DEA based production planning considering attainability and management goals with undesirable outputs

doi:10.3969/j.issn.0253-2778.2020.07.011

Abstract

Abstract: The problem of production planning in a centralized decision-making environment usually involves the participation of all subunits, and each subunit makes its own contribution to the total production. When making a production planning, the central decision-maker needs to determine the inputs and outputs quantities of each subunit based on the changes in product demand and available resources that are known or predictable in the next quarter. On the one hand, due to the relative stability of the production technology level in the short term, it is unrealistic to greatly increase production, we need to consider the attainability of the production planning, that is, the closer the production plan for the next quarter is to the production status of the current quarter, the easier it is to achieve. On the other hand, in the previous production planning research, the central decision-maker did not formulate targeted management goals for each subunit, so that the production plan cannot fully reflect the manager’s expectations of the goal, so the management goals should be taken into account, and the production planning is required to be as close as possible to the management goals. In addition, in the production process, outputs can be divided into desirable and undesirable outputs.From these perspectives, a production planning method based on data envelopment analysis(DEA) for decision-making units with undesirable outputs was proposed by considering the attainability of production planning and management goals set in advance in a centralized decision-making environment. The proposed model was illustrated by the real cases of 32 paper mills along the Huaihe River in Anhui Province, China.

Key words: data envelopment analysis, production planning, attainability, management goals, undesirable outputs

CLC Number:

C935

ZENG Wenjiang, YANG Feng. DEA based production planning considering attainability and management goals with undesirable outputs[J]. Journal of University of Science and Technology of China, 2020, 50(7): 940-949.

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附录

A.1 定理1.1证明

记
hk=(h(1)k,h(2)k,…)，hNk=(h(1)k,…, h(N)k)T，
h-Nk=(hk(N+1),h(N+2)k,…)T，

Φ(xi)N=(1(xi), 2(xi),…,N(xi))T，
Φ(xi)-N=(N+1(xi),
N+2(xi),…)T，
HN=(h1N,h2N,…,hdN)T，
H-N=(h-N1,h-N2,…,h-Nd)T，
则


H(Φ(xi)-Φ(xj))=HN(ΦN(xi)-ΦN(xj))+
H-N(Φ-N(xi)-Φ-N(xj))(A1)


由于{Φ(x1),…,Φ(xn)}互不相同，有‖Φ(xi)-Φ(xj)‖2>0.由Cauchy-Schwartz不等式，可得

‖H-N(Φ-N(xi)-Φ-N(xj))‖2≤
∑dk=1‖h-Nk‖2‖Φ-N(xi)-Φ-N(xj)‖2(A2)

因为hk∈l2, Φ(xi)∈l2，所以对∈(0,1/3), N>0，使得

‖H-N(Φ-N(xi)-Φ-N(xj))‖2≤‖Φ(xi)-Φ(xj)‖2(A3)

且

‖(ΦN(xi)-ΦN(xj))‖2≥(1-)‖Φ(xi)-Φ(xj)‖2(A4)

所以当d>O(log(n)2)时，
由Johnson-Lindenstrauss定理[15],有

‖H(Φ(xi)-Φ(xj))‖2≤
‖HN(ΦN(xi)-ΦN(xj))‖2 +
‖H-N(Φ-N(xi)-Φ-N(xj))‖2 ≤
(1+)‖ΦN(xi)-ΦN(xj)‖2+
‖Φ(xi)-Φ(xj)‖2 ≤
(1+2)‖Φ(xi)-Φ(xj)‖2(A5)

另一方面，

‖H(Φ(xi)-Φ(xj))‖2 ≥
‖HN(ΦN(xi)-ΦN(xj))‖2 -
‖H-N(Φ-N(xi)-Φ-N(xj))‖2≥
(1-)‖ΦN(xi)-ΦN(xj)‖2-
‖Φ(xi)-Φ(xj)‖2≥
(1-)(1-)‖Φ(xi)-Φ(xj)‖2-
‖Φ(xi)-Φ(xj)‖2=
(2-3+1)‖Φ(xi)-Φ(xj)‖2≥
(1-3)‖Φ(xi)-Φ(xj)‖2(A6)

由(A5)和(A6)，我们可得证.

A.2 定理1.2证明
对于ERCRPn，我们有


ERCRPn=Eπ0 ∫Rp 1CRPn(Φ(x))=1dP0(x)+
Eπ1 ∫Rp 1{CRPn(Φ(x))=0}dP1(x)=
Eπ0 ∫Rp 1{νn(Φ(x))≥1/2}dP0(x)+
Eπ1 ∫Rp 1{νn(Φ(x))<1/2}dP1(x)=
π0 ∫RpP{νn(Φ(x))≥1/2}dP0(x)+
π1 ∫RpPνn(Φ(x))<1/2dP1(x).

令Um=1{CHmn(Φ(X))=1},m=1,…,M，在给定μn(Φ(X))=θ∈[0,1] 的条件下，随机变量U1,…,UM 是独立的，都服从两点分布Bernoulli(θ). 注意到Gn,0 和Gn,1是μn(Φ(X))|{Y=0}的分布函数和μn(Φ(X))|{Y=1}的分布函数，我们可得

∫Rp Pνn(Φ(x))<1/2dP1(x)=
∫[0,1]P1M ∑Mm=1 Um<1/2 |μn(Φ(X))=θdGn,1(θ)=
∫[0,1] PT

其中，T服从参数为M和θ的二项分布，记为T～Bin(M,θ).类似地，

∫RpPνn(Φ(x))≥1/2dP0(x)=
1-∫[0,1] PT
所以，ERCRPn=∫[0,1]PT
∫[0,1]PT1/2-[[M/2]]Mg°n(1/2)+18M°n(1/2)+o1M(A7)

根据文献[23]的结论，有

∫[0,1]PT1-α-[[M α]]Mg°n(α)+α(1-α)2 M°n(α)+o1M(A8)

我们令α=1/2即可证得定理.

()
()

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