Measuring Olympic achievements in DEA: a data fitting technique subject to downside-deviation restrictions

Measuring Olympic achievements in DEA: a data fitting technique subject to downside-deviation restrictions

01206313

^{東京理科大学 関谷 和之}

05000602

東京理科大学

趙 宇

1. Introduction

Japan has become an Olympic setting since the Summer Olympic Games were first staged in Asia in 1964. After Tokyo was selected as the host city of the 2020 Summer Olympics, the Japanese Olympic Committee (JOC) set a target of a record-high 30 gold medals. However, due to the developing global situation in light of the COVID-19 pandemic, the International Olympic Committee (IOC) and JOC announced that the Tokyo 2020 Summer Olympic games are resched- uled to a date beyond 2020 but not later than summer 2021. As the upcoming Olympic Games gets closer, interest in predicting and measuring the Olympic achievements is rising. To investi- gate and predict the performance of the upcom- ing Summer Olympics, the proposed model of this study incorporates a data fitting technique of medal prediction using ordinary least squares regression (OLS) in input multiplier restrictions of the conventional DEA model.

2. Downside-deviation least squares Suppose that there arenDMUs (j= 1, . . . , n) to be evaluated. Three output variables are con- sidered: the number of gold (y1j), silver (y2j), and bronze medals (y3j) achieved in the previ- ous Olympic games. Regarding the inputs, we adopt the GDP (x_1j) and the population (x_2j) recorded in the latest year. The DEA technology converts an input vector x_j = (x_1j, x_2j, x_3j)^⊤ to an output vector y_j = (y1j, y2j, y3j)^⊤, thereby satisfying xj ≥0,xj ̸=0 and y_j ≥0,y_j ̸=0 for all j= 1, ..., n.

Let e be a column vector with all elements unity. The OLS estimator for predicting the total number of medalse^⊤y_j of DMU_j (j∈ {1, ..., n})

is formulated as the following optimization prob- lem:

(OLS: Ordinary least squares) min

Xn j=1

d²_j

s.t. d_j =v^⊤x_j+ν−e^⊤y_j (j= 1, . . . , n).

Here, the estimation (v^∗, ν^∗) presents the hy- perplane passing the barycenter of n DMUs,

1 n

P_n

j=1xj,P_n

j=1e^⊤yj

.

Since an upper bound v^⊤x_k+ν of e^⊤y_k im- plies v^⊤x_k+ν ≥ e^⊤y_k, we add n nonnegative constraints, d₁ ≥ 0, . . . , d_n ≥ 0, into the above OLS problem, which is formulated as

(DLS: Downside-deviation least squares) min

Xn j=1

d²_j

s.t. 0≤d_j =v^⊤x_j+ν−e^⊤y_j (j= 1, . . . , n) v≥0.

Let ( ˆd,v,ˆ ˆν) be an optimal solution to the DLS problem. Thus, ˆv^⊤x_k+ ˆν is one of the upper bounds of e^⊤y_k, and ˆd_k is a downside deviation of e^⊤yk. Let

a:=

Xn j=1

e^⊤y_j− ˆ

v^⊤x_j+ ˆν, (1) then we have

a= Xn j=1

dˆj.

Theorem 1. Let ( ˆd,v,ˆ ν)ˆ be an optimal solution to the DLS problem. The upper bound vˆ^⊤xj + ˆν of e^⊤yj is independent of the choice of optimal solutions to the DLS problem. The same is true of the equation (1).

2-D-4

2021年 春季研究発表会

3. A multiplier DEA model with downside-deviation restrictions We define

V =





(v, ν)

P_n

j=1 v^⊤xj +ν−e^⊤yj

≤a e^⊤y_j ≤v^⊤x_j+ν, ∀j

v≥0





 as a set of input multiplier (v, ν). Here,v^⊤x_k+ ν is referred to as a substantial medal total of DMUk for some (v, ν)∈V.

To evaluate the efficiency of Olympic achieve- ments based on the substantial medal totals, we propose the following DEA model (LP_θ) :

θk= max. u^⊤y_k v^⊤x_k+ν s.t. u^⊤y_j−

v^⊤x_j+ν

≤0, ∀j

e^⊤yj−

v^⊤xj+ν

≤0, ∀j Xn

j=1

v^⊤xj+ν−e^⊤yj

≤a

R^⊤u≥0,u≥0,v ≥0,

where R=





1 0 0

−P 1 0 0 −Q 1



. The notationsP

and Q represent the lower bounds of the ratios u1/u2 and u2/u3, respectively. If θ_k = 1, DMU_k is efficient. Otherwise, DMU_k is inefficient.

The efficiency of Olympic achievements for DMU_k in model (LP_θ) can be decomposed as

u^⊤yk

v^⊤x_k+ν = e^⊤yk

v^⊤x_k+ν ·u^⊤yk

e^⊤y_k. Here, the first component, e^⊤y_k/ v^⊤x_k+ν

is referred to asthe achievement ratio of substantial medal total. The second component u^⊤yk/e^⊤yk

is termed as theunit value index of medals.

4. Examining the target of 30 gold medals in Tokyo 2020

The input vector of Japan (xJ) in Rio 2016 is used for predicting the total number of gold medals in Tokyo 2020, which is x_J =

(xJ1, xJ2, xJ3)^⊤ = (32477.2,126958472,41)^⊤. For the input vector xJ, we derive the output possibility set from the dual problem of (LP_θ), which is

P(xJ) =















 y

y≤P_n

j=1λjyj−Rd x_J ≥P_n

j=1(λj+µj)xj

P_n

j=1(λ_j +µ_j) = 1

−aµj ≤P_n

h=1µh(e^⊤yh) d≥0, λ≥0, µ:free















 .

Provided that the target of 30 gold medals in Tokyo 2020 is feasible, the optimal value of

max{y1|y∈ P(x_J), y is nonnegative} (2) is equal or more than 30. However, the optimal value of the problem (2) is 26, which implies the target of 30 gold medals in Tokyo 2020 is infea- sible even if we investigate all possible combina- tions of medals within P(xJ).

How many medals will Japan win if 26 gold medals are actually achieved? To answer this question from a practical point of view, it is necessary characterize the possible patterns of medal won based on the patterns of past Olympic games. Consider the following linear combina- tion:

Y_J = ( ₄

X

t=1

γ^ty_J^t

γ¹≥0,· · · , γ⁴ ≥0 )

, whereγ¹, ..., γ⁴ are scalars andy¹_J, . . . ,y⁴_J are in- teger vectors of medals that Japan won in the past four Olympic games. Assume the medals Japan will achieve in Tokyo 2020 belong to Y_J. Therefore, if Japan bags 26 gold medals in Tokyo 2020, the maximal number of medal totals will be the optimal solution of the following problem:

max ( ₃

X

r=1

y_r

y∈ P(x_J), yis nonnegative )

. (3) We found that the optimal value of the problem (3) is a total of 63, and y^∗ = (y^∗₁, y^∗₂, y^∗₃)^⊤ = (26,15,33)^⊤. Thus, Japan will win 63 medals at most, which consists of 26 gold, 15 silver, and 22 bronze medals.