what is linear programming?
a technique of allocating resources in order to achieve the best results
- in profit maximising business - maximising contribution will maximise profit
- all business have some constraints (labour hrs/machine hrs)
- products have different profit margins
- products have different market demands
what do we use when we have one limiting factor?
(in relevant costing lectures)
we use ‘contribution per unit of scarce resource’ to allocate resources
- but what happens if there are more than one scarce resource/bottleneck activities?
what do we use when we have more than one limiting factor?
we use linear programming to solve problems where there is more than one limited resource!!
- we require an objective function
what is an objective function?
a statement of what is required e.g. maximise contribution, minimise costs
what are the 2 methods of solving a linear programming problem?
- solving graphically by drawing the limiting resource equations and finding the edge of the feasible region
- using software i.e. excel
how do we solve a linear programming problem?
- define the problem
- formulate the problem
- graph the model and find the optimum solution/use Excel Solver
what are the 2 major assumptions?
ADDITIVITY:
if one unit of A = 3hrs of labour, 15kg of material and if one unit of B = 2hrs, 5kgs
-> then one unit of A+B requires 5hrs of labour and 20kgs of material
DIVISIBILITY:
total resources required are directly proportional to the volume of output
- if 12 units of C requires 24 hrs of labour and 36 kgs of material
- then one unit of C requires 2 hrs and 3kgs
what are some other assumptions?
- unit variable cost is constant i.e. no bulk buying discounts, efficiency and productivity constant
- fixed costs remain the same regardless of the decision. therefore profit maximisation and contribution maximisation are the same thing
- estimates of sales demand are certain
- units of output are divisible and profit maximisation may include fractions
what are constraints?
we cannot do everything that we want to because resources are scarce
- e.g. if our labour and machine time are restricted, our output will be limited because we cannot obtain enough labour and machine time
outline the method for formulating a linear programming model
- define terms (e.g. let P = number of product P)
- work out the contribution for each product
- state the objective function (e.g. need to maximise contribution maximise the sum of contributions for each product)
- state constraints: non-negativity is a constraint (we cannot produce less than 0 items and therefore need to state each product >= 0)
- if given information on expected hrs/materials (LIMITS) formulate that too (e.g. expects to have only 600 hrs of skilled labour and 2,000 hrs of unskilled labour: skilled labour limited to 30 + 4Q + T =< 600)
- state minimum requirements
solving linear programming problems - graphical method
can solve problems using graphs but as a graph is 2-dimensional we can only use a graph if there are no more than 2 variables
- there can be a number of constraints but only 2 variables
what is the method when solving linear programming problems graphically?
- formulate linear programme model
- plot these on a graph (dont need to know how to): but an easy way is to find out H when S = 0 and find S when H = 0, don’t forget the limits too
- find the feasible area using the inequalities from constraints
- look at the vertices of the area - need to find out how much product will be produced at each point to see which produces the max contribution
- work out max contributions for each point of the feasible area
what is the easier way for solving linear programming problems graphically?
the iso-profit line !!
what is the iso-profit line?
the easiest way of working out max values. the FURTHEST away from the origin you can get within the feasible region (on the lines) the better
how do you find the optimum solution using the iso-profit line?
- plot the objective function (iso-profit line)
- see where it touches (leaves) the feasible region !
- read off the graph or solve by simultaneous equation to find values of H and S
what is the method when using the iso-profit line?
- find the gradient of the objective function (this just gives the objective function a value!)
- choose a value that is a multiple og both the variables in the equation (e.g. 2S + 3H -> multiple of both is 6 but if area is bigger i.e. in the thousands make it 6000)
- find out where it crosses the axis by setting each variable to 0 and solving the equation (e.g. working with the equation 2S + 3H = 6,000, so when S = 0, H = 2000 and when H = 0, S = 3000)
what is the optimal solution ?
the line at the point where contribution is maximised
when is a constraint binding?
if changing the constraint alters the optimal solution !
- two constraints may be binding if max contribution is obtained where they both coincide
when is a constraint non-binding?
if they do not affect the optimal solution!
- if it is beyond the optimal solution then it will be non-binding
explain how you can use Excel for linear programming
- instead of preparing a graph we can use an add-in within excel to solve linear programming problems
- still need to define and formulate the problem and then insert the equations into solver (will not be asked to create a problem solver, only interpret it !!)
what 2 kinds of results do you get from Excel Solver?
- an answer result (the table values including target cells, adjustable cells and constraints)
- sensitivity result
what does the target cell give you?
final value gives us the maximum contribution achievable with these constraints
what do the adjustable cells (aka variable cells) tell us?
final values tell us how many units of S and H to produce to achieve this maximum contribution
what does the cell value of the constraints tell us?
the quantities being used
