Output-input ratio and frontier

Chen & Ali (2002) use the output-input ratio to identify DEA frontier DMUs prior to the DEA calculation. They conclude that the output-input ratio with top-ranked performance is a DEA frontier DMU.

or

(ii)

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ −

− =

∑ ∑

∑ ∑

=

=

∈

=

=

s

r r rP

m i i iP s P

r r rT

m i i iT

y u x v y

u x v

1

1 0

Ω 1

1 0

~

~

~

~ max

~

~

μ

μ ^{. (2.2)}

Then, DMU_T is located on the VRS frontier (Chen & Ali, 2002). „

The properties allow using output-input ratio to identify the efficient DMUs without solving DEA mathematical programming problems. To illustrate the property, we consider the data set consists of 6 DMUs, D₁–D₆, each consuming one input, x₁, to produce two outputs, y₁ and y₂, as listed in Table 1. Columns 5-7 present the output-input ratios of y₁/x₁, y₂/x₁, and (y₁+y₂)/x₁, respectively. The ratios are calculated along with Theorem 2.1 by setting ~v₁=1,

~1

μ =1, and μ~₂=1 to part (i).

Table 1. Data set with 6 DMUs.

Outputs Input Efficient

DMU y₁ y₂ x₁ y₁/x₁ y₂/x₁ (y₁+y₂)/x₁ classification

D₁ 1 4 1 1 4^a 5 F

D₂ 2 4 1 2 4^a 6 E

D₃ 3 3.5 1 3 3.5 6.5 E

D₄ 4 3 1 4^a 3 7^b E

D₅ 4 2 1 4^a 2 6 F

D₆ 3 3 1 3 3 6 N

* E means efficient, F means inefficient on frontier, and N means inefficient inner frontier.

a The maximum ratio indicates the DMU is located on the frontier.

b The unique maximum ratio indicates the DMU is extremely efficient.

The ratio of y₁/x₁ indicates that D₄ and D₅ are located on the frontier, ratio of y₂/x₁ indicates that D₁ and D₂ are located on the frontier, and ratio of (y₁+y₂)/x₁ indicates that D₄ is located on the frontier. Hence, there are four DMUs, D₁, D₂, D₄, and D₅, locate on the efficient frontier. Unfortunately, the inefficient DMUs, D₁ and D₅, are also indicated. To avoid the misidentification of inefficient DMUs, Lai & Liu (2006) extend the property that allows using output-input ratio to identify the ‘extremely’ efficient DMUs without solving DEA programs.

This following Corollary will indicate that the unique maximum value of ratio (y₁+y₂)/x₁ allows us to identify D₄ is VRS extremely efficient.

Corollary 2.1 If there exist weight combinations of ~ ≥v_i 0, i=1, … , m, μ~ ≥_r 0, r=1, … , s, and u~ such that

(i)

∑ ∑

Then, DMU_T is VRS extremely efficient.

Proof: We first prove the part (i). For the weights of ~ ≥v_i 0, i=1, … , m, μ~ ≥_r 0, r=1, … , s,

It shows that weight combinations of v_i and μ_r takes the values to all constrains less than one, except the T ^th constrains, and it has optimal value to one. Therefore, following the results of Charnes et al. (1991), DMU_T is VRS extremely efficient. The proof of part (ii) is analogous to part (i) and is omitted. „

We observe that: there are m*s possible pairs of input i and output r, i∈{1, … , m} and r∈{1, … , s}. If any one of the pairs satisfies the following Corollary, DMU_T is VRS extremely efficient (Lai & Liu, 2006).

Corollary 2.2 For any given pair of i’ and r’, i’∈{1, … , m} and r’∈{1, … , s}. If there exists a weight combinations of ~ 0

' ≥

Then, DMU_T is VRS extremely efficient.

Proof: By taking ~ =v_i 0, i=1, … , m, and i≠i’, μ~ =_r 0, r=1, … , s, and r≠r’, we have:

Following the results of Corollary 2.1, DMU_T is VRS extremely efficient. „

2.3 Sensitivity and stability analysis

DEA is non-parametric because it requires no assumption on the weights of the production function. Sensitivity and stability of DMUs is an important issue in DEA. Charnes et al. (1985b) first investigate the sensitivity of single output variation on the CCR model by updating the inverse of the optimal basis matrix. Charnes & Neralic (1990) use the same technique to explore the sensitivity of the additive model for a simultaneous change in all inputs and/or all outputs of an efficient DMU. Andersen & Petersen (1993) propose the

‘extended DEA measure’ (EDM) model for ranking the efficient units. The EDM model (is also called super-efficiency model) is widely applied in the DEA sensitivity analysis. It is based on modifying DEA models in which the test DMU is excluded from the reference set.

For DMU_T is under variation, Charnes et al. (1992; 1996) provide the following formulation to compute stability regions for efficiency classifications under the additive model:

The optimal value Δ is the radius of stability under the ∞-norm. The absolute ^*_T increase of inputs and absolute decrease of outputs are considered only for DMU_T. If we use different Δ and ^I_i Δ and minimize ^O_r

∑ ∑

I , then the optimal solution provides the radius of stability under the 1-norm. The sign of the optimal value indicates the classification of the test DMU_T (Charnes et al., 1992). In the event of set Π comprising the whole DMU

In the event of set Π is a subset of Ω and DMU_T excludes in Π is under evaluation, negative also identifies inefficient; however, positive indicates that DMU_T is located above the frontier of Π, it means that DMU_T has the possibility to perform better than Π, and it is classified as an efficient candidate. Based on the results, our study suitably selects a class of portfolios with higher performance relative to the others, which is called an ‘efficient candidate group’ (ECG) within our proposed algorithm which is called the ‘filtering algorithm’ in this paper. The main frame of our filtering algorithm is:

(i) Using model (M7) to evaluate DMU_T, where Π is substituted by set ECG.

(ii) If Δ <0, DMU^*_{T T} is identified as inefficient.

Otherwise, DMU_T joins to ECG as a new membership.

Zhu (1996) uses the super-efficiency model to determine necessary and sufficient conditions for preserving efficiency of the efficient DMUs under the CCR model when data of the test efficient DMU was changed, and Seiford & Zhu (1998a) generalize the method to yield the entire stability region of the test DMU. These literatures of sensitivity and stability analysis deal with the situation in which the data variations are only applied to the test DMU.

However, possible data errors may occur in all DMU simultaneously or individually.

Thompson et al. (1994) utilize the Strong Complementary Slackness Condition (SCSC) multipliers to analyze the stability of CCR efficiency when the data for all efficient DMUs were worsened and data for all inefficient DMUs were improved simultaneously. Seiford &

Zhu (1998b) discuss the stability of efficient DMU based on a worst-case scenario in which the efficiency of the test DMU was deteriorating while the efficiency of all other DMUs were improving. They use super-efficiency models to find a range of stability for each efficient DMU to preserve efficiency when data variations occurred in all DMUs simultaneously. In the real-world problems, uncertain conditions could occur not only in single DMU or in all of DMUs but also in a particular local or regional subset of DMUs. It means that the possible data errors may occur in a subset due to the situations of local uncertainty.

In this research, we are interested in the stability of a specific efficient DMU_T while the data of a particular subset of DMUs, including DMU_T, is deteriorated simultaneously in the same value. Since either an increase of any output or a decrease of any input cannot worsen an efficient DMU, we consider the data was changed by giving upward variations in inputs or giving downward variations in outputs in a subset of DMUs.

3. Identification of Efficient Portfolios

The difficulty for using DEA to assess and select portfolios of collective projects is that there are 2^K portfolios need to be evaluated. We must spend more effort on intensive DEA calculation. The papers Ali (1990; 1992; 1994) present some properties to allow identification of efficient and inefficient DMUs without solving a mathematical programming.

To circumvent the time-consuming DEA computations, we also derive some properties to identify efficient and inefficient classes prior to the DEA calculation for streamlining the solution of DEA programs.

3.1 Single input and output problems

Now, let us first consider the special case that the projects have only one input and output. The two objectives BILP model is expressed as follows:

Maximize y= c₁w₁+c₂w₂+ … +c_Kw_K (M8) Minimize x= a₁w₁+a₂w₂+ … +a_Kw_K

Subject to w_k ∈{0, 1}, k=1, 2, … , K.

3.1.1 relationship between ratio dominance and inefficiency

Let R_k denote the ratio of the output to input for project k. That is, R_k=c_k /a_k. The relationship of dominance between two projects by the output-input ratios is defined as follows:

Definition 3.1 Project h dominates project p, if R_h > R_p. „

We shall show that if project p is dominated by project h, and a portfolio includes the dominated project p but excludes project h, then the portfolio is inefficient.

Lemma 3.1 If

2 2 1 1

a c

ac ≥ , where a₁, a₂, c₁, and c₂ are all positive. Then,

2 2 2 1

2 1 1 1

a c a a

c c a

c ≥

+

≥ + . „

This property shows that c₁ /a₁ ≥ (c₁w₁+c₂w₂+ … +c_Kw_K)/(a₁w₁+a₂w₂+ … +a_Kw_K), for all portfolio P=(w₁, w₂, … , w_K) in Ω and P≠(0, 0, … , 0). That is, P=(1, 0, … , 0) possesses the maximum output-input ratio among the 2^K possible portfolios. Note that P=(0, 0, … , 0) and P=(1, 0, … , 0) are evidenced as CCR efficiency (Ali, 1994). The following Theorem will be

used to characterize inefficient portfolios. Let e_k denote the unit row vector with 1 at the k^th component and 0 elsewhere.

Theorem 3.1 T=(w₁, w₂, … , w_K) with w_h=0 and w_p=1, is inefficient if project h dominates project p.

Proof: Let portfolios H=T−e_p and G=T+e_h. The DMUs corresponding to portfolios H, T, and G are expressed respectively as the followings:

DMU_H=(y_H, x_H),

DMU_T=(y_T, x_T)=(y_H +c_p, x_H +a_p), and

DMU_G=(y_G, x_G)=(y_H +c_h +c_p, x_H +a_h +a_p).

Let us take constant t=a_p /(a_h+a_p). It thus follows:

(1−t) x_H + t x_G = (1−t) x_H + t (x_H +a_h +a_p) (3.1)

= x_H + t (a_h +a_p) = x_H + a_p

= x_T , and

(1−t) y_H + t y_G = (1−t) y_H + t (y_H +c_h +c_p) (3.2)

= y_H + t (c_h +c_p)

= y_H + a_p (c_h +c_p)/(a_h +a_p)

> y_H + a_p (c_p /a_p) (By Lemma 3.1) = y_T .

It shows that DMU_T is convex-dominated by DMU_H and DMU_G. Therefore, DMU_T is DEA inefficient and so does portfolio T. „

This Theorem enables us to identify efficient and inefficient portfolios prior to the DEA calculation by comparing the output-input ratios of pair of projects.

3.1.2 Efficient portfolios

Without loss of generality, it is assumed that the indices of projects are arranged according to the descendant order of their output-input ratios, i.e., R₁ >R₂ > L >RK, and the

strict inequality holds here. The following Corollary uses ratio analysis to characterize the dominated portfolios, and like their correspondent DMUs, they are inefficient.

Corollary 3.1 Portfolio T=(w₁, w₂, … , w_K) is inefficient if w_k=0 and w_k+1=1 for some k.

Proof: Since R_k > R_k+1 implies that project k dominates project (k+1). Then, the result follows from Theorem 3.1. „

Corollary 3.1 indicates that a project with larger output-input ratio must be selected prior to the others. Based on the result, only the remaining (K+1) portfolios that have the possibility of VRS efficiency. They are listed in the followings:

Table 2. The portfolio lists of candidate efficiency.

Portfolio w₁ w₂ L w_K−1 w_K

The null portfolio (0, 0,…, 0) with minimum input value is clearly VRS efficient (Ali, 1994).

The other K portfolios will be shown as VRS efficient by employing model (M6) to evaluate their corresponding DMUs. These DMUs are expressed as the followings:

DMU_T=(x_T, y_T)=(a₁ +a₂ + … +a_T, c₁ +c₂ + … +c_T), T=1, 2, … , K. (3.3)

For all k >T, we have:

The equality holds only for k=T. This indicates that the optimal value to (M6) is equal to one.

Therefore, DMU_T is VRS extremely efficient for T=1, … , K. „

Hence, there are (K+1) VRS efficient portfolios obtained by using ratio techniques.

Ratio analysis is shown to be an effective method to identify the entire set of efficient portfolios for the single input and output problems. To illustrate this, let us consider the following example.

3.1.3 Example 1: single input and output case

Suppose there are five projects numbered k=1, … , 5, in a decision set. Their input and output are given in Table 3. Where the indices of projects have been arranged in descendent order of output-input ratios. All possible portfolios comprise a subset of the 5 projects are evaluated by the following unconstrained MOBILP.

Maximize y= 6 w₁+ 4.0 w₂+ 7.2 w₃+ 8 w₄+ 1 w₅ (M9) Minimize x= 4 w₁+ 2.8 w₂+ 5.6 w₃+ 9 w₄+ 2 w₅

Subject to w_k ∈{0, 1}, k=1, 2,…, 5.

According to the results of Theorem 3.2, six portfolios, (0,0,0,0,0), (1,0,0,0,0), (1,1,0,0,0), (1,1,1,0,0), (1,1,1,1,0), and (1,1,1,1,1) are identified as VRS efficient.

Table 3. The data of 5 projects for Example 1.

Project Output (c_k) Input (a_k) Ratio (R_k)

3.1.4 Problems with non-positive coefficients

The assumption that the positive coefficients a_k >0 and c_k >0 for all k=1, … , K, could be violated. Now, let us consider that the projects be partitioned based on the following six sets of indices:

I_P={ k | 1≤ k≤ K, c_k > 0 and a_k > 0}, (3.7) I_N={ k | 1≤ k≤ K, c_k < 0 and a_k < 0}, (3.8) I₀={ k | 1≤ k≤ K, c_k > 0 and a_k ≤ 0}, (3.9) I₁={ k | 1≤ k≤ K, c_k ≤ 0 and a_k > 0}, (3.10) I_C={ k | 1≤ k≤ K, c_k < 0 and a_k = 0}, (3.11) and

I_A={ k | 1≤ k≤ K, c_k = 0 and a_k < 0}. (3.12) The problem can be handled according to the following theorems.

Theorem 3.3 Portfolio H=(w₁, … , w_k−1, 0, w_k+1, … , w_K) is DEA inefficient if k∈I₀. Proof: Let T=(w₁, … , w_k−1, 1, w_k+1, … , w_K). It follows

(–x_T, y_T) = (–x_H –a_k, y_H +c_k) > (–x_H, y_H). (3.13)

This implies that portfolio H is DEA inefficient. „

Theorem 3.4 Portfolio H=(w₁, … , w_k−1, 1, w_k+1, … , w_K) is DEA inefficient if k∈I₁. Proof: Let T=(w₁, … , w_k−1, 0, w_k+1, … , w_K). It follows

(–x_T, y_T) = (–x_H +a_k, y_H –c_k) > (–x_H, y_H). (3.14)

This implies that portfolio H is DEA inefficient. „

Theorem 3.3 and 3.4 indicate that a portfolio is inefficient if it excludes a project consuming non-positive input to produce positive output, or it includes a project consuming positive input to produce non-positive output. Therefore, we have the following subsets of portfolios are inefficient:

Ω₀={P=(w₁, … , w_K) | w_k=0, for any k∈I₀} (3.15)

and

Ω₁={P=(w₁, … , w_K) | w_k=1, for any k∈I₁}. (3.16)

For the case that both a_j and c_j are non-positive occurs in model (M8). We redefine all binary variables and coefficients of objectives as the followings:

⎩⎨

⎧ − ∈Θ= ∪ ∪

= , Otherwise if

= , Otherwise if

= , Otherwise if

Then, model (M8) can be rewritten as follows:

.

The new MOBILP model (M10) has objectives with non-negative coefficients, either

≥0

ck or a_k ≥0, corresponding to all new variables w_k. We can construct the new sets of indices I , ₀ I , and ₁ I corresponding to model (M10), and it follows that I_{P C} ⊆I , I_{0 A} ⊆I , and ₁ I_N ⊆I . Then, the following sets of inefficient portfolios are characterized by using Theorem _P 3.3 and 3.4.

Ω_A={P=(w₁, … , w_K) | w_k=0 if k∈I_A} ⊆ {P=(w K₁ , , w_K)| w_k=1 if k∈I }. (3.20) ₁ and

Ω_C={P=(w₁, … , w_K) | w_k=1 if k∈I_C} ⊆ {P=(w K₁ , , w_K)| w_k=0 if k∈I } (3.21) ₀ However, the new model (M10) transforms the objectives to non-negative coefficients

and all efficient portfolios can be determined by using Theorem 3.2–3.4.

3.1.5 Algorithm for identification of efficient classification

A complete algorithm for developing all efficient portfolios is presented as follows:

Step 1. Identify sets of indices I_P, I_N, I₀, I₁, I_C, and I_A according to (3.7)–(3.12).

Step 2. Reset original indices of projects in I_N, I_C, and I_A according to equations (3.17)–(3.19).

Step 3. Identify sets of indices I , _P I , and ₀ I , and let N_{1 P}, N₀, and N₁ denote the number of elements in set I , _P I , and ₀ I , respectively. ₁

Step 4. Re-index all projects and rewrite model:

Step 4.1 Re-indexed project, w_k, from 1 to N_p for k∈I , from (N_{P P}+1) to (N_P+N₀) for k∈I , and from (N_{0 P}+N₀+1) to (N_P+N₀+N₁) for k∈I . ₁

Step 4.2 Rearrange w_k according to R₁ >R₂ >L>R_Npfor k∈I , where _P R_k =c_k/a_k. Step 4.3 Original problem (M8) is rewritten as (M10).

Step 5. Identify the set consists of N_P+1 efficient portfolios as follows:

Ω_E={P=(w K₁ , , w_K)| w_k≥w_k+1 if k<N_P, w_k=1 if k∈I , and ₀ w_k=0 if k∈I }. (3.22) ₁

3.1.6 Example 2: general two objectives BILP

Suppose there are 10 projects indexed by k=1, … , 10, in a decision set. The values of input and output are given in Table 4. The problem of portfolio evaluation in the set is modeled as (M8). The efficient portfolios can be obtained by according the following steps:

Step1. Sets of indices based on (3.7)–(3.12) are identified as follows:

I_P={1, 4, 6}, I_N={5, 9}, I₀={8}, I₁={2, 10}, I_C={3}, and I_A={7}.

Step 2. Reset original data of projects 5, 9, 3, and 7 according to (3.17)–(3.19).

Step 3. Identify sets of indices I , _P I , and ₀ I , and number of elements in these sets are N_{1 P}=5, N₀=2, and N₁=3, respectively.

I =IP _P∪I_N={1, 4, 6, 5, 9}, I =I_{0 0}∪I_C={8, 3}, I =I_{1 1}∪I_A={2, 10, 7}.

Step 4. Use Step 4.1 and 4.2 to re-index all projects as the followings:

I ={1, 2, 3, 4, 5}, P I ={6, 7}, ₀ I ={8, 9, 10}. ₁

The relationship between origin and transformed index is listed in Table 4. Then, use Step 4.3 to rewrite the original problem as the followings:

.

Table 4. The original and transformed data of 10 projects.

Original data of projects Transformed data of projects Index

(k of w_k)

Output (c_k)

Input (a_k)

Index (k of w_k)

Output (c_k)

Input (a_k)

Ratio (c_k/a_k)

6 6.0 4.0 1 6.0 4.0 1.50

4 4.0 2.8 2 4.0 2.8 1.43

9 –7.2 –5.6 3 7.2 5.6 1.30

1 8.0 9.0 4 8.0 9.0 0.89

5 –1.0 –2.0 5 1.0 2.0 0.50

8 1.0 –2.4 6 1.0 –2.4 ––

3 –1.6 0 7 1.6 0 ––

2 –3.2 1.5 8 –3.2 1.5 ––

10 –3.0 2.0 9 –3.0 2.0 ––

7 0 –2.5 10 0 2.5 ––

Step 5. Using Theorem 3.2–3.4, we have 6 efficient portfolios which is listed as follows:

) , ,

(w K₁ w₁₀ = (0,0,0,0,0,1,1,0,0,0) = (w₁, … , w₁₀) = (0,0,1,0,0,0,0,1,0,0), )

, ,

(w K₁ w₁₀ = (1,0,0,0,0,1,1,0,0,0) = (w₁, … , w₁₀) = (0,0,1,0,0,1,0,1,0,0), )

, ,

(w K₁ w₁₀ = (1,1,0,0,0,1,1,0,0,0) = (w₁, … , w₁₀) = (0,0,1,1,0,1,0,1,0,0), )

, ,

(w K₁ w₁₀ = (1,1,1,0,0,1,1,0,0,0) = (w₁, … , w₁₀) = (0,0,1,1,0,1,0,1,1,0), )

, ,

(w K₁ w₁₀ = (1,1,1,1,0,1,1,0,0,0) = (w₁, … , w₁₀) = (1,0,1,1,0,1,0,1,1,0), )

, ,

(w K₁ w₁₀ = (1,1,1,1,1,1,1,0,0,0) = (w₁, … , w₁₀) = (1,0,1,1,1,1,0,1,1,0).

3.2 Multiple inputs and outputs problems

When there are m inputs and s outputs to MOBILP (M1). Since, ratio analysis is shown to be an efficient method to identify the entire set of efficient portfolios for the case of single input and output. Based on the results of Theorem 3.2 and Corollary 2.2, the ratio analysis is capable of identifying a subset of efficient portfolios for the cases of multiple inputs and outputs. The MOBILP can be decomposed to (s×m) sub-problems by the pairs of one output and one input. There are (K+1) efficient portfolios identified by each sub-problem. Corollary 2.2 also indicates that those efficient portfolios are also efficient for the original model.

By removing the duplications, the efficient portfolios identified by employing the (s×m) sub-problems are aggregated as a subset. The subset is called the ‘seed efficient class’

(SEC). In our filtering algorithm, the frontier of ECG is the filter for the algorithm and ECG consists of those elements in SEC initially.

3.2.1 Inefficiency with project dominance relationship (PDR)

Let R_k^ri denote the ratio of the r^th output value to i^th input value of project k, where

ik rk ri

k c a

R = / . The dominance relationship between two projects by the output-input ratios is defined as follows:

Definition 3.2 Project h dominates project p, if R_h^ri ≥R^ri_p for all pairs of r and i, i = 1, … , m, and r = 1, … , s, and strict inequality holds for at least one pair of indices. „

The relationship between output-input ratios of projects and the efficiency of portfolio to the multiple inputs and outputs problems is shown in Liu & Lai (2005a).

Theorem 3.5 Portfolio T=(w₁, … , w_K) is inefficient if project h dominates project p and w_h =0 and w_p =1.

Proof: Let H =T−e_p and G =T+e_h. The DMUs corresponding to portfolios H, T, and G are expressed as follows:

DMU_H=(x_1H, … , x_mH, y_1H, … , y_sH),

DMU_T=(x_1H+a_1p, … , x_mH+a_mp, y_1H+c_1p, … , y_sH+c_sp), and

DMU_G=(x_1H+a_1h+a_1p, … , x_mH+a_mh+a_mp, y_1H+c_1h+c_1p, … , y_sH+c_sh+c_sp).

Let us take constants β₁ and β₂ as follows:

β₁= max{c_rp/(c_rh +c_rp)| r=1, … , s.}

and

β₂= min{a_ip/(a_ih +a_ip)| i=1, … , m.},

where β₁, β₂∈(0,1). Then, there exist specific indices i and r such that β₁ /β₂ =(c_rp /(c_rh +c_rp)) / (a_ip /(a_ih +a_ip))

=(c_rp / a_ip) / ((c_rh +c_rp) / (a_ih +a_ip))

< 1. (by Lemma 3.1)

It indicates that β₁<β₂. Let β be a constant between β₁ and β₂. We shall show that DMU_T is convex-dominated by DMU and DMU . Since,

(1−β) x_iH +β x_iG = x_iH + β (a_ih +a_ip) ≤ x_iH + β₁ (a_ih +a_ip) ≤ x_iH + a_ip

= x_rT, for all i = 1, 2, …, m, and

(1−β) y_rH +β y_rG = y_rH + β (c_rh + c_rp) ≥ y_rH + β₂ (c_rh + c_rp) ≥ y_rH + c_rp

= y_rT, for all r = 1, 2, …, s,

and at least one inequality holds. It shows that DMU_T is dominated by (1−β)DMU_H+βDMU_G. Therefore, DMU_T is inefficient and so does portfolio T. „

It has shown that if project p is dominated by project h and a portfolio includes the dominated project p but excludes project h, then the portfolio must be inefficient. This enables us to identify efficient and inefficient portfolios prior to the DEA calculation by using the output-input ratio of an individual project.

3.2.2 Example 3: use ratio analysis to identify SEC

A simulated data set comprising seven R&D projects in a high tech corporation is listed in Table 5. These projects are proposed to promote the product quality for the company.

Each project consumes two inputs to produce two outputs. The outputs are percentages of technical contributions to the products and direct economic contributions in product sales, while the inputs are percentages of manpower usage and finance usage with respect to the company. Suppose that the projects are neither synergistic nor interfering and the resources are fully supported. The decision-maker wants to select a class of portfolios, from all of the 128 (=2⁷) feasible portfolios, play the best practice with respect to the others.

By comparing the output-input ratios of projects, we have project 7 being dominated by project 6. Following the results of Theorem 3.5, we conclude that a portfolio is identified as inefficient if it contains project 7 but excludes project 6. That is, a portfolio is inefficient if it is expressed as the following form.

Table 5. Data set of 7 R&D projects for Example 3.

R&D project

Technical contribution

Product sales

Manpower usage

Resource usage

(k) (c_1k) (c_2k) (a_1k) (a_2k) R_k¹¹ R_k²¹ R¹²_k R_k²²

1 1.8 7.0 3.0 6.0 0.600 2.333 0.300 1.167

2 1.6 10.0 4.0 5.5 0.400 2.500 0.291 1.818

3 1.4 8.2 3.6 4.5 0.389 2.278 0.311 1.822

4 1.9 13.0 5.0 7.0 0.380 2.600 0.271 1.857

5 1.4 5.0 6.0 4.0 0.233 0.833 0.350 1.250

6 1.8 12.0 8.0 3.0 0.225 1.500 0.600 4.000

7 1.7 6.0 9.3 4.0 0.183 0.645 0.425 1.500

(w₁, w₂, w₃, w₄, w₅, 0, 1) for w_k=0 or 1, k=1, 2, 3, 4, 5. (3.23) Hence, 32 portfolios are characterized as inefficient by using ratio analysis. Now, we turn to identify efficient portfolios by using Theorem 3.2. The ratios of output 2 to input 1, sayR²¹_j , of projects are ranked as following.

21 7 21 5 21 6 21 3 21 1 21 2 21

4 R R R R R R

R > > > > > > (3.24)

It indicates that the 8 portfolios, (0,0,0,0,0,0,0), (0,0,0,1,0,0,0), (0,1,0,1,0,0,0), (1,1,0,1,0,0,0), (1,1,1,1,0,0,0), (1,1,1,1,0,1,0), (1,1,1,1,1,1,0), and (1,1,1,1,1,1,1), are efficient.

Similarly, we can rank the ratios R , _k¹¹ R , and¹²_k R to identify efficient portfolios. By _k²² removing the duplications, our ratio analysis identifies 23 efficient portfolios. These techniques identify 23 efficient and 32 inefficient portfolios prior to the DEA programs. In total, we save 55 computations for solving linear program effectively and efficiently.

3.2.3 Inefficiency with inferior project combination (IPC)

Apply additive model (M3) or (M7) to evaluate a particular project h with respect to the original K projects. The reference set is defined as Λ(h)={ k | λ^*_k>0, k = 1, 2, … , K}.

Then, a portfolio is identified to be inefficient if it composes project h and without any element in set Λ(h). That is, the portfolio comprises only inferior projects. This portfolio is called as an inferior project combination (IPC).

Theorem 3.6. Portfolio T = (w₁, w₂, … , w_K) is inefficient if w_h = 1 and

∑

k∈_Λ(h_),k≠hwk =0. Proof: The result is trivial and is omitted. „

One can use this Theorem to pre-identify some inefficient portfolios: just use model (M3) or (M7) to evaluate the K projects. It is clear that the situation occurs only if project k is inefficient with respect to the original K projects.

3.2.4 Inefficiency with total dominated relationship (TDR)

Ali (1994) defined a total dominated relationship (TDR) between DMUs. A portfolio is totally dominated if its corresponding DMU is dominated by any other DMU in Ω_D.

Definition 3.3 Portfolio T is totally dominated by portfolio H if DMU_T is dominated by DMU_H, that is, x_iT ≥ x_iH, for all i = 1, … , m, y_rT ≤ y_rH, for all r = 1, … , s, and strict inequality holds for at least one index. „

Theorem 3.7. If portfolio T is totally dominated by portfolio H for some H then portfolio T is inefficient.

Proof: The proof is omitted. „

3.3 Filtering algorithm

We propose a forward and reverse filtering algorithm to solve the unconstrained MOBILP (M1). To reduce the problem size of model (M7) and to identify inefficient portfolios effectively, we substitute the reference set Π by a group of portfolios ECG with higher performance throughout the algorithm. ECG is updated dynamically by using forward and backward filtering algorithms. An algorithm comprising three phases is presented below.

We propose a forward and reverse filtering algorithm to solve the unconstrained MOBILP (M1). To reduce the problem size of model (M7) and to identify inefficient portfolios effectively, we substitute the reference set Π by a group of portfolios ECG with higher performance throughout the algorithm. ECG is updated dynamically by using forward and backward filtering algorithms. An algorithm comprising three phases is presented below.

在文檔中 評選多項計畫的組合之高效能方法 (頁 22-0)