# Research paper on linear programming

As we have in the past, we will pose a problem and follow it through the steps of the model: We have two types of boxed sets; the Aristorocks and the Millionaire sets. We have two people working in the boxing section and one person working in the shipping department. Hire Writer There is a maximum of 20 hours of machine time available per time period. But let us check this out.

## Magic Ink: Information Software and the Graphical Interface

It may seem surprising that the product with the lower contribution margin is the more profitable to produce. What is important, however, is not the contribution margin per unit of product, but the contribution margin per unit of the limiting resource.

Contribution margin per unit of limiting resource Thus, with one limiting resource, we should produce only one product, the one with the highest contribution margin per unit of that limiting resource.

In example we should maximise the production of Product 1 which has the higher contribution margin per machine Research paper on linear programming. It would not pay to produce a mix of the two, that is, some of each product.

Two Limiting Resources production, because it may not be possible to obtain the same profitability rankings on different scarce resources. Example Suppose that, in addition to the machine constraint in Exampleboth products have to proceed through a labour-intensive finishing process.

Both products require hour of finishing labour, but there is a maximum of hours of skilled finishing labour available per time period. Having learned our lesson about calculating contribution margin per unit of scarce resource, we now proceed to calculate the contribution margin per hour of finishing labour for each product, as shown in Table Thus Product 2 is more profitable in terms of contribution per hour of finishing abour.

We are now faced with a problem of conflicting profitability rankings. There are two scarce resources, machine time and finishing time. Product 1 is more profitable in terms of machine resources consumed but Product 2 is more profitable in terms of finishing resources.

Since these profitability rankings conflict, we cannot determine the optimum action using this type of analysis. We have to use a technique known as linear programming. In general, when there are multiple constraining resources there are usually conflicting profitability rankings, and linear programming must be used to determine the optimum product mix.

Linear Programming achieve some objective such as profit maximisation or cost minimisation. It is assumed that there are constraining resources such as limited production or distribution facilities or market factors demand etc.

The word linear indicates that linearity is an assumption behind this technique. It is assumed that cost and revenue functions are linear, and that there is a linear relationship between inputs and outputs. If the output of a product is doubled, resource requirements are doubled. Care is needed when using this analysis that linearity is a reasonable assumption, and that the data are reasonably accurate.

A convenient way of introducing LP is by means of the graphical method, but this restricts us to somewhat trivial examples involving two variables. For problems with more than two variables we need to use the simplex method. There are many specialised LP computer programs employing the simplex method.

Also, these problems can be solved using most popular spreadsheets. Spreadsheet solutions will lso be illustrated. Maximisation Problems Example This example uses the data from Examplewhere we have two products and two constraints machine hours and finishing hours.

A linear programming model consists of two parts, an objective function and a set of constraints. First, we formulate the objective function. Let xl represent the number of units of Product 1 to be produced, and x2 the number f units of Product 2.

We can, however, simplify the analysis by ignoring fixed costs at this stage, and simply maximise variable profit total contribution margin: Now we turn to the constraint set. First, there is a machine constraint: Product 1 there is a maximum of 20 machine hours available per period.

This expression states that the machine time is to be fully used i. Graphically, Equation would appear as shown in Figure Linear Programming: As we have in the past, we will pose a problem and follow it through the steps of the model: We have two types of boxed sets; the Aristorocks and the Millionaire sets/5(1). A Linear Programming Approach to Max-sum Problem: A Review Toma´ˇs Werner Dept. of Cybernetics, Czech Technical University paper  formulates a linear programming relax-ation of the max-sum problem and its dual program.

The dual graphical models, also known as MRFs. This research primar-ily focused on computing the partition. Main conference: 23rd International Conference on Architectural Support for Programming Languages and Operating Systems.