where k( ) is the average yield stress evaluated at the mean reduction bar  , defined as Eq. (12):
 
where H0 is the annealed strip thickness. According to Hill's experimental results,
 
where μt is a constant; νroll is Poisson's ratio of the work roll; Εroll is Young's modulus of the work roll; R is the work roll radius; and u is the friction coefficient.
3.1.2.    Tension cost function
The tension between stands should be kept midway between its lower limit t i f min and upper limit t i f max: The lower limit is specified as the maximum value of the measured noise under mill conditions, while the upper limit is assigned on the basis of a consideration of strip skidding and tearing situations. Normally, the tension cost function is described as follows:
 
where t fi is the tension between two adjacent stands (the i th and (i+1)th stands), K2 is the weighting constant of the tension cost function, and N is the total number of stands. When N=5, the tension tf5 means the tension between the fifth stand and the coiler of the mill.
3.1.3.    Perfect shape condition
Good rolling scheduling should lead to uniformity of the tension distribution in the strip width direction, or the minimal possibility of the strip buckling during rolling. The optimum shape condition is obtained when the strip is uniformly rolled across the strip width (ignoring the insignificant lateral spread in cold rolling). This means that the deformed roll profile should perfectly match the incoming transverse strip thickness profile geometrically. For example, the input profile of the incoming strip may be described as follows:
 
The reason to choose the model with x 2 and x4 in Eq. (18) is that both the strip to be rolled and the strip after rolling are usually symmetrical about the center-line. So neither x nor x3 is considered in the model. The parameters H c, P1, and Q1 are determined by the measured strip thickness profile, where H(x ) is the entry thickness of the strip at a location of a distance x from the strip centerline, H c is the entry thickness of the strip at the strip centerline, and P1 and Q1 are constants. The output profile is shown in Fig. 2 and can be described as follows:
 
where h(x ) is the exit thickness of the rolled strip at a location of a distance x away from the strip centerline, h c is the exit thickness of the strip at its centerline, and P2 and Q2 are constants.
It can be shown that the condition for a perfect shape requires:
 
where r is the thickness reduction.
Thus, if the profile of the rolls exactly matches the incoming transverse gauge profile of the strip geometrically, a perfect exit shape will be obtained (Sabatini and Yeomans, 1968).
So the cost function for perfect shape can be described as follows:
 
Fig. 2. The transverse profile of a rolled strip.
 
where K3 is the weighting constant, and Уw(x) is the deformed roll profile, which is a function of the roll force and the roll bending force if used. It is calculated by summing the deflections due to bending, shear and work roll flattening.Сw(x) is the total work roll crown, including the machined crown, the thermal crown and the roll wear. For mills with roll-bending jacks, the deformed roll profile could be geometrically adjusted by changing the jack force, which would be reflected in the term Уw(x): The deformed work roll profile is obtained by calculating the roll deflections due to bending, shear, the effect of Poisson's ratio, the bending moment, the interference between the work and backup rolls, and the interference between the work roll and the strip, which are described in the following sections.
3.1.3.1.    The deflection due to bending.
Beam theory for the bending and shear components has been widely employed to calculate the roll deflections. A typical roll deflection model under the effect of point loads is shown in Fig. 3. 冷连轧机轧制规程英文文献和翻译(4):http://www.youerw.com/fanyi/lunwen_1229.html