# 数学代写|最优化理论作业代写optimization theory代考|MATH683

## 数学代写|最优化理论作业代写optimization theory代考|Constructing a model of the market

The model examines the market with one product. $Q$ manufacturers and L consumers of the product are participants:
$q=\overline{1, Q}$ is an index $(q)$ and $\operatorname{set}(Q)$ of the manufacturers (firms).
$l=\overline{1, L}$ is an index and set of the consumers.
To construct a mathematical model of the market, for each manufacturer and consumer let’s create a vector variable and restrictions imposed on production operation and their purposes (criteria) of functioning.
The manufacturers (firms):
$X=\left{x_{q l}(t) \geq 0, q=\overline{1, Q}, l=\overline{1, L}\right}$ is a vector variable, determining the volume of production made by the $q \in Q$ firm and sold to the $l \in L$ consumer over a particular (unspecified) period of time $t \in T$;
$p_q$ is the cost per unit established by the $q \in Q$ firm in the market (the same for all consumers);
$p_q x_{q l}$ is the amount of money received by the $q \in Q$ firm from the $l \in L$ consumer;
$f_q(X)=\sum_{l=1}^L p_q x_{q l}(t)$ is the value characterizing the amount of money received by the $q \in Q$ firm from all consumers.
Resource (costs) characteristics:
$i=\overline{1, M_q}$ is an index and set of all resources used by the $q \in Q$ firm for
production; $i=\overline{1, M_{\text {mat }}}$ is an index and set of material resources used for production $M_{m a t} \in M_q$;
$a_{i q}, i=\overline{1, M_{\text {mat }}}$ is the cost of the $i \in M_{\text {mat }}$ resource per unit of output produced by the $q \in Q$ firm.

We assume there is a linear, functional dependence of costs which increase at output:
$$g_i(X(t))=\sum_{l=1}^L a_{i q} x_{q l}(t), i=\overline{1, M_{\text {mat }}}$$
where $g_i(X)$ is the cost of the $i \in M_{\text {mat }}$ resource for the whole volume of production.
The following calculations are carried out similarly:
The cost of labour per unit of production. $i=\overline{1, M_{l a b}}$ is an index and set of labour resources used for production $M_{\text {lab }} \in M_q$;

## 数学代写|最优化理论作业代写optimization theory代考|The solution to vector optimization problems underpins

Let’s examine the solution to VOP $(9.6 .6)-(9.6 .10)$, representing a mathematical model of the one-product market. Solving the problem (9.6.6)-(9.6.10) with equivalent criteria on the basis of the normalization of criteria and the principle of a guaranteed result, we will calculate:
an optimum point of $X^o=\left{x_{q l}(t), q=\overline{1, Q}, l=\overline{1, L}\right.$
which determines the output of a product sold by each manufacturer to each consumer;
values of criterion functions $f_k\left(X^o(t)\right), k=\overline{1, K}, K=Q \cup L$ including $f_q\left(X^o(t)\right), q=\overline{1, Q}$, which determine the income (benefits) of each manufacturer; $f_l\left(X^o(t)\right), l=\overline{1, L}$, which determines the expenditure of each buyer.
The maximal relative estimation $\lambda^o$, where $\lambda^0 \leq \lambda_k\left(X^o\right), k=\overline{1, K}, X^o \subset S$.
where $\lambda_k\left(X^o\right)=\frac{f_k\left(x^0\right)-f_k^0}{f_k^-f_k^0}, k=\overline{1, K}$ is normalized criterion (the relative estimate), in which $f_k^$ is the best solution against criterion $f_k^0, k \in K$ which is the worst one accordingly, and $K=Q \cup L$ is the set of criteria.
The maximal relative estimation $\lambda^{\circ}$ can be interpreted as a maximum level of the mutual interests of all manufacturers and consumers at relative units. Any increase in the interests (criterion) of any manufacturer or consumer changes things for the worse for all the rest of the market’s participants. The received point is optimum against Pareto.

A vector function $F_1(X(t))=\left{f_q\left(X^o(t)\right), q=\overline{1, Q}\right}$ is a function of supply (offer); a vector function $F_2(X(t))=\left{f_l\left(X^o(t)\right), l=\overline{1, L}\right}$ is a function of demand.

Let’s look into the methodology for solving a vector problem (9.6.6)(9.6.10), simulating the one-product market in the form of test examples.

# 最优化理论代考

## 数学代写|最优化理论作业代写optimization theory代考|Constructing a model of the market

$q=\overline{1, Q}$ 是一个索引 $(q)$ 和 $\operatorname{set}(Q)$ 的制造商（公司)。
$l=\overline{1, L}$ 是消费者的索引和集合。

$X=\backslash$ left ${X$ {q $\mid}(t) \backslash$ geq $0, q=\backslash$ overline ${1, Q}, I=\backslash$ overline ${1, L} \backslash$ right $}$ 是一个向量变量，决定了生产商的生产量 $q \in Q^{\prime}$ 公司并出售给 $l \in L$ 消费者在特定 (末指定) 时间段内 $t \in T$ ； $p_q$ 是由制定的每单位成本 $q \in Q$ 在市场上坚定（所有消费者都一样）； $p_q x{q l}$ 是收到的金额 $q \in Q$ 公司从 $l \in L$ 消费者;
$f_q(X)=\sum_{l=1}^L p_q x_{q l}(t)$ 是表征收到的货币数量的值 $q \in Q$ 来自所有消费者的坚定。

$i=\overline{1, M_q}$ 是所有资源使用的索引和集合 $q \in Q$ 生产公司
$; i=\overline{1, M_{\text {mat }}}$ 是用于生产的材料资源的索引和集合 $M_{\text {mat }} \in M_q$;
$a_{i q}, i=\overline{1, M_{\text {mat }}}$ 是成本 $i \in M_{\text {mat }}$ 单位产出的资源 $q \in Q$ 公司。

$$g_i(X(t))=\sum_{l=1}^L a_{i q} x_{q l}(t), i=\overline{1, M_{\text {mat }}}$$

## 数学代写|最优化理论作业代写optimization theory代考|The solution to vector optimization problems underpins

$\$ X^{\wedge} 0=\backslash$left$\left{X_{-}{q \mid}(t), q=\backslash\right.$overline${1, Q}$, I = \overline${1, L} \backslash$right$。$whichdeterminestheoutputofaproductsoldbyeachmanufacturertoeachconsumer; valuesofcri , whichdeterminetheincome(benefits)ofeachmanufacturer;$\mathrm{f}{-}$\$\backslash$left$\left(\mathrm{X}^{\wedge} \mathrm{O}(\mathrm{t}) \backslash\right.$right),$\mathrm{I}=\backslash$\overline${1, \mathrm{~L}}$, whichdeterminestheexpenditureofeachbuyer. Themaximalrelativeestimation$\lambda^{\wedge} \mathrm{O}$isnormalizedcriterion(therelativeestimate), inwhich$\mathrm{f}{-} \mathrm{k}^{\wedge}$isthebestsolutionagainstcriterion$\mathrm{f}^{\wedge} \mathrm{k}^{\wedge} 0, \mathrm{k} \backslash$in$\mathrm{K}$whichistheworstoneaccordingly, and$\mathrm{K}=\mathrm{Q} \backslash \mathrm{}$Listhesetofcriteria. Themaximalrelativeestimation$\backslash \mathrm{lambda}{ }^{\wedge}{\backslash \mathrm{irc}} \ 可以解释为所有制造商和

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