage of the customers request for more than 1 SKU per order.
In case of a backorder resulting from an order line for multiple parts, available parts are delivered and parts that are not in stock are backordered (implying partial delivery). However, employees from order desk(the department responsible for sending the items to the customer)indicated that in almost all cases where more than 1 part per order line is requested, the part that was not in stock is canceled in the days after.
4.2.2 Assumptions
The assumptions made so far are:
Failure of each SKU i occurs according to an independent Poisson process
Lead times to the central warehouse are assumed to be constant
There is ample repair capacity
The lead times from the central warehouse/inbetween hubs to the inbetween hubs/local warehouses are assumed to be constant
All orders are handled First Come First Serve
A continuous review base stock policy (S-1,S) is applied for all SKUs at the local warehouses and inbetween hubs, and an (Q,R) policy is applied for all SKUs at the central warehouses
The local warehouses are only resupplied by the inbetween hubs and the inbetween hubs are only resupplied by the central warehouses.