(b) A toy company manufactures two types of dolls, a basic version doll-A and a deluxe version doll-B. Each doll of type B takes twice as long to produce as one of type A, and the company would have time to make maximum of 1000 per day. The supply of plastic is sufficient to produce 1000 dolls per day (both A & B combined). The deluxe version requires a fancy dress for which there are only 500 per day available. If the company makes a profit of Rs 3.00 and Rs 5 per doll, respectively on doll A and B, then how many of each doll should be produced per day in order to maximise the total profit. Formulate this problem.
Linear programming focuses on obtaining the best possible output (or a set of outputs) from a given set of limited resources. The LPP is a class of mathematical programming where the functions representing the objectives and the constraints are linear. Optimization refers to the maximization or minimization of the objective functions. You can define the general linear programming model as follows: Maximize or Minimize Z = c1 x1 + c2 x2 +………….. +cn xn
subject to the constraints,
a11 x1 + a12 x2 + …………….+ a1n xn ~ b1
a21 x1 + a22 x2 +……………..+ a2n xn ~ b2
am1x1 + am2 x2 +……………. +amn xn ~ bm
and x1 > 0, x2 > 0, xn > 0.
Where cj, bi and aij (i = 1, 2, 3, ….. m, j = 1, 2, 3…….. n) are constants determined from the technology of the problem and xj (j = 1, 2, 3 n) are the decision variables. Here ~ is either < (less than), > (greater than) or = (equal). Note that, in terms of the above formulation the coefficient cj, aij, bj are interpreted physically as follows. If bi is the available amount of resources i, where aij is the amount of resource i, that must be allocated to each unit of activity j, the “worth” per unit of activity is equal to cj.
Let X1 and X2 be the number of dolls produced per day of type A and B, respectively. Let the A require t hrs.
So that the doll B require 2t hrs.
So the total time to manufacture X1 and X2 dolls should not exceed 2000t hrs. Therefore, tX1 + 2tX2 ≤ 2000t
Other constraints are simple. Then the linear programming problem becomes: Maximize p = 3 X1 + 5 X2
Subject to restrictions,
X1 + 2X2 ≤ 2000 (Time constraint)
X1 + X2 ≤ 1500 (Plastic constraint)
X2 ≤ 600 (Dress constraint)
And non-negatively restrictions
X1, X2 ≥ 0
Q 2. What are the advantages of Linear programming techniques?
1. The linear programming technique helps to make the best possible use of available productive resources (such as time, labor, machines etc.) 2. It improves the quality of decisions. The individual who makes use of linear programming methods becomes more objective than subjective. 3. It also helps in providing better tools for adjustment to meet changing conditions. 4. In a production process, bottle necks may occur. For example, in a factory some machines may be in great demand while others may lie idle for some time. A significant advantage of linear programming is highlighting of such bottle necks. 5. Most business problems involve constraints like raw materials availability, market demand etc. which must be taken into consideration. Just we can produce so many units of product does not mean that they can be sold. Linear programming can handle such situation also.
6. It helps in attaining the optimum use of productive factors.
7. It improves the quality of decisions. The individual who makes use of linear programming methods becomes more objective than subjective.
8. It also helps in providing better tools for adjustment to meet changing conditions.
9. It highlights the bottlenecks in the production processes.
10. Most business problems involve constraints like raw materials availability, market demand etc. which must be taken into consideration. Just we can produce so many units of product does not...