6.1 An Introduction to Cost-effectiveness Analysis

Although the previous chapter on cost-benefit analyses stated that if the so-called net present value (NPV) is positive, a company should implement the new safety investment, it is not always realistic to assume that companies are able to implement all the safety investments with positive NPV, as they will face budget limitations. Safety managers looking for efficient safety investments but facing constraints that prevent them from implementing all recommendations stemming from cost-benefit analyses may find cost-effectiveness analyses very useful. Essentially, the approach gives an idea of whether an investment is “affordable” or not.

But the question arises as to what “affordable” means with respect to safety. If the budget available for safety were infinite, it would be possible to invest so much in safety that almost all occupational and transport incidents and accidents would be eliminated. There would be no tension at all between productivity and safety, and organizations would follow the high-reliability principles at all time. Industry could be much more automated with respect to the “risky” jobs, further avoiding and reducing incidents and accidents. However, regretfully, reality tells a different story: the budgets are not infinite, not for governments and not for private organizations. A lot of safety measures are very expensive and cannot be purchased by many companies. Or, at least, many companies are very careful about spending and allocating the available safety resources. The challenge, then, is to work out how best to spend the money, so that the quantity and quality of safety can be maximized. This is the objective of a cost-effectiveness analysis.

Boardman et al. [1], however, indicate that cost-effectiveness analyses may circumvent three well-known problems associated with cost-benefit analyses. The first boils down to the problems of monetization of certain costs and benefits. For instance, people may be willing to predict the number of lives saved by a safety investment, but they may be unwilling to put a value on human life. The second problem stems from incompleteness of costs and benefits, and the resulting fact that not all impacts of a safety investment are monetized. Third, analysts may be dealing with intermediate impacts whose linkage to certain end-goals or preferences is not fully clear.

This book takes the viewpoint that cost-effectiveness analyses involve computing cost-effectiveness ratios (CERs) and using these ratios to select safety investments or policies that are most effective. This can be done without explicit constraints, but usually in organizations, carrying out a cost-effectiveness analysis is most useful when considering a certain budget constraint (see Section 6.3).

Although monetized benefits are clearly the easiest to use (and to process in further calculations) and to interpret, it should be clear that a cost-effectiveness analysis does not strictly require the monetization of benefits. Nonetheless, a cost-effectiveness exercise always involves: (i) costs measured in monetary units (e.g., euros); and (ii) effectiveness (corresponding to the benefits associated with the costs), which may be measured in units such as lives saved, amount of chemical spill avoided, employees' competences improved, and the like. The ratio of the two measures can then be determined, which can be employed as a basis for ranking alternative investments or policies.

6.2 Cost-effectiveness Ratio

Consider two options, i and j. Note that one of these options may be the current situation (i.e., doing nothing, no investment). The CER of investment i relative to investment j, CER_ij, is given by the following formula:

where C_i represents the costs accompanying option i and E_i represents the number of effectiveness units produced by option i.

The simplest application of this formula occurs when a single safety investment is being assessed as an addition to the status quo. The CER then simply becomes “costs divided by effectiveness.” For instance, a company may wish to know whether an extra safety investment (on top of the existing health and safety investments) of €1 000 000 leading to two avoided fatalities over a period of 5 years should be made. The CER (calculated per year) would then be (€1 000 000/5 years)/2 avoided fatalities, or €100 000 per year per avoided fatality. By itself, this CER does not indicate whether this safety investment is efficient. It does, however, indicate that other safety investments costing more than €100 000 per year per avoided fatality are less efficient than this investment.

The application of the CER becomes more complicated when choosing among multiple investment options. To illustrate this, consider several safety investment options (SIOs) for improving lost time injury figures, as shown in Table 6.1.

SIO, safety investment option ; LTI, lost time injury.

If management is interested in the incremental CER where the comparison option is “no investment” (thus with zero safety costs and zero effectiveness), column 4 of Table 6.1 can be used. This ratio is also referred to as the “average CER.” The smallest average ratio is SIO₁. If safety management seeks to compare different SIOs with each other, SIO₁ can be used as the starting point for comparisons. For calculating the incremental CER, as has been done in the last column of Table 6.1, figures are calculated relative to the SIO, each time providing the smallest incremental increase (see Table 6.1).

Assume now a SIO that costs €2400 per employee, and delivers an effectiveness of 6.5. In this case, SIO₂ and this new SIO have the same costs, but SIO₂ offers a larger gain (effectiveness). Hence, SIO₂ strictly dominates this new SIO, and it would never make sense to choose this new option. In fact, this dominance of options can be illustrated graphically, as shown in Figure 6.1.

The dominance of SIO₂ over the new SIO is shown in Figure 6.1, plotting the cost of each investment option on the vertical axis and its effectiveness on the horizontal axis. Holding cost (or effectiveness) constant, one should obviously always prefer an option that has a larger effectiveness (or lower cost). In fact, the figure shows that the segments connecting the different options, which in theory have slopes equal to their CERs, map out a frontier of the best possible outcomes. The more toward the bottom right of the figure, the better (i.e., the more optimized). The initial step in comparing different options using CERs should therefore always be to remove all options that are dominated from further consideration. In absence of more information about the decision maker's preferences, there is no option preferred over another one if one considers all options situated on the frontier. Hence, new information such as budget constraint or cost limitation needs to be introduced into the problem formulation.

6.3 Cost-effectiveness Analysis Using Constraints

Several approaches are feasible for imposing constraints on the cost-effectiveness problem formulation. Two practices are typically available to carry out a cost-effectiveness analysis with constraints: (i) a minimum acceptable level of effectiveness (Eff_min) and (ii) a maximum acceptable cost or the use of a so-called safety budget (Bu_tot).

If a minimum level of effectiveness is imposed, either the costs or the CER may be minimized. This translates into the following possible problem formulations:

or:

If only the costs are minimized, such as in the first problem formulation, the decision-maker does not value additional units of effectiveness. In the case of the second problem formulation, the most cost-effective option is determined that also satisfies the effectiveness constraint. This latter case will usually lead to higher costs than the former case.

If the constraint of a maximum safety budget is used, then one of two options may be selected: that leading to the largest number of units of effectiveness, or the one yielding the lowest CER. The two problem formulations are as follows:

or:

In the first problem formulation of this limited budget approach, cost savings beyond the budget are not valued, which for some applications may be a disadvantage. In the second problem formulation, the imposed budget constraint is met in the most cost-effective way.

In the remainder of this chapter, the choice of a combination/portfolio of safety investments with maximizing effectiveness and under safety budget constraint is further elaborated. In principle, it would also be possible to work out the other problem formulations with constraints in an analogous way, i.e., using the same theoretical stepwise approach. Clearly, different input information from that described in the next section will be required in the case of the other problem formulations.

6.4 User-friendly Approach for Cost-effectiveness Analysis under Budget Constraint

6.4.1 Input Information

In industrial practice, companies are faced with budget limitations. Consistent with the previous section, the available yearly budget for prevention related to safety is called Bu_tot. When possible prevention investments exceed this budget, they cannot be carried out. Therefore, only the preventative measures having a cost within Bu_tot will be considered in the approach explained hereafter (see also Reniers and Sörensen [2]).

Based on the risk matrix approach (see, e.g., Meyer and Reniers [2]), a risk matrix that is divided into four consequence grades and five likelihood grades can be used. Consequence grades can be expressed in financial terms, while likelihood grades are expressed in the number of times per year that a risk leads to an accident in an organization (e.g., based on generic historical information). Financial losses linked to accident consequences, may be estimated through direct and indirect costs that might occur due to the risk, such as those described in Chapter 5.

Table 6.2 illustrates the risk matrix used in the remainder of this section. Every cell of the risk matrix corresponds to a risk class. For each cell (and thus each risk class), the financial consequence value is multiplied by the likelihood value, and the total expected yearly costs per risk class are determined and shown.

Likelihood (per year)	Cell assignments (in €/year)
< 1	7 500	75 000	750 000	2 500 000
< 10−1	750	7 500	75 000	250 000
< 10−2	75	750	7 500	25 000
< 10−3	7.5	75	750	2 500
< 10−4	0.75	7.5	75	250
Consequence classes/financial impact (€) →	< 7 500	< 75 000	< 750 000	< 2 500 000

The numbers shown in Table 6.2 are illustrative in a sense that every company may develop its own risk matrix with its preferred numbers, applicable to the company. As an example, to guarantee the usefulness of the matrix for the method explained for type I risks, a cut-off consequence class with a maximum financial impact of €2 500 000 is used. The matrix can be designed for type II risks in an analogous way, but the remainder of this section is focused on type I risks. Every type I risk that is part of the cost-effectiveness safety investment study is assumed to be subdivided into one of the risk classes shown in Table 6.2. Cox [4] indicates that a certain risk in each of the cells of any risk matrix is not equally large (or small) due to the classification into risk classes. Hence, a risk cell may contain different varieties of risks. This does not create a problem with the approach, as the aim is to compare groups or bundles of risks, not individual risks.

A discretization of the risk matrix into n cells is illustrated in Figure 6.2. Every risk cell is numbered from 1 to n (in the example, n = 20).

Although not strictly needed for carrying out the cost-effectiveness analysis, the discretization of the risk matrix is presented to provide a better understanding of the method that is elaborated, and the numbering will be used to explain the approach. The risk matrix can be refined by relating the risk classes (thus the risk cells) to a cost-benefit analysis. In this way, a decision support instrument can be developed which can be used to determine, taking a certain safety budget into account, the risk reduction measures or precaution measures leading to the most optimal and cost-effective result within an organization. To be able to carry out the approach, certain actions need to be fulfilled by the user and certain input information is also needed. All risks should have been classified into one of the risk cells of the risk matrix.

Every cell i corresponds to a potential expected cell cost C_i, determined by, (Eq. (2.1)):

where:

1. = expected costs resulting from an accident related to a risk from risk cell i;
2. = likelihood corresponding to risk cell i;
3. = financial impact (consequences) corresponding to risk cell i.

Table 6.2 illustrates the “expected accident cost” figures (expressed in €/year) for the risk matrix (which is used for illustrative purposes). Other risk matrix configurations are, of course, possible in real-life industrial practice. Evidently, it is also possible for organizations to use actual (real) accident cost figures (to calculate such figures, the reader is referred to Chapter 5), which may lead to more accurate results using the approach.

When precautionary investments are made to decrease risks situated within cell i toward cell j (note that j is characterized by lower consequences and/or likelihood), the potential cell costs become C_j. The expected hypothetical benefits in that case can be calculated as C_i − C_j.

The required information for application of the approach is shown in Table 6.3.

n is the total number of cells in the risk matrix

is the number of cells where risks do exist for the organization ()

Ci is the potential expected cell cost (i.e., risk)

Butot is the available yearly budget for prevention related to safety

CoPij is the costs of prevention for going from risk cell i to risk cell j, whereby

When the data from Table 6.3 are known, it is possible to use the approach (discussed in the next section) to determine the most cost-effective prevention measures bundle, where the constraint of a maximum budget is imposed, and the costs and effectiveness (expressed in monetary units) of an option are compared with the situation of no investment. Notice, as already mentioned, that the other problem formulations are only variants of the problem formulation used in this book, and that an analogous approach for a different cost-effectiveness analysis (i.e., a cost-effectiveness analysis with a different problem formulation) can be easily worked out by an organization.

It should be noted that in this illustrative example, the information needed is at a risk matrix cell level, and as such, no individual risk information is needed, only aggregated risk information. Hence, the scope of the suggested approach is on an aggregated scale and requires the user to estimate, in aggregated and composite terms, the prevention costs to go from one risk matrix cell to any other (lower) risk matrix cell (irrespective of the different kinds of risk that may be situated within the cells). This aggregated level is an approximation of real circumstances, and it is possible that in industrial practice more accurate and detailed information will be available which can be used to carry out the cost-effectiveness study. Using more detailed information will lead to better results and more optimal decision-making.

6.4.2 Approach Cost-effectiveness Working Procedure and Illustrative Example

The first step in the first part of the approach is the categorization of risks into the risk classes of the risk matrix. Assume that N_c risk cells (out of the n risk cells in total) contain one or more risks. Safety costs to go from risk cell i to risk cell j (note that j < i) are written as CoP_ij. If the safety costs are higher than the yearly prevention budget, Bu_tot, no investment will be made in these safety measures, and hence these safety costs are excluded at the beginning of the approach. The expected hypothetical benefits corresponding to a decrease in risk cell from i to j are calculated by subtracting C_j from C_i (as explained before).

Following an analysis and preliminary work, a list of possible SIOs will have been drawn up. Using this list, the optimal safety investment portfolio can be determined using a mathematical optimization. In its simplest form, determining the optimal safety investment portfolio is equal to solving a “knapsack problem.” The knapsack problem derives its name from the fact that a person having to fill his/her fixed size knapsack with the most valuable items faces a similar problem. The knapsack problem is one of the most fundamental problems in combinatorial optimization and has many applications, e.g., in stock portfolio management, as well as many extensions.

In the basic version of this problem, a set of decision variables, x_i, is defined where variable x_i (corresponding to measure i) takes on a value 1 if this measure i is chosen as part of the portfolio, and 0 if it is not. A mathematical formulation of the knapsack problem is as follows:

The first equation expresses the total benefit from the selected portfolio (B), which should be maximized. The second equation expresses the fact that the total cost of the selected measures should not exceed the budget. The third constraint implies that a measure is either fully taken or not taken at all.

A number of assumptions are thus implicitly made in this formulation:

• a measure is either taken or not (it cannot be partially taken);
• the total benefit of all measures taken is the sum of the individual benefits of the chosen measures;
• the total cost of all measures taken is the sum of the costs of the individual measures;
• measures can be independently implemented, without consequences for the other measures.

Some of these assumptions are not completely realistic. In the following section, this observation will be discussed.

Although the knapsack problem is non-deterministic polynomial-time hard (NP hard),¹ it can be solved efficiently even for very large instances [5]. The advantage of using the knapsack formulation is that it can be solved by standard off-the-shelf commercial software for mixed-integer programming. Moreover, even spreadsheet software such as Excel (very popular in many organizations) include a solver that can be used to optimize the safety measures portfolio using the approach described.

6.4.3 Illustrative Example of the Cost-effectiveness Analysis with Safety Budget Constraint

Consider the following example to illustrate the approach. Notice that the annual budgeting is assumed to be designed in such a way that no reservations can be made for capital-intensive items. Based on the risk matrix displayed in Table 6.1 and the basic information of Tables 6.4 and 6.5, the execution of the approach is explained.

n = 20

Nc = 6; risk cells 3, 7, 10, 12, 13, and 15 in Figure 6.2

Ci as in risk matrix in Figure 6.1

Butot = €50 000

CoPij as in see Table 6.5

Prevention measure ij	(Illustrative) costs of prevention for going from i to j (CoPij) (€)	Hypothetical benefits for going from i to j (€)
Start = Risk cell 3
3 2	35	67.5
3 1	42	74.25
Start = Risk cell 7
7 6	325	675
7 5	460	742.5
7 3	295	675
7 2	420	742.5
7 1	590	749.25
Start = Risk cell 10
10 9	330	675
10 6	350	675
10 5	390	742.5
10 2	400	742.5
10 1	880	749.25
Start = Risk cell 12
12 11	13 500	17 500
12 10	13 750	24 250
12 9	14 800	24 925
12 8	13 000	22 500
12 7	15 000	24 250
12 6	16 500	24 925
12 5	26 000	24 992.5
12 4	13 900	24 750
12 3	17 000	24 925
12 2	27 500	24 992.5
12 1	38 000	24 999.25
Start = Risk cell 13
13 9	410	675
13 5	550	742.5
13 1	700	749.25
Start = Risk cell 15
15 14	31 000	67 500
15 13	36 650	74 250
15 11	29 880	67 500
15 10	38 000	74 250
15 9	52 000	74 925
15 7	41 440	74 250
15 6	48 990	74 925
15 5	64 450	74 992.5
15 3	50 000	74 925
15 2	62 250	74 992.5
15 1	88 000	74 999.25

An optimal allocation of safety measures with a maximum of one prevention measure from each of the risk cells which can be assigned needs to be determined. As explained before, to solve this problem, four conditions have to be met: (i) the total benefit of measures taken needs to be maximized; (ii) the available budget constraint needs to be respected; (iii) a maximum of one decrease per risk cell is allowed; and (iv) a measure can be taken, or not. These conditions translate into the following mathematical expressions:

1. i.
2. ii.
3. iii.
4. iv. .

Solving these equations for the unknown x_ij yields the optimal solution of the illustrative example represented by Tables 6.4 and 6.5. In this solution the measures taken are shown in Table 6.6 with a total cost of €49 987 and a total hypothetical benefit of €97 499.25. The total hypothetical profit for the illustrative example is thus equal to €47 512.25.

It should be stressed that this illustrative example only serves to explain how to use the knapsack software to determine an optimal allocation of safety resources. It is possible, and recommended, that exact figures be used to determine the hypothetical benefits, simply by calculating all the real expected costs per year of the risk cells for the feasible scenarios in the organization, and thus creating a “real matrix” with real cell assignment figures (instead of general figures).

Furthermore, it is possible to further include more advanced conditions for real-case problems and situations with the method. The next section elaborates on what refinements might be carried out.

6.4.4 Refinements of the Cost-effectiveness Approach

In general, the portfolio of safety investments/measures chosen by a company is subject to a number of extra constraints, expressing relationships between these investments/measures. Fortunately, these relationships are generally easily added to the knapsack approach, usually by introducing additional constraints. This section discusses some of these relationships and shows how they can be expressed in the approach using additional constraints.

6.4.4.1 Binary Relationships

If risk cell r is decreased, risk cell t also has to be decreased and vice versa. This situation occurs when measures are mutually dependent on each other and taking one measure without the other makes no sense. An example is when the use of a new device that enhances safety requires training. It does not make sense to install the device without the training, and it does not make sense to give the training without installing the device.

This relationship between risk cell decreases from r to s and from t to u can be expressed in the approach by the extra constraint

Another (less flexible) way to include this relationship in the approach is to combine risk cell decreases r → s and t → u in a single risk cell decrease.

Suppose, for example, that measures 15 → 13 and 13 → 9 either need to be taken together or not at all in our illustrative example. We add constraint to the knapsack problem and get solution with total cost €48 012 and total hypothetical benefit €94 747. The total hypothetical profit in this case would be €44 735.

Another situation that might occur is the following: if risk cell r is decreased, risk cell t also has to be decreased, but the reverse is not true. As an example, to prevent fire from spreading between departments, a company is considering installing a fire-resistant door. The length of time the door resists a fire can be increased by adding an extra layer of fireproof coating. Clearly, applying the coating without installing the door makes no sense, but the reverse does.

The relationship between risk cell decrease r → s (installing the door) and risk cell decrease t → u (installing the fireproof coating) can be expressed as:

Suppose that when measure 15 → 13 is taken, measure 3 → 2 also needs to be taken. The constraint is added, the problem is resolved and the optimal solution is with a total cost of €49 980 and a total hypothetical benefit and profit, respectively , or €97 492.5 and €47 512.5.

Yet another possible situation is: either risk cell r or risk cell t needs to be decreased, but not both risk cells at the same time. This situation can occur if two measures duplicate each other's effects and the company judges it superfluous to invest in both measures simultaneously. For example, a machine can be protected by a concrete casing or a steel casing, but not by both.

This can be mathematically expressed as follows:

Another possibility is: either risk cell r or risk cell t, or both, needs to be decreased. Such a situation can occur, for example, if the company management has decided that it will install either smoke detectors or fire doors, but may also decide to install both. Translated into a mathematical constraint, this situation can be included in the safety measures allocation problem as follows:

Yet another feasible situation is: if risk cell t is decreased, risk cell r cannot be decreased, and vice versa. But the possibility exists that both measures are not taken. This situation occurs, for example, when management has decided that smoke detectors might be installed, but two types are available and only one type will be selected at most.

This can be expressed as follows:

6.4.4.2 Other Relationships

In principle, all relationships between measures can be expressed as constraints in the knapsack problem. Essentially, the decision of whether to decrease risk cell i can be seen as literal in a propositional logic system, in which logical relationships are expressed by the operators NOT (risk cell i is not decreased), AND (risk cell i and risk cell j are both decreased), OR (risk cell i or risk cell j is decreased), and IMPLICATION (if risk cell i is decreased, then risk cell j is decreased). These operators can be used to create arbitrarily complex relationships to express the most complex logical relationships between safety measures.

For instance, if both the automatic fire door and the alarm system are installed, and the electricity system is not upgraded, then either a backup generator should be installed or a link to an additional power system should be purchased. Each of these relationships can be converted to constraints of the knapsack problem. This section is restricted to one example. For a more elaborate discussion, including details on how to transform a logical relationship to constraints, refer to Martello et al. [5]., Cavalier et al. [6], Mendelsohn [7] and Raman and Grossmann [8].

Consider, for example, the following measures:

1. – an automatic fire door is installed (e.g., );
2. – an alarm system is installed (e.g., );
3. – the electricity system is upgraded (e.g., );
4. – a backup generator is installed (e.g., );
5. – a link to an additional electricity system is installed (e.g., ).

The condition that if both the automatic fire door and the alarm system are installed, and the electricity system is not upgraded, then either a backup generator should be installed or a link to an additional power system should be purchased is logically equivalent to:²

This can be converted into its “conjunctive normal form”:

which translates to the following two constraints:

6.4.4.3 Non-additivity

For some situations, the benefits or costs of measures are not simply additive. Suppose, for instance, that two fire doors can be installed in series to prevent fire from spreading to the next room. Clearly, the effect of installing one door instead of none will be larger than the effect of installing two doors instead of one. In other words, there will be a diminishing rate of return on the second door.

This can be easily handled by identifying such situations and creating “virtual” measures in the cost-benefit table to represent the action of taking both measures. To ensure that each measure is only taken once, some additional constraints are also necessary.

As an example, suppose that the effect of combining risk cell decreases 3 → 1 and 7 → 3 in the example does not yield a benefit of 74.25 + 675 = 749.25, but rather yields only 640. Suppose further that the cost of implementing both 3 → 1 and 7 → 3 is not 42 + 295 = 337, but that a discount of €30 is given.

This can be handled by adding an extra risk cell decrease with cost 307 and hypothetical benefit 640. Additionally, constraints are necessary to ensure that this extra measure is not taken if either 3 → 1 or 7 → 3 are taken. The additional constraints translate mathematically into:

Additionally, we need to ensure that measures 3 → 1 and 7 → 3 are not both chosen at the same time:

In this case, resolving our illustrative example leads to an optimal solution, where the following risk reductions are carried out: 3 → 2, 12 → 8, and 15 → 13. The solution displays a total cost equal to €49 992 and a total hypothetical benefit of €97 817.5 and a total hypothetical profit of €47 825.5.

6.5 Cost-effectiveness Calculation Often Used in Industry

A popular method used in industry to determine the cost-effectiveness of a safety investment is to calculate the expected hypothetical benefit (calculated via definition (ii) in Section 4.2.4), which is usually called the “reduced risk” in industry, and dividing it by the cost of the safety investment for achieving the reduction in risk. Using this approach, it is possible in a very simple way to calculate investments that are more cost-effective than other investments. A well-known example of this approach is that of Kinney and Wiruth [9], where an effectiveness factor is defined and, depending on the value, investment is recommended or not. However, this method does not allow one to determine optimal budget allocations at all, and does not utilize constraints for the problem formulation. In this sense, it should instead be considered as an industrial practice of common sense, but it is too simplistic to be considered a true cost-effectiveness analysis. It should be considered more as a kind of cost-benefit analysis (see also Section 5.5.1).

6.6 Cost–Utility Analysis

Although the terms are sometimes used interchangeably, economists distinguish between cost-effectiveness analyses and cost–utility analyses. A cost–utility analysis can be considered a specific type of cost-effectiveness analysis in which utility measures are used in the analysis.

In operational safety economics, the purpose of a cost–utility analysis can be seen as an estimate of the ratio between the cost of a safety investment and the (theoretical) benefit it produces expressed in utils (for type I risks) or in quality-adjusted accident probabilities (QAAPs) (for type II risks) (see also Chapter 3). The outcome of a cost–utility analysis is a cost per util or QAAP gained (this is also called ‘the incremental cost-effectiveness ratio (or ICER) in health economics).

An advantage of the cost–utility analysis is that it allows comparison across different safety investments and policies, which are otherwise difficult to compare, by using a common unit of measure for both types of prevention investments, the “util” for type I prevention, and the “QAAP” for type II prevention. However, the downside is that measuring utils or QAAPs is very difficult due to their being largely subjective, and this may prove to be even more difficult than expressing all benefits in monetary values and carrying out a “monetary” cost-effectiveness analysis. Hence, if the utils or the QAAPs are not collected or determined properly by an organization, then the whole cost–utility analysis can be open to question. Nevertheless, a cost–utility analysis, when conducted well, can be helpful for safety decision-makers.

6.7 Conclusions

Optimizing prevention investments and making investment decisions in a cost-efficient way is essential for corporations. To this end, a user-friendly knapsack-based approach to take cost-efficient prevention decisions has been described in this chapter. The approach employs some essential data that can be easily determined by any organization and displayed using a risk matrix. Prevention costs are weighed against hypothetical benefits following the preventive measures taken, and the most cost-efficient preventive measures are determined following the knapsack algorithm, given a certain prevention budget available.

References

1. [1] Boardman, A.E., Greenberg, D.H., Vining, A.R., Weimer, D.L. (2011). Cost-Benefit Analysis. Concepts and Practice, 4th edn. Pearson Education Inc., Upper Saddle River, NJ.
2. [2] Reniers, G., Sörensen, K. (2013). An approach for optimal allocation of safety resources: using the knapsack problem to take aggregated cost-efficient preventive measures. Risk Analysis, 33(11), 2056–2067.
3. [3] Meyer, T., Reniers G. (2013). Engineering Risk Management. De Gruyter, Berlin.
4. [4] Cox, L.A. (2008). What's wrong with risk matrices. Risk Analysis, 28(2), 497–512.
5. [5] Martello, S., Plisinger, D., Toth, P. (2000). New trends in exact algorithms for the 0-1 knapsack problem. European Journal of Operational Research, 123(2). 325–332.
6. [6] Cavalier, T.M., Pardalos, P.M., Soyster, A.L. (1990). Approaching and integer programming techniques applied to propositional calculus. Computers and Operations Research, 17(6) 561–570.
7. [7] Mendelsohn, E. (1997). Introduction to Mathematical Logic, 4th edn. Chapman & Hall, London.
8. [8] Raman, R., Grossmann, I.E. (1991). Relation between milp modelling and logic inference for chemical process synthesis. Computers and Chemical Engineering, 15(2), 73–84.
9. [9] Kinney, G. and Wiruth, A. (1976) Practical Risk Analysis for Safety Management. NWC Technical Publication 5865. Naval Weapons Center, China Lake, CA.