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Abstract

This article is literatures review about five articles, which apply linear programming to Finance, accounting and economics. The mathematical method is found of crucial importance in those fields. The paper shows how theoretical inference in linear programming throws light upon realistic practice, and how empirical evidence supports those theories.

Keywords: finance; accounting; economics; linear programming; investment analysis

Linear Programming in Finance

“Application of Linear Programming to Financial Budgeting and the Costing of Funds” explored how to allocate funds in an enterprise by applying linear programming. As Charnes, Cooper and Miller analyzed, at least three problems are to be considered to solve the allocation problem: 1) Plans for production, purchases, and sales under certain structure of the firm’s assets, in order to maximize its profit or reach other objectives. 2) The change of the firm’s profit per unit change in the structure of the assets. 3) Opportunity cost of the firm’s funds. The article starts with a simple example with one commodity and one warehouse. Let B be the fixed warehouse capacity, A be the initial stock of inventory in the warehouse, xj be the amount to be sold in period j, yj be the amount to be sold in period j, pj be the sales price per unit in period j, and cj be the purchase price per unit in period j, then we have

due to the cumulative sales constraint;

due to the warehouse capacity constraint;

due to the buying constraint;

due to the selling constraint; and

with our goal of maximizing

The dual problem is also obvious. It is to minimize

subject to

and to

where

As we learned, “dual theorem of linear programming” says that the two optimal values of the original problem and the dual problem should be equal. Using this theorem, the authors then reached a new method of evaluating assets. Because

,

we have

in which the two sides must have the same units of measure. So it is now obvious that t*k represents the value per unit of net warehouse capacity and u*k represents the value per unit of initial inventory in the warehouse. Similarly, consider the financial problem, which has liquidity constraints as

where j-εrepresents payments and j-r represents receipts, M0 is the initial cash available and M is the balance the firm desired the maintain. By examining the dual problem of this, we can find corresponding dual variables for the problem called, say, vk. Again, from the equality we found before, we can learn that the two sides of the equation have the same units of measure. It is then seen that the v’s should be dollars per unit time per dollar invested.

The valuation of assets or investments is of crucial importance to any business. So far, by simply applying the dual theorem, Charnes, Cooper and Miller have created a new method of evaluating assets or investments. This method of evaluating is also easy to find out answers. It is intelligent to examine the units of measure rather than try to solve the specific problems. The interesting thing is that in realistic problems, we can find true meanings of theoretical dual variables.

Then the authors mixed the two former problems together to see a more realistic case – a warehouse problem with financial constraints. So the following new constraints are added:

Now if we define

We’ll get the new dual problem:

Here, V1 is the incremental cumulative internal yield rate. Or it is the opportunity cost the capital invested – “it shows the net amount to which an additional dollar invested in the firm will accumulate if left to mature to the end of the planning horizon.” This is also easy to understand in terms of economics, maximizing profit can be the same as minimizing the opportunity costs.

The article then went through several practical problems using the dual variable evaluating method....