Deterministic Operations Research Some Examples

A summary of deterministic operations research models in linear programming, inventory theory, and dynamic programming. 1 Linear Programming A mathematical model of the problem is developed basically by applying a scientific approach as described earlier. There are a number of activities to be performed and each unit of each activity consumes some amount of each type of a resource. Resources are available in limited quantities. A measure of performance (an effectiveness or ineffectiveness measure) is defined according to the objective(s) of the management on the activities of concern. “Allocating the available amounts of resources to these activities in an effective manner” is transformed into a set of mathematical expressions that may result in a mathematical programming model of the type: max f(x1; x2; : : : ; xn) (1) subject to 1 gi(x1; x2; : : : ; xn) bi for i = 1; 2; : : : ; r (2) gi(x1; x2; : : : ; xn) bi for i = r + 1; r + 2; : : : ; q (3) gi(x1; x2; : : : ; xn) = bi for i = q + 1; q + 2; : : : ;m (4) xj 0 for all j = 1; 2; : : : ; n (5) In the problem (1) through (5), xj represents the level of activity to be employed (a decision variable) for j = 1; 2; : : : ; n, and the function f(x1; x2; : : : ; xn) is the measure of effectiveness based on the values of the decision variables xj and is called the objective function. The mathematical problem (1) through (5) then requires determining the value of each xj in such a way that the objective function attains its maximum value without exceeding the available amounts of the resources, and satisfying the minimum requirements and conditions specified by the constraints (2) through (5). . Each constraint function gi(x1; x2; : : : ; xn) represents the usage of resource and the right hand side value bi is the available amount of the resource i = 1; 2; : : : ; r. Further, for i = r + 1; r + 2; : : : ; q gi(x1; x2; : : : ; xn) shows the consumption of a certain ingredient i, and its corresponding right hand side value bi represents the minimum requirement. There may also be some conditions imposed on the activities and they are represented by equalities (4). Any maximization problem can be converted to a minimization problem by simply multiplying the objective function by ( 1) and vice versa. In a similar fashion any constraint of the form (2) can be converted to the form (3) by simply multiplying by ( 1) and vice versa. An equality constraint can be expressed as two inequality constraints. In particular, if the objective function and the constraint functions are linear, then such a mathematical programming problem is called a linear programming (LP) problem. A typical LP model looks like as given below. max 1x1 + 2x2 + : : :+ nxn 2 subject to a11x1 + a12x2 + : : :+ a1nxn b1 a21x1 + a22x2 + : : :+ a2nxn b2 am1x1 + am2x2 + : : :+ amnxn bm a(m+1)1x1 + a(m+1)2x2 + : : :+ a(m+1)nxn bm+1 ar1x1 + ar2x2 + : : : + arnxn br a(r+1)1x1 + a(r+1)2x2 + : : : + a(r+1)nxn = br+1 as1x1 + as2x2 + : : :+ asnxn = bs xj 0 for j = 1; 2; : : : ; n 1.1 Basic Assumptions of Linear Programming Problems and Examples In linear programming problems, the objective function and the constraint functions are linear. That implies that the relationships between the components of the real system can be expressed or approximated by linear expressions with a high degree of certainty . Further, the measure of performance is represented by a linear function of the decision variables. The linearity condition therefore requires satisfying the following basic assumptions. 1. Proportionality and Additivity: Each unit of an activity consumes a specified amount of a resource, and the increased level of activity increases the consumption of the resource in a direct proportion. Further, the total amount of resource consumed by all the activities is the