Linear programming is also a very useful tool that can be used to deal with problems in manufacturing industry, such as the product-mix problem. In this situation, the objective of the manager is to determine the production levels that will allow the company to meet the product demand requirements, given limitations on labor capacity, machine hour’s capacity and so on, at the same time, to make the cost of production to minimum. The solutions that we calculated by using linear programming would satisfy the given limitations and to enable all resources used in manufacturing-raw materials, labors, facilities are employed most efficiently. The application of linear programming for production scheduling is beneficial because those problems will recur in the next production period; so managers just have to resolve the problem according to different limitations. Therefore ant unnecessary costs would be reduced.
Disadvantages for linear programming
Although the linear programming technique can be applied to a wide range of different problems, there are some disadvantages that we should bear in mind before using this method.
The first disadvantage is that the linear programming technique miscalculate the total costs occur in the production process. By this I mean the method assumes that outputs is associated by a certain proportion, if the amount of inputs doubles, then the amount of outputs and costs would be doubled simultaneously, But in reality, in a production process, the total costs are not changed in the same degree as the inputs changes due to the property of the fixed costs
The second disadvantage is that in some cases, the optimal solution we get by using the linear programming technique is not realistic. This refers to two types of problems: infeasibility and unbounded ness. Infeasibility occurs when there is no solution to the linear programming problem that satisfies all the constraints. This problem can arise due to management’s high expectations or due to too many limitations that have been placed on the problem. Another case is where the solution might be unbounded. This means that the value of the solution to maximization or minimization problem is infinitely large or small respectively, without violating any of the constraints. This happens due to one or more constraints have been inadvertently omitted in the problem formulation. For the unbounded result, management has to change the objective function to find a realistic solution
Finally, linear programming assumes that the production process is fully under control and the total profit is just simply the sum of all the objectives. However, in practice, the production process can be influenced by many factors, which means that there are other losses and costs occur, in this case the resources would not be allocated and used as what they were expected, and therefore leads to a miscalculating in the total profit.
Part B
Formulate this problem as a linear programming
X= number of cows
Y= number of pigs
MAXIMISE: 40X +20Y (Objective Function)
Constraints:
(1)3X+Y ≤ 9
(2)X+Y ≥ 4
(3)X ≤ 4
(4)Y ≤ 6
X, Y ≥ 0
To solve the problem, we have to use the following equations;
Line 1 3X+Y=9
Line 2 X+Y=4
Line 3 X=4
Line 4 Y=6
Use there four equations to draw the graph as below:
(Hand-Drawing)
The best Mix of purchase and the max-profit
Suppose that the maximum profit is £80 (it could be any figure as the slope of iso profit line keeps not change, just easy for calculation)
then the objective function would be :
40X+20Y=80
When X=0, Y=4 Point (0,4)
When Y=0, X=2 Point (2, 0)
Connect the two points we get the iso profit line (Shown in green)
The area shadowed in orange is the range of mixtures that satisfy all the limitations in this problem. To find out the optimal solution,, we shift the iso profit line upwards to north-east through the feasible solution region until it cut the extreme of the area, the intersection is our optimal solution which is point A(2, 6). To get the maximum profit, we plug the figure into the objective function 40X+20Y, so we get a max-profit of £160.
This solution is found by vision. To ensure it is the correct answer, we could examine it by plug other extreme points into the objective function and compare the results that we get.
For B (0, 6) the intersection of line 4 and y axis, we get a profit of 20*6+40*0=£120
For C (0, 4), the intersection of line 2 and y axis, we get a profit of 20*4+40*0=80
For D (2.5, 1.5), the intersection of lines 1 and 2, we can not get any reasonable solution as the amount of animals can not be half. As we can see that point A (1, 6) gives highest profit compare with others. So we can conclude that if the farmer wants to maximize his profit, the best option for him is to buy 1 cow and 6 pigs.
Dual or shadow prices
Because constraints 3X+Y ≤ 9 and Y ≤ 6 are the only tow constraints that would enlarge the feasible solution region if the right-hand value of the in- equation is increased, which mean adds value to profits. The changes in the right hand side value in other constraints will not affect the profit as the changes do not enlarge the feasible solution region, which means they are zero dual constraints. So we use these two constraints to calculate the shadow price.
If we increase the number of bushels by 1 at the right-hand side of the first constraint 3X+Y = 9, then, 3X+Y=10, and keep Y=6 as the same, solve these equations simultaneously, we get X=4/3, Y=6, plug them into objective function 40X+20Y we get a profit of £173.33. The difference between the two maximum profits is 173.33-160=£13.33, so the dual price for increasing/decreasing 1 unit of bushel is £13.33.
Now we increase the number of pigs by 1 at the right-hand side of Y ≤ 6, then, Y=7, and keep 3X+Y = 9 as the same, solve them simultaneously, we get X=2/3, Y=7, plug them into objective function 40X+20Y we get a profit of £166.67. The difference between this profit and the original max-profit is 166.67-160=£6.67, which means the dual price for increasing/decreasing in purchasing 1 pig, is £6.67.
Economic meaning for shadow price
The economics meaning of shadow price is the improvement in the optimal value of the objective function per unit increase in the right-hand side of the constraint. In a profit maximization problem, the dual price is the same as the shadow price. Managers could get information from the performance of each constraint and therefore make decisions on any changes in a particular input factor or resource in order to increase profit. In this case, to get more profit, the farmer is recommended to increase the number of bushels rather than increase the amount of pigs.
Reference:
‘The Quantitative methods for business decision with cases’, Lawrence L. Lapim, 6th Edition, Dryden, Chapter 9.
‘An introduction to management science-quantitative approaches to decision making’, David R.Anderson, Dennis J. Sweeney, Thomas A. Williams. . 6Th Edition, West, Chapter 4.