Page 23 - IJOCTA-15-2

A. R´acz / IJOCTA, Vol.15, No.2, pp.215-224 (2025)
Since the minimization objective function 5 • Minimization of involved bars.
includes the variables WL j and these variables
are limited only by the mentioned constraint 6, Decision maker is able to weight these goals
they will be assigned to the smallest value allowed by setting the appropriate value of the correspod-
by the constracted lower bound. ing parameters CC and CW.
Beside the previously defined constraints (2)
Remarks:
- (4)
If LOl j < W, then in (7) W − LOl j > 0,
thus 0 < M · (1 − Y R j ). Since Y Rj is a bi- m
X
nary variable, the previous inequality force it to x ij r i + LOl j = l j j = 1, 2, . . . , n
be 0. If Y R j = 0, constraints (6) looks like i=1
LOl j − WL j ≤ 0, that is LOl j ≤ WL j . LOlj
Y L j ≥ j = 1, 2, . . . , n
If LOl j ≥ W, then (6) does not limit the value l j
of Y R j , so there is no lower bound for WL j , ex- n
X
cept the sign restriction 0 <= WL j . x ij = 1 i = 1, 2, . . . , m
Because of the minimization in (5) and the j=1
fact that WL j is not contained by any other con- and (6) - (7)
straint, so it does not affect the value of any other
variables, optimal value of it is the smallest pos-
LOl j − WL j ≤ M · Y R j j = 1, 2, . . . , n
sible value which is LOlj if the leftover is under
the waste limit and 0 if it is reusable. W − LOl j ≤ M · (1 − Y R j ) j = 1, 2, . . . , n.

2.4. Extending the model with the We need to introduce the following new con-
minimization of affected bars straint:

It may also be reasonable to consider how many
m
P
bars are involved in serving the orders. Thus, in Y U j ≥ i=1 x ij j = 1, 2, . . . , n. (9)
the final version, we have extended the model so M 2
that its objective contains the minimization of the
bars used during the process too. where M 2 is a constant that is large enough
P n
to scale x ij into the interval [0, 1]. M 2 = m
A new binary variable is therefore introduced i=1
could be a suitable choice. Thus, constraint (9)
that shows if the correspoding bar is used or not.
sets a strictly positive lower bound for the binary
This variable is labeled by Y U j (see Table 1) and
variable Y U j , if there is an order assigned to it.

1 if B j is used,
Y U j = 3. Implementation and limitations
0, else.
In order to investigate the limitations of our
Let us extend the objective function according model, we implemented it in Python using Py-
to the previous formula. omo universal framework 14 and Gurobi solver. 15
Our goal was to describe the connection be-
n m
X X tween the size of the model and the required run-
min CC · (x ij ) − 1 + Y L j
time. Thus, we generated inputs for simulation
j=1 i=1
in three category of stock number of bars in stock
n
X {B 1 , . . . , B n }:
+ CW · WL j (8)
• Low stock, when n ≤ 80. (Symbol: ◦ )
j=1
n • Moderate stock, when 80 < n ≤ 100.
X
+ Y U j (Symbol: □ )
j=1 • High stock, when 100 < n. (Symbol: ▽ )
Figure 1. shows the relation between the num-
The above function is essentially a multiob-
ber of orders (horizontzal axis) and the runtime in
jective linear function containing the linear com-
seconds (vertical axis) for random generated data
bination of three objectives:
grouped by stock category. Figure shows that, de-
• Minimization of the cost of cuts. pending on stock category, sooner or later there
• Minimization of loss from waste. is an exponential increase in runtime.
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