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# 数学代写|运筹学代写Operations Research代考|KMA255 An Allocation Problem

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## 数学代写|运筹学代写Operations Research代考|An Allocation Problem

Before we discuss the allocation problem, we first look at shortest-path problems from a different angle. The shortest-path problems we have dealt with so far can be seen as typical examples of decision problems with a series of decisions. In sequential decision problems, a variable (phase variable) is needed to indicate how many decisions have been taken so far or how many still need to be taken. The phase variable is often a time variable and increases (or decreases) by one after every decision. Apart from the concept of phases, other essential features of sequential decision problems are the concepts of state and state transition. State variables are needed to describe the current situation. It is not always clear how to specify state variables; this sometimes requires some ingenuity. In order to find a suitable description of the state in a specific application, it is advisable to put oneself in the decision maker’s shoes and ask what information is needed to make a decision. Once the phase variables, state variables, and effects of the decisions on the states have been identified, the value function can be defined, and a recursive relation can be established.

Dynamic programming can sometimes also be applied to static decision problems that do not require a series of decisions. As an illustration, consider the following allocation problem:
$$\begin{array}{ll} \text { Maximize } & \sum_{k=1}^n r_k\left(x_k\right) \ \text { subject to } & \sum_{k=1}^n b_k x_k \leq B \ \text { and } \quad & 0 \leq x_k \leq m_k, \quad x_k \text { integer for } k=1, \ldots, n, \end{array}$$
where the $b_k$, the $m_k$, and $B$ are given positive numbers. This optimization problem occurs when a budget of $B$ euros is available for investing in $n$ projects. Allocating $x_k$ euros to project $k$ leads to a revenue of $r_k\left(x_k\right)$ and reduces the budget by $b_k x_k$ euros. No more than $m_k$ euros may be allocated to project $k$. The problem is finding an allocation that maximizes the total gain.

## 数学代写|运筹学代写Operations Research代考|The General Shortest-Path Problem

The basic principle of dynamic programming can also be used to develop an efficient algorithm for calculating the shortest paths in networks in general form. Consider a directed network with $N$ nodes. We use the symbols $x$ and $y$ to indicate nodes. For each existing arc $(x, y)$ in the network, we set
$$c(x, y)=\text { cost for passing through arc }(x, y) .$$
Suppose that the objective is to calculate minimum-cost paths from a given node $s$ to every other node $x$. If all costs $c(x, y)$ are nonnegative, we can use Dijkstra’s algorithm, discussed in Chapter 3. If some costs $c(x, y)$ are positive and some are negative, then the problem is complicated by the fact that a cycle (that is, a path from a node to itself) can exist with negative total cost. In this case, a path with cost $-\infty$ can be constructed by passing through the cycle an infinite number of times. How can we design an algorithm that uncovers the existence of a negative cycle and gives the optimal paths if such a cycle does not exist? The following observations give the key to the algorithm:

1. An optimal path from node $s$ to any other node $x$ exists only if there is no cycle with negative cost.
2. If an optimal path from node $s$ to node $x$ exists, then there exists an optimal path from $s$ to $x$ that consists of at most $N-1 \operatorname{arcs}$, where $N$ is the number of nodes.

## 数学代写|运筹学代写Operations Research代考|An Allocation Problem

Maximize $\quad \sum_{k=1}^n r_k\left(x_k\right)$ subject to $\quad \sum_{k=1}^n b_k x_k \leq B$ and $\quad 0 \leq x_k \leq m_k, \quad x_k$ integer for $k=1, \ldots, n$,

## 数学代写|运筹学代写Operations Research代考|The General Shortest-Path Problem

$$c(x, y)=\text { cost for passing through arc }(x, y) .$$

1. 来自节点的最佳路径 $s$ 到任何其他节点 $x$ 仅当不存在具有负成本的循环时才存在。
2. 如果从节点的最佳路径 $s$ 到节点 $x$ 存在，则存在最优路径 $s$ 到 $x$ 最多包括 $N-1 \operatorname{arcs}$ ，在哪里 $N$ 是节点 数。

## MATLAB代写

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