8. Data Envelopment Analysis in Enterprise Risk Management

For a set of data including a supply chain needing to select a repository for waste dump siting, we have 12 alternatives with four criteria. Criteria considered include cost, expected lives lost, risk of catastrophe, and civic improvement. Expected lives lost reflects workers as well as expected local (civilian bystander) lives lost. The hierarchy of objectives is:

The alternatives available, with measures on each criterion (including two categorical measures) are given in Table 8.1:

Models require numerical data, and it is easier to keep things straight if we make higher scores be better. So we adjust the Cost and Expected Lives Lost scores by subtracting them from the maximum, and we assign consistent scores on a 0–100 scale for the qualitative ratings given Risk and Civic Improvement, yielding Table 8.2:

Nondominated solutions can be identified by inspection. For instance, Nome AK has the lowest estimated cost, so is by definition nondominated. Similarly, Wells NE has the best expected lives lost. There is a tie for risk of catastrophe (Newark NJ and Epcot Center FL have the best ratings, with tradeoff in that Epcot Center FL has better cost and lives lost estimates while Newark NJ has better civic improvement rating, and both are nondominated). There are also is a tie for best civic improvement (Newark NJ and Gary IN), and tradeoff in that Gary IN has better cost and lives lost estimates while Newark NJ has a better risk of catastrophe rating, and again both are nondominated. There is one other nondominated solution (Rock Springs WY), which can be compared to all of the other 11 alternatives and shown to be better on at least one alternative.

Nondominance can also be established by a linear programming model. We create a variable for each criterion, with the decision variables weights (which we hold strictly greater than 0, and to sum to 1). The objective function is to maximize the sum-product of measure values multiplied by weights for each alternative site in turn, subject to this function being strictly greater than each sum-product of measure values time weights for each of the other sites. For the first alternative, the formulation of the linear programming model is:

s.t.

For each j from 2 to 12: +0.0001 This model was run for each of the 12 available sites. Non-dominated alternatives (defined as at least as good on all criteria, and strictly better on at least one criterion relative to all other alternatives) are identified if this model is feasible. The reason to add the 0.0001 to some of the constraints is that strict dominance might not be identified otherwise (the model would have ties). The solution for the Newark NJ alternative was as shown in Table 8.3:

The set of weights were minimum for the criteria of Cost and Expected Lives lost, with roughly equal weights on Risk of Catastrophe and Civic Improvement. That makes sense, because Newark NJ had the best scores for Risk of Catastrophe and Civic Improvement and low scores on the other two Criteria.

Running all 12 linear programming models, six solutions were feasible, indicating that they were not dominated {Nome AK, Newark NJ, Rock Springs WY, Gary IN, Wells NE and Epcot Center FL}. The corresponding weights identified are not unique (many different weight combinations might have yielded these alternatives as feasible). These weights also reflect scale (here the range for Cost was 60, and for Lives Lost was 110, while the range for the other two criteria were 100—in this case this difference is slight, but the scales do not need to be similar. The more dissimilar, the more warped are the weights.) For the other six dominated solutions, no set of weights would yield them as feasible. For instance, Table 8.4 shows the infeasible solution for Duquesne PA:

Here Rock Springs WY and Wells NE had higher functional values than Duquesne PA. This is clear by looking at criteria attainments. Rock Springs WY is equal to Duquesne PA on Cost and Lives Lost, and better on Risk and Civic Improvement.

Value-at-risk (VaR) methods are popular in financial risk management. ¹¹ VaR models were motivated in part by several major financial disasters in the late 1980s and 1990s, to include the fall of Barings Bank and the bankruptcy of Orange County. In both instances, large amounts of capital were invested in volatile markets when traders concealed their risk exposure. VaR models allow managers to quantify their risk exposure at the portfolio level, and can be used as a benchmark to compare risk positions across different markets. Value-at-risk can be defined as the expected loss for an investment or portfolio at a given confidence level over a stated time horizon. If we define the risk exposure of the investment as L, we can express VaR as:

A rational investor will minimize expected losses, or the loss level at the stated probability (1 − α). This statement of risk exposure can also be used as a constraint in a chance-constrained programming model, imposing a restriction that the probability of loss greater than some stated value should be less than (1 − α).

The standard deviation or volatility of asset returns, σ, is a widely used measure of financial models such as VaR. Volatility σ represents the variation of asset returns during some time horizon in the VaR framework. This measure will be employed in our approach. Monte Carlo Simulation techniques are often applied to measure the variability of asset risk factors. ¹² We will employ Monte Carlo Simulation for benchmarking our proposed method.

Stochastic models construct production frontiers that incorporate both inefficiency and stochastic error. The stochastic frontier associates extreme outliers with the stochastic error term and this has the effect of moving the frontier closer to the bulk of the producing units. As a result, the measured technical efficiency of every DMU is raised relative to the deterministic model. In some realizations, some DMUs will have a super-efficiency larger than unity. ¹³

Now we consider the stochastic vendor selection model. Consider N suppliers to be evaluated, each has s random variables. Note that all input variables are transformed to output variables, as was done in Moskowitz et al. ¹⁴ The variables of supplier j (j=1,2…N) exhibit random behavior represented by = , where each (r = 1 , 2 ,  …  , s) has a known probability distribution. By maximizing the expected efficiency of a vendor under evaluation subject to VaR being restricted to be no worse than some limit, the following model (1) is developed:

s.t.

For each j from 2 to 12: Prob{ +0.0001}≥(1-α)

Because each is potentially a random variable, it has a distribution rather than being a constant. The objective function is now an expectation, but the expectation is the mean, so this function is still linear, using the mean rather than the constant parameter. The constraints on each location’s performance being greater than or equal to all other location performances is now a nonlinear function. The weights w _i are still variables to be solved for, as in the deterministic version used above.

The scalar α is referred to as the modeler’s risk level, indicating the probability measure of the extent to which Pareto efficiency violation is admitted as most α proportion of the time. The α _j (0 ≤ α _j  ≤ 1) in the constraints are predetermined scalars which stand for an allowable risk of violating the associated constraints, where 1 − α _j indicates the probability of attaining the requirement. The higher the value of α, the higher the modeler’s risk and the lower the modeler’s confidence about the 0th vendor’s Pareto efficiency and vice-visa. At the (1 − α)% confidence level, the 0th supplier is stochastic efficient only if the optimal objective value is equal to one.

To transform the stochastic model (1) into a deterministic DEA, Charnes and Cooper ¹⁵ employed chance constrained programming. ¹⁶ The transformation steps presented in this study follow this technique and can be considered as a special case of their stochastic DEA, ¹⁷ where both stochastic inputs and outputs are used. This yields a non-linear programming problem in the variables w _i , which has computational difficulties due to the objective function and the constraints, including the variance-covariance yielding quadratic expressions in constraints. We assume that follows a normal distribution N(, B_jk), where is its vector of expected value and B_jk indicates the variance-covariance matrix of the jth alternative with the kth alternative. The development of stochastic DEA is given in Wu and Olson (2008). ¹⁸

We adjust the data set used in the nuclear waste siting problem by making cost a stochastic variable (following an assumed normal distribution, thus requiring a variance). The mathematical programming model decision variables are the weights on each criterion, which are not stochastic. What is stochastic is the parameter on costs. Thus the adjustment is in the constraints. For each evaluated alternative y _j compared to alternative y _k :

w_cost( y _{j cost} – z*SQRT(Var[y _{j cost} ]) + w _lives y _{j lives +} w _risk y _{j risk +} w _imp y _{j imp} ≥

w_cost( y _{k cost} – zSQRT(Var[y _{k cost} ] + 2*Cov[y _{j cost} ,y _{k cost} ]

+ Var[y _{k cost} ] + w _lives y _{k lives +} w _risk y _{k risk +} w _imp y _{k imp}

These functions need to include the covariance term for costs between alternative y _j compared to alternative y _k .

Table 8.6 shows the stochastic cost data in billions of dollars, and the converted cost scores (also billions of dollars transformed as $100 billion minus the cost measure for that site) as in Table 8.2. The cost variances will remain as they were, as the relative scale did not change.

The variance-covariance matrix of costs is required (Table 8.7):

The degree of risk aversion used (α) is 0.95, or a z-value of 1.645 for a one-sided distribution. The adjustment affected the model by lowering the cost parameter proportional to its variance for the evaluated alternative, and inflating it for the other alternatives. Thus the stochastic model required a 0.95 assurance that the cost for the evaluated alternative be superior to each of the other 11 alternatives, a more difficult standard. The DEA models were run for each of the 12 alternatives. Only two of the six alternatives found to be nondominated with deterministic data above were still nondominated {Rock Springs WY and Wells NE}. The model results in Table 8.8 show the results for Rock Springs WY, with one set of weights {0, 0.75, 0.25, 0} yielding Rock Springs with a greater functional value than any of the other 11 alternatives. The weights yielding Wells NE as nondominated had all the weight on Lives Lost.

One of the alternatives that was nondominated with deterministic data {Nome AK} was found to be dominated with stochastic data. Table 8.9 shows the results for the original deterministic model for Nome AK.

The stochastic results are shown in Table 8.10:

Wells NE is shown to be superior to Nome AK at the last set of weights the SOLVER algorithm in EXCEL attempted. Looking at the stochastically adjusted scores for cost, Wells NE now has a superior cost value to Nome AK (the objective functional cost value is penalized downward, the constraint cost value for Wells NE and other alternatives are penalized upward to make a harder standard to meet).

DEA evaluates alternatives by seeking to maximize the ratio of efficiency of output attainments to inputs, considering the relative performance of each alternative. The mathematical programming model creates a variable for each output (outputs designated by u _i ) and input (inputs designated by v _j ). Each alternative k has performance coefficients for each output (y _ik ) and input (x _jk ). The classic Charnes, Cooper and Rhodes (CCR) ¹⁹ DEA model is:

s.t. For each k from 1 to 12:

The Banker, Charnes and Cooper (BCC) DEA model includes a scale parameter to allow of economies of scale. It also releases the restriction on sign for u _i , v _j .

s.t. For each k from 1 to 12:

u _i  , v _j  ≥ 0, γ unrestricted in sign

A third DEA model allows for super-efficiency. It is the CCR model without a restriction on efficiency ratios.

s.t. For each l from 1 to 12: for l ≠ k

The traditional DEA models were run on the dump site selection model, yielding results shown in Table 8.11:

These approaches provide rankings. In the case of CCR DEA, the ranking includes some ties (for first place and 11th place). The nondominated Nome AL alternative was ranked tenth, behind dominated solutions Turkey TX, Duquesne PA, Yakima Flats WA, and Duckwater NV. Nome dominates Anaheim CA and Santa Cruz CA, but does not dominate any other alternative. The ranking in tenth place is probably due to the smaller scale for the Cost criterion, where Nome AK has the best score. BCC DEA has all dominated solutions tied for first. The rankings for 7th through 12 reflect more of an average performance on all criteria (affected by scales). The rankings provided by BCC DEA after first are affected by criteria scales. Super-CCR provides a nearly unique ranking (tie for 11th place).

The importance of risk management has vastly increased in the past decade. Value at risk techniques have been becoming the frontier technology for conducting enterprise risk management. One of the ERM areas of global business involving high levels of risk is global supply chain management.

Selection in supply chains by its nature involves the need to trade off multiple criteria, as well as the presence of uncertain data. When these conditions exist, stochastic dominance can be applied if the uncertain data is normally distributed. If not normally distributed, simulation modeling applies (and can also be applied if data is normally distributed).

When the data is presented with uncertainty, stochastic DEA provides a good tool to perform efficiency analysis by handling both inefficiency and stochastic error. We must point out the main difference for implementing investment VaR in financial markets such as banking industry and our DEA VaR used for supplier selection is that the underlying asset volatility or standard deviation is typically a managerial assumption due to lack of sufficient historical data to calibrate the risk measure.