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OBJETIVOS ESTRATÉGICOS FINANCIERA PROCESOS APRENDIZAJE CLIENTES
problem is the ability of the management to convert the problem into a mathematical model which represents the problem. Here a number of assumptions has to be made first, while trying to convert the real life situation into a scientific model written on a piece of paper. Except the problem is very clearly conceptualised by the experts it will not represent the real world problems and any solution will give misleading results. It is a complex process finding the solution, using the model is again a very time-consuming, complicated and cumbersome process.
However, the computers and their software can help decision-makers a great deal in this.
Steps involved in formulation of the goal programming model are as follows.
Step 1 Identification of decision-variable and constraints. This is the vital step in finding a solution to the problem. Clear identification of all the decision variables and environment conditions which are the constraints in the equations on the RHS have to be determined. RHS constraints are:
(i) Available resources as specified in the problem
(ii) Goals specified by the decision-maker
Step 2 Formulation of objectives or goals of the problem. As discussed earlier, an organisation could have more than one objective. Some of these could be:
(i) Maximise profits.
(ii) Maximise gain of share-holders.
(iii) Maximise machine utilisation (iv) Maximise manpower utilisation
(v) Maximise mean time between failures (MTBF) of machines
(vi) Minimise operation costs
(vii) Minimise operational time of the machine
(viii) Minimise overall time of production of the production
(ix) Optimise use of raw material (x) Satisfy social responsibilities (xi) Maximise quality of the product
(xii) Satisfy many government rules and other legal requirements.
Step 3 Formulation of the constraints. The constraints of the problem must be formulated. A constraint represents relationship between different variables in a problem. It could be the relationship between the decision-variables and the goals or objectives selected to be satisfied in order of priority.
Step 4 Identify least important and redundant goals. This is done to remove them from the problem which helps in simplifying the problem to some extent. This is again based on the judgment of the management.
Step 5 Establishing the objective function. Objective function has to be established based on the goals selected by the decision makers. Priority weight factors have to be allotted to deviational variables. The goal process models can be mathematically represented as
Minimise Objective function Z =
m
i i
i d d
W
1
1
Subject to the constraints
n
i
i i j i
i i j
ijX d d b where i n and x d d i j a
1
0 ,
, , , ...
, 3 , 2 , 1
where j is the decision variable.
Wi = is the weight of goal i
di = degree of underachievement of goal i
di = degree of overachievement of goal i
As seen earlier, goal programming attempts at full or partial achievement of goals in order of priority. Low priority goals are considered only after the high priority goals have been considered. This is very difficult to decide as contribution of a particular goal to the overall well-being of an organisation is very difficult to determine. The concept of underachievement of goals or overachievement of goals may
be understood as the most important. Selected goal continues to remain in the problem unless and until the achievement of a lower priority goal would cause the management to fail to achieve a higher priority goal.
Example 1
ABC Ltd produces two types of product P-1 and P–2 using common production facilities which are considered scare resources by the company. The scarce production facilities are in the two departments of Machining and Assembling. The company is in a position to sell whatever number it produces as their brand enjoys the market confidence. However, the production capacity is limited because of the availability of the scarce resources.
The company wants to set a goal maximum daily profit, because of its other problems and constraints and would be satisfied with #2000 daily profit. The details of processing time, capacities of each of the departments and unit profit combinations of products P1 and P2 are given in the table below:
Type of product
Time to process each product (Hours)
Profit
contribution per unit
P1 3 1 200
P2 2 1 300
Time available (hours) per day
100 50
The company wishes to know the product mix that would get them the desired profit of #2000 per day. Formulate the problem as goal programming model.
Solution
Let X1 be the number of units of P1 to be produced Let X2 be the number of units of P2 to be produced
di = the amount by which actual profit will fall short of #2000/day
di+ = the amount by which actual profit will exceed the desired profit of
# 2000/day
Minimise Z = di di
Subject to 3 X1 + 2 X2 < 100 (Machine hours constraint) X1 + X2 < 50 (Assembly hours constraint)
and 200 X1 + 300 X2 + di di = 2000. (Desired profit goal constraint)
where X1, X2, di di > 0 Example 2
The manufacturing plant of an electronic firm produces two types of television sets, both colour and black and white. According to the past experience, production of either a colour or a black and white set requires an average of one hour in the plant. The plant has a normal production capacity of 40 hours a week. The marketing department reports that, because of the limited sales opportunity, the maximum number of colour and black-and-white sets that can be sold are 24 and 30 respectively for the week. The gross margin from the sale of a colour set is # 80, whereas it is #40 from the black-and-white set.
The chairman of the company has set the following goals arranged in the order of their importance to the organisation.
(i) First, he wants to avoid an under-utilisation of normal production capacity (on lay offs of production workers).
(ii) Second, he wants to sell as many television sets as possible. Since the gross margin from the sale of colour television set is twice the amount from a black-and –white, he has twice as much desire to achieve sale for colour sets as for black-and-white sets.
(iii) Third, the chairman wants to minimise the overtime operation of the plant as much as possible. Formulate this as a goal programming problem and solve it.
Solution
Let X1 and X2 denote the number of colour TV sets and number of black-and-white TV sets for production respectively.
(i) The production capacity of both types of TV sets is given by X1 + X2 + di di = 40 di and di are deviational variables.
(ii) The sale capacity of two types of TV sets is given by X1 + d2 d2 = 24
X2 + d3 d3 = 30
where d2 d2 are the deviational variables representing under-achievements of sales towards goals and d3d3 represent deviational variables of over-achievement of sales goals.
(iii) Let P1 and P2 be the priority of the goals, complete mathematical formulation of goal programming is
Minimise Z = p1d1 + 2 p1 d2- + p2d3- + p2d1+
Subject to the constraints
X1 + X2 + di di = 40, X1 + d2 d2 = 24
X3 + d3 d3 = 30 and X1 , X2, d1, d2 , d3 d1, d2 , d3 0