=Paper=
{{Paper
|id=Vol-2914/Paper36.pdf
|storemode=property
|title=On Solving Unconditional Optimization Problems in the Wolfram Mathematica System
|pdfUrl=https://ceur-ws.org/Vol-2914/paper36.pdf
|volume=Vol-2914
|authors=Vladimir R. Kristalinskii
}}
==On Solving Unconditional Optimization Problems in the Wolfram Mathematica System==
https://ceur-ws.org/Vol-2914/paper36.pdf
383
On Solving Unconditional Optimization Problems in the
Wolfram Mathematica System*
Vladimir R. Kristalinskii 1[0000-0003-0644-1731]
1 Smolensk State University, Smolensk, Russia
kristvr@rambler.ru
Abstract. The article is devoted to the method of solving problems of uncondi-
tional optimization in the Wolfram Mathematica system, which can be used when
conducting classes in theoretical and applied disciplines in technical universities.
Algorithms for solving mathematical optimization problems are very complex.
The use of electronic computing technology makes it possible to implement these
algorithms but often imposes very high requirements on the competence of the
person who wants to implement these algorithms. The systems of computer math-
ematics that have appeared in recent decades allow us to solve this problem. They
have an interface that allows a specialist in any application field to solve these
problems without resorting to programming in the classical sense of the word.
One such system is the Wolfram Mathematica system.
Keywords: Wolfram Mathematica System, Problems of Unconditional Optimi-
zation.
1 Introduction.
The simplest example of an optimization problem is the unconditional optimization
problem. In this case, we look for the largest or smallest value of the function and the
point at which this value is reached. The need to solve such problems often arises when
solving a variety of technical tasks [2,3]. Interest in algorithms for solving optimization
problems continues [4-10]. Computer mathematics systems in this case, as in other
cases, allow you to focus on the mathematical model, and its implementation is carried
out through the system.
The purpose of this article is to study the capabilities of the Wolfram Mathematica
system for solving unconditional optimization problems.
The solution of such tasks can be carried out by the methods of differential calculus.
The Wolfram Mathematica system allows you to solve such problems, including when
the parameters are set as letter coefficients.
* Copyright 2021 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
384
1 Solving Inventory Management Problems in the Mathematica
System.
Consider the problem of managing a single-product stock [2]. Even if there is no short-
age, the stock is replenished at a time. Total expenses per unit of time consist of specific
operating and non-production expenses. For simplicity, assume that operating costs are
by definition proportional to the level of average inventory
hx
zexp l
2 ,
where h is the operating costs per unit of time associated with storing a unit of in-
ventory, and non – production costs are inversely proportional to the length of the order
cycle
kd
zn
x ,
where k is the total non – production costs. Then the specific total expenses are equal
hx kd
zA
2 x
where d is the intensity of demand.
The optimal stock level is found from the condition
z f x min
Finding the derivative of the function zA(x) using the Mathematica system
u[x]=(h*x)/2+(k*d)/x
u1[x]=D[u[x],x]
Receive result
h/2–(d k)/x^2
Equate the derivative to zero and solve the resulting equation with respect to x
Solve[u1[x]==0,x]
{{x–(( 2 d k )/ h )},{ x(( 2 d k )/ h )}}
Choosing a positive one from the two obtained roots, we get
2kd
x
h
385
Now let's assume that the stock is replenished by output with an intensity of produc-
tion per unit of time r that exceeds the intensity of demand d(r>d). then the specific
total costs are expressed by the formula.
d
hx 1
zB
r kd
.
2 x
Find the derivative of the function zB(x) using the Mathematica system.
u2[x]=(h*x*(1–d/r))/2+(k*d)/x
u3[x]=D[u2[x],x]
Receive result
1/2 h (1–d/r) – (d k)/x^2
Equate the derivative to zero and solve the resulting equation with respect to x
Solve[u3[x]==0,x]
{{x–(( 2 d k r )/ dh hr ])},{ x (( 2 d k r )/ dh hr ])}
Choosing a positive one from the two obtained roots, we get
2kdr
x .
h r d
Let's assume that the required inventory level is no longer available and the inven-
tory is replenished at the same time. In this case, the total costs additionally include the
costs zd due to the lack of the required inventory level, and the unit total costs are de-
termined by the expression
h x v bv 2
2
kd
zB
2x x
where v is the value of the deficit, and b is the specific losses from the deficit per
unit of time.
We find partial derivatives of function z(x,v) with respect to x and v using the Wolf-
ram Mathematica system.
gg[x,v]=(h*(x–v)^2+b*v^2)/(2*x)+(k*d)/x
gg1[x,v]=D[gg[x,v],x]
gg2[x,v]=D[gg[x,v],v]
386
We equate the obtained derivatives to zero and solve the resulting system of equa-
tions for x and v
Solve[{gg1[x,v]==0,gg2[x,v]==0},{x,v}]
Let's simplify the result and get the answer.
FullSimplify[%]
{{x–((Sqrt[2] Sqrt[d] Sqrt[b (b+h)] Sqrt[k])/(b Sqrt[h])),v–((Sqrt[2] Sqrt[d]
Sqrt[h] Sqrt[k])/Sqrt[b (b+h)])},{x (Sqrt[2] Sqrt[d] Sqrt[b (b+h)] Sqrt[k])/(b
Sqrt[h]),v (Sqrt[2] Sqrt[d] Sqrt[h] Sqrt[k])/Sqrt[b (b+h)]}}
Choosing from the two obtained values x and v positive, we get
2dkb b h 2kdh
x ,v .
b h b b h
Now let's consider the case of a deficit with a uniform replenishment of the stock.
In this case, the unit total expenses are defined by the expression
2
d
h x 1 v bv 2
r
zГ
kd
d x
2 x 1
r .
We find partial derivatives of function z(x,v) for x and v using the Wolfram Mathe-
matica system.
f[x,v]=(h*(x*(1–d/r)–v)^2+b*v^2)/(2*x*(1–d/r))+k*d/x
ff1[x,v]=D[f[x,v],x]
ff2[x,v]=D[f[x,v],v]
We equate the obtained derivatives to zero and solve the resulting system of equa-
tions for x and v
Solve[{ff1[x,v]==0,ff2[x,v]==0},{x,v}].
Let's simplify the result and get the answer.
FullSimplify[%]
{{x–((I Sqrt[2] Sqrt[d] Sqrt[b+h] Sqrt[k] Sqrt[r])/Sqrt[b h (d–r)]),v (I Sqrt[2]
Sqrt[d] Sqrt[k] Sqrt[b h (d–r)])/(b Sqrt[b+h] Sqrt[r])},{x (I Sqrt[2] Sqrt[d]
Sqrt[b+h] Sqrt[k] Sqrt[r])/Sqrt[b h (d–r)],v–((I Sqrt[2] Sqrt[d] Sqrt[k] Sqrt[b h
(d–r)])/(b Sqrt[b+h] Sqrt[r]))}}
387
The resulting result may be somewhat confusing, since the resulting expressions
contain an imaginary unit (I). However, remembering that by condition r>d , hence
d r i r d .
Multiplying by i, we get
2dkr b h 2dkh r d
x , v
bh r d r b h
These examples illustrate the ability of the Wolfram Mathematica system to work
with expressions in a General way. The latter case also shows that the results obtained
with its help may require additional study and analysis.
2 New Tools for Solving Optimization Problems in the Latest
Versions of Mathematica.
The Mathematica system allows you to solve such problems directly using built-in
methods. The main methods for solving such problems using Wolfram Mathematica
are widely known and discussed, for example, in [1]. However, the system is evolving
and new features are emerging. For example, in version 11 of the system, the Bayesi-
anMinimization command was added. This command makes it possible to solve several
new minimization problems along with well-known ones.
You can use this command to solve the usual problem of minimizing a function on
an interval.
bo=BayesianMinimization[(#–2)^2+ 1&,Interval[{0,7}]]
In this example, we minimize the function
f x x 2 1
2
on the interval [0;7]. Receive result
BayesianMinimizationObject[Minimum configuration: 2.03
Minimum value: 1.]
Let's present the results of calculations in a more convenient form
bo["MinimumConfiguration"]
2.03379
bo["MinimumValue"]
1.01162
Let's build graphs of the simulated and modeling functions (see Fig. 1).
388
25
20
15
10
5
function
1 2 3 4 5 6 7 model
Fig. 1. The schedule is built with Mathematica.
Here we see that it is better to use the usual Minimize function to solve this problem.
However, the BayesianMinimization team has features that previously existing Teams
did not have. We can look at the set of values that were used to calculate the minimum
bo["EvaluationHistory"]
Configuration Value
6.30544 19.5368
3.91381 4.66267
3.13817 2.29542
1.07336 1.85866
2.16175 1.02616
1.86308 1.01875
1.88728 1.01271
0.0977517 4.61855
6.82761 24.3058
3.04529 2.09263
2.05689 1.00324
4.27194 6.16173
4.80517 8.86897
3.09127 2.19086
0.924357 2.15701
We can also perform function minimization on a discrete set.
bo=BayesianMinimization[(#^2-5*#+2)&,{5.7,1.1,3.4,6.8,2.3,1.2}]
bo["MinimumConfiguration"]
2.3
bo["MinimumValue"]
–4.21
We can minimize the function on a set constructed using a random number generator
bo1=BayesianMinimization[ (#^2-5*#+4)&,RandomReal[{0,10}]&]
bo1["MinimumConfiguration"]
389
2.54493
bo1["MinimumValue"]
–2.24796
All these new features allow us to expand the range of problems that can be consid-
ered with students when studying problems whose mathematical models are reduced to
problems of unconditional optimization.
4 Conclusion.
Along with the examples discussed in this article, the system allows you to solve a wide
range of different optimization problems: linear programming problems (in particular,
the transport problem), the assignment problem, and dynamic programming problems.
Methods for solving multi-criteria optimization problems using the Mathematica sys-
tem are developed.
The Mathematica system also allows you to solve a large number of problems related
to the most diverse areas of mathematics and applied disciplines. In the latest versions
of the system, it includes some artificial intelligence algorithms (in particular, image
recognition). All this makes it an indispensable tool for teaching students of mathemat-
ical and engineering specialties, as well as for conducting scientific research.
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