Linear Programming: The Graphical Method

THE GRAPHICAL METHOD

The graphical method of linear programming is used for problems involving two products.

Formulating the problem

Let us suppose that WX manufactures two products, A and B. Both products pass through two production departments, mixing and shaping. The organisation’s objective is to maximise contribution to fixed costs.

Graphing the problem

A graphical solution is only possible when there are two variables in the problem. One variable is represented by the x axis of the graph and one by the y axis. Since non-negative values are not usually allowed, the graph shows only zero and positive values of x and y.

Graphing equations and constraints

A linear equation with one or two variables is shown as a straight line on a graph. Thus y = 6 would be shown as follows.

Finding the optimum allocation of resources

The optimal solution can be found by ‘sliding the iso-contribution (or profit) line out’.

Having found the feasible region (which includes all the possible solutions to the problem) we need to find which of these possible solutions is ‘best’ or optimal in the sense that it yields the maximum possible contribution.

Look at the feasible region of the problem faced by WX (see the solution to the question above). Even in such a simple problem as this, there are a great many possible solution points within the feasible area. Even to write them all down would be a time-consuming process and also an unnecessary one, as we shall see.

THE GRAPHICAL METHOD USING SIMULTANEOUS EQUATIONS

Instead of a ‘sliding the contribution line out’ approach, simultaneous equations can be used to determine the optimal allocation of resources, as shown in the following example.

The optimal solution can also be found using simultaneous equations.

An organisation manufactures plastic-covered steel fencing in two qualities: standard and heavy gauge. Both products pass through the same processes involving steel forming and plastic bonding.

Slack and surplus

Slack occurs when maximum availability of a resource is not used. Surplus occurs when more than a minimum requirement is used.

If, at the optimal solution, the resource used equals the resource available there is no spare capacity of a resource and so there is no slack.

If a resource which has a maximum availability is not binding at the optimal solution, there will be slack.

SENSITIVITY ANALYSIS

Once a graphical linear programming solution has been found, it should be possible to provide further information by interpreting the graph more fully to see what would happen if certain values in the scenario were to change.

  • What if the contribution from one product was Rwf1 lower than expected?
  • What if the sales price of another product was raised byRwf2?
  • What would happen if less or more of a limiting factor were available, such as material?

 

Sensitivity analysis with linear programming can be carried out in one of two ways.

  • By considering the value of each limiting factor or binding resource constraint
  • By considering sale prices (or the contribution per unit)

 

Limiting factor sensitivity analysis

We use the shadow price to carry out sensitivity analysis on the availability of a limiting factor.

Shadow prices

The shadow price of a resource which is a limiting factor on production is the amount by which total contribution would fall if the organisation were deprived of one unit of the resource. The shadow price also indicates the amount by which total contribution would rise if the organisation were able to obtain one extra unit of the resource, provided that the resource remains an effective constraint on production and provided also that the extra unit of resource can be obtained at its normal variable cost.

So in terms of linear programming, the shadow price is the extra contribution or profit that may be earned by relaxing by one unit a binding resource constraint.

Suppose the availability of materials is a binding constraint. If one extra kilogram becomes available so that an alternative production mix becomes optimal, with a resulting increase over the original production mix contribution of RWF2, the shadow price of a kilogram of material is RWF.

Note, however, that this increase in contribution of RWF2 per extra kilogram of material made available is calculated on the assumption that the extra kilogram would cost the normal variable amount.

 

Note the following points.

  • The shadow price therefore represents the maximum premium above the basic rate that an organisation should be willing to pay for one extra unit of a resource.
  • Since shadow prices indicate the effect of a one unit change in a constraint, they provide a measure of the sensitivity of the result.
  • The shadow price of a constraint that is not binding at the optimal solution is zero.
  • Shadow prices are only valid for a small range before the constraint becomes nonbinding or different resources become critical.

Depending on the resource in question, shadow prices enable management to make better informed decisions about the payment of overtime premiums, bonuses, premiums on small orders of raw materials and so on.

Sales price sensitivity analysis

Sales price sensitivity analysis is carried out by changing the slope of the ‘iso-contribution’ line.

CHAPTER ROUNDUP

  • The graphical method of linear programming is used for problems involving two products.
  • The steps in the graphical method are as follows.

− Define variables.

− Establish objective function.

− Establish constraints.

− Draw a graph of the constraints.

− Establish the feasible region.

− Determine the optimal product mix.

  • The optimal solution can be found by ‘sliding the iso-contribution (or profit) line out’.
  • The optimal solution can also be found using simultaneous equations.
  • Slack occurs when maximum availability of a resource is not used. Surplus occurs when more than a minimum requirement is used.
  • The shadow price of a resource which is a limiting factor on production is the amount by which total contribution would fall if the organisation were deprived of one unit of the resource. The shadow price also indicates the amount by which total contribution would rise if the organisation were able to obtain one extra unit of the resource, provided that the resource remains an effective constraint on production and provided also that the extra unit of resource can be obtained at its normal variable cost.
  • Sales price sensitivity analysis is carried out by changing the slope of the ‘isocontribution’ line.
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