Optimisation of resources for managing competing risks (May 2009)

Ujjwal R Bharadwaj*; John B Wintle**; V V Silberschmidt***; John D Andrews***

* Corresponding Author; TWI Ltd, Granta Park, Cambridge CB21 6AL
** TWI Ltd, Cambridge, UK
***Loughborough University, UK

Paper presented at ESIA 10 - Engineering Structural Integrity Assessment: Present goals - future challenges. Manchester, UK. 19 - 20 May 2009.

The model for managing risk presented in this paper was developed to address the problem of how many parts of different kinds of a major industrial enterprise should advance in order for stock to operate at optimum efficiency given that failures may occur in service. The particular application was a fleet of cargo ships where parts of different kinds are required to keep ships available for service. For example, spare propellers are expensive items to hold in stock, but the consequential costs of not having a spare when required can also be expensive. There is an optimum number of propellers that it is worth holding in order to balance the cost of stocking too many against the risk of a stock-out. Such risk optimization techniques, in general, find the optimum trade-off between the cost of a risk mitigating measure and the expected value of risk without that measure. They have the potential of being applied more widely within structural integrity and asset management, in any situation where competing risks need to be managed within finite resources. They may therefore find application in areas such as risk based inspection and more generally in other areas of risk management.

Introduction

The level of spares in an inventory has a direct bearing on machine availability. The availability of a machine is a function of the mean time to correct a failure which in turn depends upon, among other factors, the time to obtain a spare (to conduct repairs) or a replacement. The level of spares in a spares inventory is constrained by the cost of holding stock and the penalty of being out of stock. In a competitive climate companies strive to keep their spares inventory at an optimum level to minimize the costs involved. This paper presents a new approach - a risk based approach - to spares inventory management aimed at establishing an optimum level of spares such that financial benefit is maximized within accepted levels of risk, and the remaining (residual) risks are clearly identified. The paper presents a framework that enables consistent and auditable decision making in spares inventory management.

Risk based approaches are used in many sectors of industry and for prioritizing different types of actions; for example, there are risk based approaches in the process industry to manage maintenance and inspection and there are standards or guidance documents to implement these approaches (ASME^[1], API^[2], API^[3] and EEMUA^[4]). As opposed to other approaches, in a risk based approach, actions are based on the risk estimate of various options. In the current context, this means maintaining an inventory at an optimal level depending on the risk profile of the spares in which the likelihood of a failure to meet the demand for a spare is considered in conjunction with the consequences of the failure to meet that demand. The optimal level is such that financial benefits are optimized given risk associated constraints.

The paper discusses some of the unique features of spares inventories vis-à-vis other types of inventories. There is also a section on typical costs associated with inventories followed by a brief note on the main principles underlying various current approaches to spares inventory management. The main body of the paper then presents the risk based methodology and the basic model that has been created to implement that methodology. This is followed by possible areas of further research and conclusions.

Spare parts inventories

Spare parts inventories are maintenance inventories; they are used by maintenance personnel to keep machines available and exist to meet an internal demand for spares. They perform a different function compared to other inventories such as Work-in-progress (WIP) inventories and Finished Product Inventories. WIP inventories smooth out irregularities in production flow. These irregularities are caused by factors such as changes in product mix, equipment breakdowns, differences in production rates, between processes and material handling. Finished Product Inventories provide a buffer stock to protect against lead time demand, differences in quality levels, differences in machine production rates, labor troubles, scheduling problems, gap between capacity and demand and other well established production problems, Kennedy.^[5]

Characteristics of spare parts inventories: Spares inventories are hugely influenced by maintenance policies rather than customer usages that dictate WIP or Finished Product Inventories. For scheduled maintenance, the demand for spares is relatively more predictable and it may be possible to order parts to arrive just in time for use and indeed not stock such parts at all. For unplanned maintenance, a lack of some safety stock often means that the consequences of not keeping some safety stock include production loss. There are other factors such as the amount of redundancy within a system, availability of reliability information from condition monitoring equipment, dependency of failure events, possibility of demands being met by cannibalism and the effect of parts or machine obsolescence on the level of stock holding. There has also been research illustrating how other factors such as the organizational context of inventories, especially the responsibilities and authorities of the persons concerned, have a bearing on inventory management, Zomerdijk and Jan.^[6]

Some typical costs associated with inventories

Ordering and setting up costs: These are fixed costs that do not depend on the size of the order. For example, ordering costs would include paperwork and billing associated with the order. For parts made in-house, ordering and set up costs would include the cost of labor, setting up and shutting down the associated machinery.

Unit purchasing cost: This is the variable unit cost of a part or a component. If the part is manufactured in-house, it includes the variable labor cost, the overhead cost, and the raw material cost needed to produce a single unit. If this part is ordered from an external source, then the unit purchase cost must include the shipping cost.

Holding or carrying cost: These are essentially the inventory costs expressed in monetary value per unit part per year. It includes storage cost, insurance cost, taxes on inventory, and cost due to the possibility of spoilage, theft, or obsolescence. However, usually the most significant of the holding cost is the opportunity cost incurred by tying up capital in inventory. The opportunity cost is the return the company would expect on an investment elsewhere rather than in stock-holding.

Stockout costs: When a demand for a product or a part is not met on time, a stockout is said to have occurred. If it is acceptable for demands to be met at a later date, no matter how much later, it is said that demands may be back-ordered. In the current context, risk is the combination of the probability of a stockout event and its consequence, where a stockout is an event in which a spare is not available on demand. If it is necessary for demands to be met on time, and if this is not achieved, then the scenario is a lost case one. In the current context, a lost case may result in production loss, regulatory penalty and other consequences such as loss of goodwill. Usually it is more difficult to measure the cost of a stockout rather than ordering, purchasing or holding costs.

Approaches to inventory management

There are different approaches to Inventory Management. Prasad^[7] categorizes inventory models into two: Economic Order Quantity (EOQ) and Materials Requirement Planning (MRP). Under these, he classifies about ninety inventory models. The basic model in an EOR method determines, subject to a number of assumptions, an ordering policy that minimizes the yearly sum of ordering cost, purchasing cost, and the holding cost of a part in the inventory. The basic model in an MRP based method considers the relationship between a component that is demanded and other associated (sub) components that also need to be available in order to fulfill that demand.

Winston^[8] classifies models as Deterministic EOQ Models, Probabilistic Models and other recent models such as MRP, Just-in-time (JIT) and Exchange Curves. Nahmias^[9] looks specifically at repairable inventory systems and classifies existing models into three general classes: continuous review, periodic review and models based on cyclic queuing systems. The Risk Based approach presented here is unique in that it does not completely fall in any of these categories although it might have some elements of the approaches listed above.

The risk based approach to spare parts inventory management

In the current context, the following terms have a special meaning: Risk is the combination of the probability of a stock out event and its consequence where a stockout is an event when a spare is not available on demand. Qualitative Risk Analysis broadly covers methods that use engineering judgment and experience as the bases for the analysis of probabilities and consequences of failure. Failure Modes, Effects, and Criticality Analysis (FMECA) and Hazard and Operability Studies (HAZOPs) are examples of qualitative risk analysis that become quantitative risk analysis when consequences and failure probability values are estimated. Quantitative Risk Analysis a) identifies and delineates the combinations of events that, if they occur lead to an undesired event b) estimates the frequency of occurrence for each combination c) estimates the consequences. The approach shown here is a semi-quantitative approach that captures best estimates from experts as well as raw historical data. The risk referred here is relative risk which is a comparative risk of components or equipment in relation to each other.

In the method described below, a risk profile of the spares is obtained by considering the likelihood of a failure to meet the demand for a spare in conjunction with the consequences of the failure to meet that demand. This risk profile is then used to find the optimal level of inventory such that financial benefit is maximized given an identified acceptable risk level.

A basic model to implement the approach

Underlying concepts

The model has two parts: Part 1 establishes baseline values, and Part 2 optimizes values. The implementation of the approach is shown by way of an example shown in Figure 1.

Fig.1. Minimize Total Risk subject to given Budget

Part 1: Obtaining baseline values:

This part of the model aims at establishing baseline values for certain parameters for the purpose of optimizing in the second part of the model. The parameters with their descriptions are as follows:

i = unique part number for identification;

n_i(max) = Expected maximum number of spares required in a planning period, based on historical data, expert opinion, generic data or a mix of these. There are guidelines/ procedures for combining data from various sources (Clemen and Winkler^[10], ASME^[11]).

α_i = ratio of part i in stock to the expected demand depending on α_i(ref);

α_i(ref) = reference (baseline) value of α_i for planning purpose (to obtain baseline values);

n_i = α_i(ref) *F_i = number of parts held in stock, rounded to the nearest integer;

C_i = net unit cost of part i;

CoF_i = consequence of a stockout for part i;

β _i = Stockout frequency estimate for part i, assuming stock outs are proportional to stock levels

where P(x_i ) = the probability of a stockout given that a) a quantity x_i of that part may be required during the timeframe i.e. demand for that part b) n_i is the quantity in stock c) n_i(max) being the maximum expected requirement of part i; this is assuming that the demand for spares follows a uniform distribution.

RV_i = Risk value associated with part i

In the model shown above, qualitative estimates of CoF_i , VH, H, M, L, VL have been assigned values 100, 80, 60, 40 and 20 respectively. In a more advanced model, these values would be a weighted average of values obtained by considering a number of consequence or impact factors. One such possible scheme is shown in the Figure 5.

Part 2: Obtaining optimized values:

This part of the model contains optimized values obtained after using a linear optimizing tool using the values in Part 1 of the model, wherever applicable. The optimized values contain the subscript 'o'. For example, n_io is the optimized value of units of part i to be held in the inventory.

Working of the model:

At the outset, in Part 1 of the model, an appropriate value for α_i(ref) is assumed; in Figure 1, this is 0.90. α_i(ref) (shown in the highlighted box with a small circle towards its top right corner) is a percentage of the expected spare parts demand, necessitated by failures, to be held in the inventory. This starting assumption is necessary to find baseline values for Total Stock Value (TSV) and the Total Risk Value (TRV) of the inventory, say, TSV_b and TRV_b respectively where,

The values above are, in essence, baseline or reference values that establish what is an acceptable level of overall risk and the associated cost of stock holding at that level. Having established these baseline values as a starting point, we then move on to Part 2 of the model in which linear optimization is carried out. In the model demonstrated here, the Linear Programming is through the Solver add-in to Microsoft Excel.

The optimization here is as follows:

Minimize TSV such that TRV ≤ TRV_b . All values are integers and greater than zero. The optimized values for TSV and TRV are TSV_o and TRV_o respectively.

Figure 1 shows the risk profile of various components in the last column. The figure shows Stockout Risk values associated with α_i(ref) 0.90 both before and after optimization.

Modes of operation:

(A) Minimize Total Risk

Figure 1 shows the results when Total Risk is minimized subject to given budget. As seen the Total Risk Value reduces from 87 before optimization to 16 after optimization.

(B) Minimize Total Cost

Figure 2 shows the results when Total (inventory) Cost is minimized subject to a reference level of (tolerable) Total Risk. As shown in the figure, the Total Cost comes down to 304500 from 422500, given a Total Risk Value tolerance of 87.

Fig.2. Minimise Total Cost subject to a tolerable level of Total Risk Minimise Total Cost graph

Although optimization has been carried out as described, there may be some components for which the risk profile may be considered too high to be acceptable to the decision maker. For example, as shown in Figure 1, the RV for part 3 i.e. RV ₃ o is 16; this is an expensive part of VL consequence, but the likelihood value of a stockout relative to other parts is high at β_3o. Similarly, in Figure 2, Part 3 has a Stockout likelihood of 1.00 (100%). Such a value may be deemed too high by the decision maker especially when it is substantially different to that implied by α_i(ref) that was assumed initially in Part 1 of the model as part of establishing baseline values. It may be noted here that α_i(ref) = 0.9 implies a reference stock level that expects to meet 90% of the expected maximum demand, i.e., other things being equal, 10% of expected maximum demand is expected not to be met: stockout would be at 10%.

(C) Minimize Total Risk or Total Cost subject to maximum Individual Risk constraints

To rectify this, one more constraint is added to the above optimization process by way of adding a maximum acceptable stockout likelihood value for each of the parts, β_i(max). This constraint is shown in the highlighted box with two concentric circles towards its top right corner in Figure 3. In Figures 1 and 2, β_i(max) is mentioned but the value is 1.0 so that, in effect, it is not a constraint. Imposing β_i(max) constraint means that however low impact a failure to meet a demand for a part is, a minimum stock level will be maintained. Figures 3 and 4 repeat the optimisation with same values as in Figures 1 and 2 with the added constraint of β_i(max) = 0.3. As seen in Figures 3 and 4, the additional constraint ensures that however Low consequence a part maybe and however expensive it may be, it will be stocked within the stipulated level (0.7 or 70% of expected maximum demand, in this case) of tolerable Risk both at a system level (Total Risk Value) and at the individual or component level as indicated by the various β_io values. However, as shown in the Figures, this extra risk mitigation effected by the constraint β_i(max) comes at a price. Figures 3 and 4 show that the Optimised Total Risk value is now 32 up from 16, and the Optimised Total Cost is 375500 up from 304500.

Fig.3. Minimize Total Risk subject to given Budget and individual Risk constraints (β_i(max))

Fig.4. Minimize Total Cost subject to a given tolerable level of Total Risk and individual Risk constraints (β_i(max))

Further research

The consequence or impact of a failure to meet a demand for a spare is a qualitative assessment in the model described above. It is worth noting that the same model will work by directly putting in the likely impact cost (in monetary terms) of a failure to meet a demand for a particular spare. However, it is at times difficult to quantify the full implications of such an event. Therefore, such qualitative assessments, despite the subjectivity involved, are often the best way to fully estimate the impact of an event. One can make such estimates more precise by fine tuning the consequences part of the risk estimate. For example, CoF (Consequence of Failure to meet a demand i.e., a stockout) can be a weighted sum of various consequences factor values such as: extra cost of procuring a part on an urgent basis and the lead time under such circumstances, availability of technical personnel to effect repairs, knock on effect of failure to meet a part on the general availability of the system and the risk of obsolescence of a part or the machine itself. Figure 5 shows how such an approach can be developed. The approach consists of the following main steps: (1) Identify factors that impinge on a failure to meet a demand for a spare- impact factors (IF) - for each spare under consideration. In Figure 5, these are lead time, availability of technical staff, knock-on failures and associated demands, potential for machine or spare redundancy and obsolescence (2) Ascribe weights to each of these impact factors to reflect their relative importance. In the table, these are 10, 10, 60, 15, and 5 for IF1 through to IF5 respectively. (3) Determine relative importance values for VH, H, M, L or VL representing Very High, High, Medium, Low and Very Low. These are shown to be 100, 80, 60, 40 and 20 respectively. (4) The spare is then assessed qualitatively- VH, H, M, L or VL - under each of the impact factors identified in step (1). The net result of this exercise is an aggregate Impact Value. This is '84' in the example shown in the Figure 5. This process is repeated for each spare that is under consideration in an inventory. The relative weights or values in step (2) and step (3) may or may not be the same for all parts. Indeed, impact factors also may differ with parts.

This model takes an expected maximum demand for a period as given; the model in its current form is thus based on periodic reviews. The forecast can be made continuous by using a moving average method for the demand trend which can make this a continuous review model that may be more advantageous in some circumstances. The model can also be configured to take in probabilistic distributions as demand forecast. A more advanced version of this model is being developed along these lines.

Fig.5. Deriving Consequence Value for a part from a number of Consequence estimates

Conclusions

This paper applies risk based principles to spares inventory management. It extends the risk based approach that is well established in other areas of industry. The example illustrates a technique of managing risk within user specified constraints as applicable to Spares Inventory. Risks can be minimized subject to a given Budget (Maximum Total Cost), and Total Costs can be minimized subject to a specified tolerable level of Risk. The optimisation can be done at a system level (Total Cost and Total Risk) and it can be done at a component level too, thus affording the user to manage risks at different levels.

The approach outlined here has the potential to increase plant or system availability and manage business as well as operational risks. It is thus of interest to a number of industry stakeholders including operators, maintenance personnel, regulators and insurance companies. The methodology can be used in other areas that impinge on the structural integrity and asset management such as Inspection and Maintenance where competing risks (of failure) need to be managed within finite resources.

Acknowledgements

This research has been funded by the Centre for Innovative and Collaborative Engineering (CICE) at Loughborough University, UK in collaboration with TWI Ltd at Cambridge, UK via the Engineering Doctorate (EngD) scheme. Inspiration for developing this methodology came from a real life problem in the shipping sector brought to the notice of the lead author, courtesy of Lloyd's Register, London, UK.

References

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