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# 数学代写|运筹学代写Operations Research代考|MATH4202 Allocation Problems

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## 数学代写|运筹学代写Operations Research代考|Allocation Problems

Allocation problems are one of the most prominent areas of application in linear programming. All models in this class have in common that they deal with the allocation of scarce resources to (economic) activities. At times, more than one scarce resource exists, in which case the modeler can choose any one of them as the basis for the mathematical formulation. This will be further elaborated upon below. Due to the many different types of allocation problems, we will present two such scenarios below.

First consider an investment allocation problem. Problems of this nature were first formulated by Markowitz in the early 1950s as nonlinear optimization problems. Here, we will discuss a linear version of the problem. In this problem, the decision maker has to decide how much of the scarce resource (money) to allocate to different types of investments. As we will see below, this observation already indicates how to define the variables.

In our numerical example, the investor has $\$ 300,000$that can be invested. In addition to the money at hand, it is possible to borrow up to$\$100,000$ at $12 \%$ interest. This money can be used for leveraging (borrowing to invest). The investor has narrowed down the choices to six alternatives, shown in Table 2.6. The table also shows the expected annual interest or dividend for the investment alternatives, the expected annual increase of the value of the investment, and an indication of the risk of the investment (per dollar).

## 数学代写|运筹学代写Operations Research代考|Blending Problems

All blending problems have in common that they take a number of given ingredients or raw materials and blend them in certain proportions to the final products. In other words, in the process we are creating something new by mixing existing materials. Typical examples of blending are coffees, teas, whiskeys, tobaccos, perfumes, gasoline, and similar products. Clearly, there are some rules for the blending process. For instance, in order to ensure a specific taste, it may be required that a blend includes at least a certain proportion of a raw material.

The process is shown in Fig. 2.1. On the left, there are $m$ buckets with given quantities of raw materials, while on the right, there are $n$ empty buckets that are to be filled with known quantities of the blends. In the blending process, we take, one at a time, a certain number of scoops from each of the raw materials and transfer them into the buckets on the right. Once sufficient quantities have been transferred to the “Product” buckets on the right, all that is left to do is stir, package, and sell.

This figure not only demonstrates the actual process, but it also allows us to see how the variables in blending problems are to be defined. What we need to know in any specific blending problem is how many “scoops” of each raw material goes into each of the “Product” buckets. In more general terms, we define $x_{i j}$ as the quantity of raw material $i$ that goes into product $j$.

## MATLAB代写

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