4.1. The ECM Algorithm
In this subsection, we develop a new solution algorithm to solve the energy consumption minimum problem (P4), which is formulated in
Section 3.2. Before that, we need to analyze the optimal equations of (P4), such as [
33]. First, constraint (
20) is one of the optimal equations, so it is necessary to explore the relationship between
and
. The previous section noted that
is monotonically increasing with respect to
, and as shown in
Figure 2, for a given
,
is a monotonically decreasing function of
.
Then, we analyze the feasible region of
and
. Since
is decreasing with respect to
, when
takes the maximum value of
takes the minimum value,
. Similarly, when
takes the maximum value of
, the corresponding
takes the minimum value,
. Therefore, the feasible regions of
and
can be obtained as below:
and
can be uniquely solved by constraint (
20) due to the decreasing relationship between
and
.
Next, another optimal equation is derived below. We define an auxiliary function
as follows
where the specific form of
and
can be found in [
36]. According to Numerical Facts 4.1 and 4.2 in [
33], it can be known that
is continuously differentiable on
, and it is monotonically increasing with respect to
, with its value in
, which is also obtained in
Figure 3.
Meanwhile, another auxiliary function
has the following formula:
It is continuously differentiable on and monotonically increasing with respect to , with its value in .
It should be noted that the energy consumption parameters and machine efficiencies directly affect the objective function in problem (P4); thus, we analyze the property of (P4) under different parameters with respect to and . Since is an implicit function of , i.e., , then is a function of only. Similarly, in function , is an implicit function of , which can be expressed as . For further analysis, the following notations are defined.
- (i)
.
- (ii)
- (iii)
.
Then, the relationship between and on and and on are obtained, respectively. For example, when , and when . Similarly, when , we have , and when . Therefore, when or , or is always established; however, when , there may be two cases of and . When or , we always have or , when ; both and are possible.
Clearly, the optimality equation requires the partial derivative of the objective function
E with respect to
, i.e.,
Thus, the following Lemma is another optimality equation. In the following analysis, let and .
Lemma 1. The equality is the sufficient and necessary condition for , where and .
Proof. For
, the partial derivative of E with respect to e1 in (28) can be abbreviated as
If , a similar result can be obtained in the same way. □
It is necessary to explore the properties of . By Lemma 1, we get that if . We have mentioned the properties of the function above, then mainly analyzed the monotonicity of , that is, the properties of and . For further analysis, we denote the corresponding value when is . We discuss the relationship between and ; for instance, if , the only case that exists would be . If , then only exists. However, both and may exist if . The objective function E of problem (P4) can be expressed as since is an implicit function of . To sum up, the problem (P4) can be analyzed in three cases, i.e., and . For the first case , the properties of the problem (P4) are as follows
Observation 1. The monotonicity and local optimum of are listed when and ,
- (i)
When ;
If and hold simultaneously, has one local minimum (see Figure 4a). If and hold simultaneously, has one local maximum (see Figure 4b). If and hold simultaneously, is monotonically decreasing (see Figure 4c). If and hold simultaneously, is monotonically increasing (see Figure 4d).
- (ii)
When , stands for the left limit of ;
If , has one local minimum and one local maximum, and the minimum point is to the left of the maximum (see Figure 5a). If , is monotonically decreasing (see Figure 5b).
- (iii)
When , is monotonically decreasing (see Figure 6).
Figure 4.
The energy consumption with respect to when . (a) One local minimum. (b) One local maximum. (c) Monotonically decreasing. (d) Monotonically increasing.
Figure 4.
The energy consumption with respect to when . (a) One local minimum. (b) One local maximum. (c) Monotonically decreasing. (d) Monotonically increasing.
Figure 5.
The energy consumption with respect to when . (a) One local minimum and maximum. (b) Monotonically decreasing.
Figure 5.
The energy consumption with respect to when . (a) One local minimum and maximum. (b) Monotonically decreasing.
Figure 6.
The energy consumption with respect to when .
Figure 6.
The energy consumption with respect to when .
As for the case , the properties of the problem (P4) are shown in the following observation,
Observation 2. The monotonicity and local optimum of are listed as follows when and :
- (i)
when ;
If such that , has one local maximum and one local minimum, and the local maximum is to the left of the minimum (see Figure 7a). If for , we have , then is monotonically increasing (see Figure 7b).
- (ii)
When , and stand for the left and right limits of ;
If , , and hold, where , has two local maximums and one local minimum, and the local minimum is in between the two maximums (see Figure 8a); If , , and hold, where , has one local maximum (see Figure 8b).
- (iii)
When , has one local maximum (see Figure 9).
Figure 7.
The energy consumption with respect to when . (a) One local minimum and maximum. (b) Monotonically increasing.
Figure 7.
The energy consumption with respect to when . (a) One local minimum and maximum. (b) Monotonically increasing.
Figure 8.
The energy consumption with respect to when . (a) One local minimum and two local maximums. (b) One local maximum.
Figure 8.
The energy consumption with respect to when . (a) One local minimum and two local maximums. (b) One local maximum.
Figure 9.
The energy consumption with respect to when .
Figure 9.
The energy consumption with respect to when .
For the case , the properties of the problem (P4) are shown in Observation 3.
Observation 3. The monotonicity and local optimum of are listed as below when and :
- (i)
When , is monotonically increasing (see Figure 10). - (ii)
When ;
If and hold simultaneously, has one local maximum (see Figure 11a). If and hold simultaneously, is monotonically increasing (see Figure 11b).
- (iii)
When , has one local maximum (see Figure 12).
Figure 10.
The energy consumption with respect to when .
Figure 10.
The energy consumption with respect to when .
Figure 11.
The energy consumption with respect to when . (a) One local maximum. (b) Monotonically increasing.
Figure 11.
The energy consumption with respect to when . (a) One local maximum. (b) Monotonically increasing.
Figure 12.
The energy consumption with respect to when .
Figure 12.
The energy consumption with respect to when .
Similarly, the energy consumption of problem (P4) is analyzed with respect to in the same way as with respect to and is not repeated here.
Based on the conclusion of the analysis of problem (P4) from the above three observations, we propose a new ECM algorithm, the pseudo-code of which is shown in Algorithm 1.
Algorithm 1: Energy consumption minimization (ECM) algorithm. |
Input: - 1:
calculate the feasible regions of and by ( 20), i.e., and ); - 2:
if
then - 3:
find the optimal value from the candidate solution set . - 4:
obtain the optimal system configuration according to ( 19). - 5:
else - 6:
if then - 7:
find the optimal value from the candidate solution set . - 8:
obtain the optimal system configuration according to ( 19). - 9:
else - 10:
obtain the non-boundary candidate solution with ( 20), ( 23), ( 25) - 11:
find the optimal value from the candidate solution set . - 12:
obtain the optimal system configuration according to ( 19). - 13:
end if - 14:
end if
Output: The optimal solution , the candidate solution and the total energy consumption E.
|
4.2. The ES Algorithm
In
Section 4.1, we consider the energy consumption minimization problem when
and
are decision variables and the buffer capacity
N is a fixed value. In this subsection, we fix
as e to solve what value of
N should be taken to minimize energy consumption. In other words, problem (P4) is extended to
N and
as decision variables, i.e.,
It is worth noting that the objective function
E has only one variable,
, and constraint (27) reflects the relationship between the two variables. In order to solve this problem, we first explore the constraint (27). From the analysis of the monotonicity properties of the
Q-function in [
35], it can be seen that
Q is strictly decreasing in
N. According to (27),
increases monotonically with
Q. Thus,
N is monotonically decreasing in
; see
Figure 13.
Next, the derivative of the objective function
E with respect to
is
Clearly, is a monotonically decreasing linear function in , and when , the value is greater than 0; otherwise, the value is less than 0.
Then, we find that the objective function E increases first and then decreases when , E has a maximum value, which means when or , E has a minimum value. A new algorithm is designed to solve problem (P5) with the pseudo-code shown in Algorithm 2.
Summarizing the above theoretical analysis, the framework of the present research in dealing with the energy-saving two-machine geometric serial machine line is shown in
Figure 14.
Figure 13.
The behavior of N with respect to . (a) . (b) .
Figure 13.
The behavior of N with respect to . (a) . (b) .
Algorithm 2: Energy-saving (ES) algorithm. |
Input: - 1:
ifthen - 2:
obtain the optimal buffer capacity . - 3:
else - 4:
find the optimal value from the candidate solution set . - 5:
obtain the optimal buffer capacity according to (18). - 6:
end if
Output: The optimal solution , the candidate solution and the total energy consumption E. |