**6. Comparison of the three algorithms**

In order to compare the three algorithms, a scenario simulation was generated with 15 risks and 3 responses per risk. The simulation draws the probabilities and damages according to the following rules:


For all i=1,….15, j=1,…,3

62 Risk Management – Current Issues and Challenges

Risk Number

Risk Number

Ranked Risk

Ranked Risk

seen later.

The numeric example is generated by a simulation that will be described later. Table 2

In Table 3, the ranked risk =1 means the first risk to respond. The first risk that is handled is Risk number 2, since its Qi is 145 (from Table 2). The response is selected as the highest savings solution. Total handling of the 6 risks requires a budget of 90.1 and saves 236.7 in

Table 4 shows that the selection order is different from MDR. However, the accumulated savings converges to the same amount, since at the end both algorithms use the same

Most Dangerous Risk (MDR)

Budget Savings Accumulated

Budget Savings Accumulated

Savings

Savings

Number Cost Accumulated

1 2 1 31.06 31.06 96.31 96.31 2 3 2 14.02 45.08 58.76 155.07 3 6 2 37.98 83.06 13.94 169.01 4 1 3 0.67 83.72 34.34 203.35 5 5 3 1.54 85.27 22.40 225.75 6 4 2 4.83 90.10 10.99 236.73

Most Profitable Response (MPR)

Number Cost Accumulated

1 2 1 31.06 31.06 96.31 96.31 2 3 2 14.02 45.08 58.76 155.07 3 1 3 0.67 45.75 34.34 189.40 4 5 3 1.54 47.29 22.40 211.80 5 6 2 37.98 85.27 13.94 225.75 6 4 2 4.83 90.10 10.99 236.73

Table 5 shows that the BSR uses different response options and therefore converges to different accumulated savings. In this example, the BSR is the worst option out of the 3 algorithms, although this result does not represent the most common situation, as will be

includes all the information needed for applying the algorithms MDR, MPR and BSR.

Tables 3, 4, 5 present the MDR, MPR and BSR solutions accordingly.

expected damages, plus the cost of applying the responses.

response plans. The difference is in the selection order.

**Table 3.** Solution of the example using the MDR algorithm

**Table 4.** Solution of the example using the MPR algorithm

Response

Response

The following chart (Figure 1) shows an example of the behavior of the three algorithms, while the budget increases, step by step.

Figure 1 is an example of a typical situation in which, for a limited budget the BSR is the best algorithm, while for an unlimited budget, the other algorithms can produce better results. This phenomenon holds in most of the simulation examples, but there are cases where the BSR is better for all budgets and cases.

In order to compare the three algorithms, 100 simulations were generated. For each simulation, the maximum needed budget was calculated. (Since the Cij are drawn, the required budget is stochastic and different in each simulation). For each simulation, the savings was calculated for an investment of 20%, 40%, 60%, 80% and 100% of the budget.

For each percentage investment, the savings was calculated for each algorithm. Later, the best algorithm was defined as the successor, for each specific budget, and the frequency of its success was calculated. The following table (Table 6) summarizes the number of successes of each algorithm

Table 6 shows that for a low budget (20 to 60 percent) the BSR is the best algorithm, while for an unlimited budget the MPR behaves better. In many cases, the MPR and MDR behave the same and reach the same savings.

Selecting a Response Plan Under Budget Constraints 65

One limitation of the current paper is that estimating the probabilities and damages for each risk and response is usually considered to be a very difficult task. However, it is required by most of the risk management standards. Tools, like mathematical models and simulations, are available for this task and there are already many projects that include these estimations. Another limitation is that we assume that responses with a negative expected savings cannot be selected. However, in reality, there are responses, like insurance, that are based on negative

A third limitation is the dependencies among risks. It might be that a delay in one task is not critical, while a delay in a second task, together with delay in the first task, might prove to

In this article, we describe a method for how to allocate a risk management budget among the possible mitigation or transfer plans. In most of today's literature, the risk management plan usually ranks the risks and recommends handling those with high rankings. Almost no consideration is given to either response plans or response feasibility. This study proposes three heuristic algorithm approaches to budget allocation, and demonstrates the method, including a sensitivity analysis of the budget constraint. The results are encouraging and

The model is based on the expected damage, and assumes we will always prefer to reduce

A simulated scenario with 15 risks and 45 response plans was demonstrated. The most important lesson learned from the example tested in the study is that the solution is mainly influenced by the response plan, and not only by the expected damage of the risk, as all of the ranking methods recommend. Moreover, for a limited budget, the BSR is usually the best algorithm, while for an unlimited budget the MDR or MPR algorithms are more

expected damages plus their cost. It does not discuss the question of risk taking.

*Faculty of Management of Technology, HIT - Holon Institute of Technology, Holon, Israel* 

project risks and opportunities. Project Management Institute.

Knowledge. Project Management Institute. Newtown Square, PA.

[1] Wideman R.M. 1992. Project and Program Risk Management: A guide to managing

[2] PMI Standards Committee, 2008. A Guide to the Project Management Body of

expected savings (otherwise, insurance company would not sell insurance policies).

2. Limitations

be a severe problem.

help define rules about risk management budgeting.

**8. Conclusion** 

preferable..

A. Gonen

**Author details** 

**9. References** 

**Figure 1.** The savings of each algorithm regarding the limited budget

The main conclusion from Table 6 is that there is no optimal heuristic algorithm. Moreover, if only part of the risks budget can be handled, it is recommended to use the BSR algorithm.


**Table 6.** Distribution of success of each algorithm
