To satisfy Constraint 3 (yield cannot deviate too far from baseline yield), the difference between a yield and its baseline yield is called “deviation yield,” denoted as WE.

WE=W-B

With the yield W constrained to [0,1] and a fixed baseline yield B, the theoretical range of the deviation yield WE is [-1,1]. However, this range is too broad. Because inputs and outputs are adjusted daily in production, we aim for today’s yield to closely approximate yesterday’s. When calculating the deviation yield boundary range, we can consider the input-to-output conversion rate of the last day (row) as a benchmark. Using this benchmark and the baseline yield, we can calculate the deviation yield boundary BD. We call this method the “Error Limitation Method.” The calculation process is as follows:

Assume the deviation yield boundary BD has the following form:

Where d_ij represents the lower bound of the deviation yield for the i-th input and j-th output, while u_ij represents the upper bound.

The last row of input and output data:

Xr_m=[x_m1,x_m2,…,x_mn]

Yr_m=[y_m1,y_m2,…,y_mn]

(1) Calculate theoretical output, YTr_m:

YTr_m=Xrm*B

(2) Calculate the difference (DFr_m) between theoretical and actual output:

DFr_m=Yr_m-YTr_m

(3) Calculate the ratio of DFr_m to total input (Pr_m)

Pr_m=DFr_m/sum(Xr_m)

(4) Calculate the deviation yield boundary BD:

Set a proportion for the BD’s allowable adjustment, denoted as pg.

The reference adjustment values d’_mj and u’_mj for BD are calculated based on Pr_m and pg, as follows:

d’_mj=if(p_mj>0,p_mj*(1-pg),p_mj*(1+pg))

u’_mj=if(p_mj>0,(p_mj*(1+pg),p_mj*(1-pg))

Where p_mj is the j-th element of Pr_m.

Originally, this boundary could be used for the deviation yields of all corresponding inputs. However, we need to ensure that all yields are within the range of [0,1] and that when the baseline yield is 0, the yield is also 0, and the boundary upper and lower limits must also be 0.

d_ij and m_ij must satisfy the following four inequalities:

d_ij+b_ij ≥0

u_ij+b_ij ≥0

d_ij+b_ij ≤1

u_ij+b_ij ≤1

Calculate BD using the inequalities and the baseline yield B:

SPL routine:

	A	B
1	[[30,8],[31,7],[38,10]]	/X
2	[[2,13,23],[3,15,20],[11,13,24]]	/Y
3	[[0,0.5,0.5],[0.55,0.05,0.4]]	/B
4	0.1	/rg
5	=A1.m(-1).sum()	/Total Xrm
6	=mul([A1.m(-1)],A3).~	/Theoretical output, YTrm
7	=A2.m(-1)–A6	/Difference between actual and theoretical output, DFrm
8	=A7.(~/A5)	/Ratio of DFrm to total Xrm, Prm
9	=A8.([if(~>0,~*(1-A4),~*(1+A4)),if(~>0,~*(1+A4),~*(1-A4))])	/D’rm,U’rm
10	=A3.(~.((r=A9(#),(if(~==0,[0,0],if(~+r(1)>1,[1-~,1-~],if(~+r(2)<0,[-~,-~],[max(-~,r(1)),min(1-~,r(2))])))))))	/Deviation yield boundary BD

Calculation result example:

Input X:

Output Y:

Baseline yield B:

Boundary adjustment ratio rg:

rg=0.1

Calculated boundary BD: