³Profit Maximization through Optimization Technique´
Faculty PROF.
Section
OBJECTIVE
y
y
Allocation of Share with the help of Operation Research Research Technique. Profit Maximization of HDFC
Research Technique.
Bank through Operation
ACKNOWLEDGEMENT
This project on OPERATIONAL RESEARCH would not have been possible without the help of many people. We would like to thank our project guide PROF AKHILESH TIWARI for his kind support & guidance towards the project.
Introduction Linear programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear equations. More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Given a polyhedron and a realvalued affine function defined on this polyhedron, a linear programming method will find a point on the polyhedron where this function has the smallest (or largest) value if such point exists, by searching through the polyhedron vertices. Linear programs are problems that can be expressed in canonical form: Maximize Subject to where
represents the vector of variables (to be determined), and
are vectors of (known) coefficients and
is a (known) matrix of
coefficients. The expression to be maximized or minimized is called the the obje object ctiv ivee func functi tion on (
in this this case case). ). The The equa equati tion onss
are are
the constraints which specify a convex polytope over which the objective function is to be optimized. Linear programming can be applied to various fields of study. It is used most extensively in business and economics, but can also be utilized for some engineering problems. Industries that use linear
programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.
The standard form of the linear programming problem is used to develop the procedure for solving a general programming problem. A general LPP is of the form Max (or min) Z = c1x1 + c2x2 + « +cnxn x1, x2, ....xn are called decision variable.
The Application Areas of Linear Programming The Application Areas of Linear Programming are: 1. Transportation Problem 2. Military Applications 3. Operation of System Of Dams 4. Personnel Assignment Problem 5. Other Applications: (a). manufacturing plants, (b). distribution centres, (c). production management and manpower management.
Basic Concept of Linear Programming Problem Objective Function: The Objective Function is a linear function of variables which is to be optimised i.e., maximised or minimised. e.g., profit function, cost function etc. The objective function may be expressed as a linear expression. Constraints: A linear equation represents a straight line. Limited time, labour etc. may be expressed as linear inequations or equations and are called constraints. Optimisation: A decision which is considered the best one, taking into consideration all the circumstances is called an optimal decision. The process of getting the best possible outcome is called optimisation. Solution of a LPP: A set of values of the variables x 1, x2,«.xn which satisfy all the constraints is called the solution of the LPP.. Feasible Feasible Solution: A set of values of the variables x 1, x2, x3,«.,xn which satisfy all the constraints and also the non-negativity conditions is called the feasible solution of the LPP. Optimal Solution: The feasible solution, which optimises (i.e., maximizes or minimizes as the case may be) the objective function is called the optimal solution. Important terms Convex Region and Nonconvex Sets.
Mathematical Formulation of Linear Programming Problems There are mainly four steps in the mathematical formulation of linear programming problem as a mathematical model. We will discuss formulation of those problems which involve only two variables. 1. Identify the decision variables and assign symbols x and y to them. These decision variables are those quantities whose values we wish to determine. 2. Identify the set of constraints and express them as linear equations/inequations in terms of the decision variables. These constraints are the given conditions. 3. Identify the objective function and express it as a linear function of decision variables. It might take the form of maximizing profit or production or minimizing cost. 4. Add the non-negativity restrictions on the decision variables, as in the physical problems, negative values of decision variables have no valid interpretation.
Advantages of Linear Programming i. The linear programming technique helps to make the best possible use of available productive resources (such as time, labour, machines etc.) ii. 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.
Limitations of Linear Programming (a). Linear programming is applicable only to problems where the constraints and objective function are linear i.e., where they can be expressed as equations which represent straight lines. In real life situations, when constraints or objective functions are not linear, this technique cannot be used. (b). Factors such as uncertainty, weather conditions etc. are not taken into consideration.
Graphical Method of Solution of a Linear Programming Problem The graphical method is applicable to solve the LPP involving two decision variables x 1, and x2, we usually take these decision variables as x, y instead of x 1, x2. To solve an LPP , the graphical method includes two major steps. a) The determination of the solution space that defines the feasible solution (Note that the set of values of the variable x 1, x2, x3,....xn which satisfy all the constraints and also the non-negative conditions is called the feasible solution of the LPP). b) The determination of the optimal solution from the feasible region. There are two techniques to find the optimal solution of an LPP. Corner Point Method and ISO- PROFIT (OR ISO-COST).
History of linear programming The problem of solving a system of linear inequalities dates back at least as far as Fourier, after whom the method of Fourier-Motzkin elimination is named. Linear programming arose as a mathematical model developed during the second world war to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Postwar, many industries found its use in their daily planning. The founders of the subject are Leonid Kantorovich, a Russian mathematician who developed linear programming problems in 1939, George B. Dantzig, who published the simplex method in 1947, and John von Neumann, who developed the theory of the duality in the same year. The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior point method for solving linear programming problems. Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the Simplex algorithm. The theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked.
Uses Linear programming is a considerable field of optimization for several reasons. Many practical problems in operations research can be expressed as linear programming problems. Certain special cases of linear programming, such as network flow problems and multicommodity flow problems are considered important enough to have generated much research on specialized algorithms for their solution. A number of algorithms for other types of optimization problems work by solving LP problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as duality, decomposition, and the importance of convexity of convexity and its generalizations. Likewise, linear programming is heavily used in microeconomics and company management, such as planning, production, transportation, technology and other issues. Although the modern management issues are everchanging, most companies would like to maximize profits or minimize costs with limited resources. Therefore, many issues can boil down to linear programming problems.
Standard form S t
f
i t
usual usual and most most intuiti
form of descr i ing a
linear linear programmi programming probl problem. It It consi consists of t of t e foll follow owiing t ree par ts: y
linear functi function to be maxi maximized ized A linear
e.g. maxi maximi e y
Probl roblem
constrai constraints of t of t e foll follow owiing form
e.g.
y
Non-negati on-negati
e vari variabl ables
e.g.
The probl problem is usuall usuall expressed in mat i f
m,
and then becomes: becomes:
maxi maximi e sub jec jectt to Other forms, such as mi minimi ation tion probl problems, probl problems wit with h const cons trai raints on alt alterna ernati ti e forms, as well well as probl problems invol nvolving negati negative ve var iabl ables can al always be rewr itten itten into an equi equival valent ent probl problem in st standard form. Exampl xample
hat a farmer has a pi piece of farm land, say A square Suppose that k ilome ilomettres large, to be pl plant anted wit with h eit either her wheat wheat or bar ley or some combi combinati nation on of t of the two. The farmer has a lim limited ited permi permissi ssi bl ble amount amount
F of
fertilizer and P of insecticide which can be used, each of which is
required in different amounts per unit area for wheat ( F 1, P 1) and barley ( F 2, P 2). Let S 1 be the selling price of wheat, and
S 2
the price of
barley. If we denote the area planted with wheat and barley by x by x1 and x 2 respectively, then the optimal number of square kilometres to plant with wheat vs barley can be expressed as a linear programming problem: maximize subject to
(maximize the revenue ² revenue is the "objective function") (limit on total area) (limit on fertilizer) (limit on insecticide) (cannot plant a negative area).
Which in matrix form becomes:
maximize
subject to
Augmented form (slack form) Linear programming problems must be converted into augmented form before being solved by the simplex algorithm. This form introduces non-negative slack non-negative slack variables to replace inequalities with equalities in the constraints. The problem can then be written in the following block matrix form: Maximize Z Maximize Z in: in:
where
are the newly introduced slack variables, and Z and Z is is the
variable to be maximized. Example The example above is converted into the following augmented form: maximize
(objective function)
subject to
(augmented constraint) (augmented constraint) (augmented constraint)
where
are (non-negative) slack variables, representing in this
example the unused area, the amount of unused fertilizer, and the amount of unused insecticide. In matrix form this becomes: Maximize Z Maximize Z in:
Duality See also: Dual linear program Every linear programming problem, referred to as a primal problem, can be converted into a dual problem, which provides an upper bound to the optimal value of the primal problem. In matrix form, we can express the primal the primal problem as: maximize
subject to
, with the corresponding
symmetric dual problem: problem : minimize minim ize
subject to
.
An alternative primal formulation is: maximize
subject to
, with the corresponding
asymmetric dual problem: problem : minimize minimiz e
subject to
.
There are two ideas fundamental to duality theory. One is the fact that (for the symmetric dual) the dual of a dual linear program is the original primal linear program. Additionally, every feasible solution for a linear program gives a bound on the optimal value of the objective function of its dual. The weak duality theorem states that the objective function value of the dual at any feasible solution is always greater than or equal to the objective function value of the primal at any feasible solution. The strong duality theorem states that if the primal has an optimal solution, x*, then the dual also has an optimal T
T
solution, y*, such that c x*=b y*. A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible (See also Farkas' lemma).
Example Revisit the above example of the farmer who may grow wheat and barley with the set provision of some A land, F fertilizer and P insecticide. Assume now that unit prices for each of these means of production (inputs) are set by a planning board. The planning board's job is to minimize the total cost of procuring the set amounts of inputs while providing the farmer with a floor on the unit price of each of his crops (outputs),
S 1
for wheat and
S 2
for barley. This corresponds to the
following linear programming problem: (minimize the total cost of the means minimize
of production as the "objective function")
subject
(the farmer must receive no less than
to
S 1
for his wheat)
(the farmer must receive no less than S 2
for his barley)
(prices cannot be negative). Which in matrix form becomes:
minimize
subject to The primal problem deals with physical quantities. With all inputs available in limited quantities, and assuming the unit prices of all outputs is known, what quantities of outputs to produce so as to maximize total revenue? The dual problem deals with economic
values. With floor guarantees on all output unit prices, and assuming the available quantity of all inputs is known, what input unit pricing scheme to set so as to minimize total expenditure? To each variable in the primal space corresponds an inequality to satisfy in the dual space, both indexed by output type. To each inequality to satisfy in the primal space corresponds a variable in the dual space, both indexed by input type. The coefficients that bound the inequalities in the primal space are used to compute the objective in the dual space, input quantities in this example. The coefficients used to compute the objective in the primal space bound the inequalities in the dual space, output unit prices in this example. Both the primal and the dual problems make use of the same matrix. In the primal space, this matrix expresses the consumption of physical quantities of inputs necessary to produce set quantities of outputs. In the dual space, it expresses the creation of the economic values associated with the outputs from set input unit prices. Since each inequality can be replaced by an equality and a slack variable, this means each primal variable corresponds to a dual slack variable, and each dual variable corresponds to a primal slack variable. This relation allows us to complementary slackness. Another example Sometimes, one may find it more intuitive to obtain the dual program without looking at program matrix. Consider the following linear program:
minimize subject to
, , ,
We have m + n conditions and all variables are non-negative. We shall define m + n dual variables: y variables: y j and s and si. We get: minimize subject to
, , , ,
Since this is a minimization problem, we would like to obtain a dual program that is a lower bound of the primal. In other words, we would like the sum of all right hand side of the constraints to be the maximal under the condition that for each primal variable the sum of its coefficients do not exceed its coefficient in the linear function. For example, x example, x1 appears in n + 1 constraints. If we sum its constraints' coefficients we get be at most c1. As a result we get:
. This sum must
maximize subject to
, , ,
Note that we assume in our calculations steps that the program is in standard form. However, any linear program may be transformed to standard form and it is therefore not a limiting factor.
Covering-Packing Dualities Covering-Packing Covering-Packing Dualities Covering pr problems
Packing problems
Minimum Set Cover
Maximum Set Packing
Minimum Vertex Cover
Maximum Matching
Minimum Edge Cover Maximum Independent Set A covering LP is a linear program of the form minimize subject to such that the matrix
and the vectors and are non-negative.
The dual of a covering LP is a packing LP, a linear program of the form maximize subject to such that the matrix
and the vectors and are non-negative.
Examples Covering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms.
[1]
For example, the LP relaxations of the
set packing problem, the independent set problem, and the matching problem are packing LPs. The LP relaxations of the set cover problem, the vertex cover problem, and the dominating set problem are covering LPs.
Finding a fractional coloring of a graph is another example of a covering LP. In this case, there is one constraint for each vertex of the graph and one variable for each independent set of the graph.
Complementary slackness It is possible to obtain an optimal solution to the dual when only an optimal solution to the primal is known using the complementary slackness theorem. The theorem states: Suppose that x = (x1, x2, . . ., xn) is primal feasible and that y = (y1, y2, . . . , ym) is dual feasible. Let (w1, w2, . . ., wm) denote the corresponding primal slack variables, and let (z 1, z2, . . . , zn) denote the corresponding dual slack variables. Then x and y are optimal for their respective problems if and only if x jz j = 0, for j = 1, 2, . . . , n, wiyi = 0, for i = 1, 2, . . . , m. So if the ith slack variable of the primal is not zero, then the ith variable of the dual is equal zero. Likewise, if the jth slack variable of the dual is not zero, then the jth variable of the primal is equal to zero. This necessary condition for optimality conveys a fairly simple economic principle. In standard form (when maximizing), if there is slack in a constrained primal resource (i.e., there are "leftovers"), then additional quantities of that resource must have no value. Likewise, if there is slack in the dual (shadow) price non-negativity constraint requirement , i.e., the price is not zero, then there must scarce supplies (no "leftovers").
Theory Geometrically, the linear constraints define a convex polytope, which is called the feasible region. It is not hard to see that every local optimum (a point x point x such that for every unit direction vector d vector d with with positive objective value any every > 0 it holds that x that x + d d is is infeasible) is also a global optimum. This holds more generally for convex programs: see the KKT theorem. There are two situations in which no optimal solution can be found. First, if the constraints contradict each other (for instance, x 2 and x 1) then the feasible region is empty and there can be no optimal solution, since there are no solutions at all. In this case, the LP is said to be infeasible. infeasible . Alternatively, the polyhedron can be unbounded in the direction of the objective function (for example: maximize x maximize x1 + 3 x 3 x2 subject to x to x1 0, x 2 0, x1 + x2 10), in which case there is no optimal solution since solutions with arbitrarily high values of the objective function can be constructed. Barring these two conditions (which can often be ruled out when dealing with specific LPs), the optimum is always attained at a vertex of the polyhedron (unless the polyhedron has no vertices, for example in the feasible bounded linear program
; polyhedra
with at least one vertex are called pointed). However, the optimum is not necessarily unique: it is possible to have a set of optimal solutions covering an edge or face of the polyhedron, or even the entire polyhedron (this last situation would occur if the objective function were constant on the polyhedron). The vertices of the polyhedron are also called basic feasible solutions. solutions . The reason for this choice of name is as follows. Let d denote d denote the
dimension, i.e. the number of variables. Then the following theorem holds: for every vertex x* vertex x* of the LP feasible region, there exists a set of d of d inequality inequality constraints from the LP such that, when we treat those d constraints d constraints as equalities, the unique solution is x* is x*.. Thereby we can study these vertices by means of looking at certain subsets of the set of all constraints (a discrete universe), rather than the continuous universe of LP solutions. This principle underlies the simplex algorithm for solving linear programs.
gorithms Algori
of linear near const constrai raints on two var iabl ables produces a regi reg ion of A ser ies of li possi possi bl ble val values for t for those var iabl ables. Solvabl vable probl problems will will have a feasi feasi bl ble regi region in the shape of a si simpl mple pol polygon.
Thee simpl Th simplex ex algo algori rithm thm of Dan Dantt ig The simplex algorithm, developed by George Dantzig, solves LP problems by constructing a feasible solution at a vertex of the polyhedron and then walking along a path on the edges of the polyhedron to vertices with non-decreasing values of the objective function until an optimum is reached. In many practical problems, "stalling" occurs: Many pivots are made with no increase in the objective function.
[2]
In rare practical problems, the usual versions of
the simplex algorithm may actually "cycle". researchers developed new pivoting rules
[3]
To avoid cycles,
[4]
In practice, the simplex algorithm is quite efficient and can be guaranteed to find the global optimum if certain precautions against cycling are cycling are taken. The simplex algorithm has been proved to solve "random" problems efficiently, i.e. in a cubic number of steps (Borgwadt, Todd), which is similar to its behavior on practical problems
[5]
However, the simplex algorithm has poor worst-case behavior: Klee and Minty constructed a family of linear programming problems for which the simplex method takes a number of steps exponential in the problem size.
[6]
In fact, for some time it was not known whether the
linear programming problem was solvable in polynomial time (complexity class P).
The ellipsoid algorithm, following Khachiyan This long standing issue was resolved by Leonid Khachiyan in 1979 with the introduction of the ellipsoid method, the first worst-case polynomial-time algorithm for linear programming. To solve a problem which has n variables and can be encoded in L input bits, this
4
algorithm uses O( n L) pseudo-arithmetic operations on numbers with O( L) digits. Khahiyan's algorithm and his convergence analysis have predecessors, notably iterative methods developed by Naum Z. Shor and by Arkadi Nemirovski and D. Yudin.
Interior point methods, following Karmarkar Khachiyan's algorithm was of landmark importance for establishing the polynomial-time solvability of linear programs. The algorithm had little practical impact, as the simplex method is more efficient for all but specially constructed families of linear programs. However, it inspired new lines of research in linear programming with the development of interior point methods, which can be implemented as a practical tool. In contrast to the simplex algorithm, which finds the optimal solution by progressing along points on the boundary of a polyhedral set, interior point methods move through the interior of the feasible region. In 1984, N. Karmarkar proposed a new interior point projective method for linear programming. Karmarkar's algorithm not only improved on Khachiyan's theoretical worst-case polynomial bound 3.5
(giving O(n L)). L)). Karmarkar also claimed that his algorithm exhibited dramatic practical performance improvements over the simplex method, which created great interest in interior-point methods. Since then, many interior point methods have been proposed and analyzed. Early successful implementations were based on affine scaling variants of the method. For both theoretical and practical properties, barrier function or path-following methods are the most common recently.
Comparison of interior-point methods versus simplex algorithms The current opinion is that the efficiency of good implementations of simplex-based methods and interior point methods are similar for routine applications of linear programming.
[7]
LP solvers are in widespread use for optimization of various problems in industry, such as optimization of flow in transportation networks.
[8]
Open problems and recent work There are several open problems in the theory of linear programming, the solution of which would represent fundamental breakthroughs in mathematics and potentially major advances in our ability to solve large-scale linear programs. y
y
Does LP admit a strongly polynomial-time algorithm? Does LP admit a strongly polynomial algorithm to find a strictly complementary solution?
y
Does LP admit a polynomial algorithm in the real number (unit cost) model of computation?
This closely related set of problems has been cited by Stephen Smale as among the 18 greatest unsolved problems of the 21st century. In Smale's words, the third version of the problem "is the main unsolved problem of linear programming theory." While algorithms exist to solve linear programming in weakly polynomial time, such as the ellipsoid methods and interior-point techniques, no algorithms have yet been found that allow strongly polynomial-time performance in the number of constraints and the number of variables. The
development of such algorithms would be of great theoretical interest, and perhaps allow practical gains in solving large LPs as well. y
Are there pivot rules which lead to polynomial-time Simplex variants?
y
y
Do all polyhedral graphs have polynomially-bounded diameter? Is the Hirsch conjecture true for polyhedral graphs?
These questions relate to the performance analysis and development of Simplex-like methods. The immense efficiency of the Simplex algorithm in practice despite its exponential-time theoretical performance hints that there may be variations of Simplex that run in polynomial or even strongly polynomial time. It would be of great practical and theoretical significance to know whether any such variants exist, particularly as an approach to deciding if LP can be solved in strongly polynomial time. The Simplex algorithm and its variants fall in the family of edgefollowing algorithms, so named because they solve linear programming problems by moving from vertex to vertex along edges of a polyhedron. This means that their theoretical performance is limited by the maximum number of edges between any two vertices on the LP polyhedron. As a result, we are interested in knowing the maximum graph-theoretical diameter of polyhedral graphs. It has been proved that all polyhedra have subexponential diameter, and all experimentally observed polyhedra have linear diameter, it is presently unknown whether any polyhedron has superpolynomial or even superlinear diameter. If any such polyhedra exist, then no edgefollowing variant can run in polynomial or linear time, respectively. Questions about polyhedron diameter are of independent mathematical interest.
Simplex pivot methods preserve primal (or dual) feasibility. On the other hand, criss-cross pivot methods do not preserve (primal or dual) feasibility --- they may visit primal feasible, dual feasible or primaland-dual infeasible bases in any order. Pivot methods of this type have been studied since the 1970s. Essentially, these methods attempt to find the shortest pivot path on the arrangement polytope under the linear programming problem. In contrast to polyhedral graphs, graphs of arrangement polytopes are known to have small diameter, allowing the possibility of strongly polynomial-time criss-cross pivot algorithm without resolving questions about the diameter of general polyhedra.
Integer unknowns If the unknown variables are all required to be integers, then the integer linear problem is called an integer programming (IP) or or integer programming (ILP) problem. In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in many practical situations (those with bounded variables) NP-hard. 0-1 integer programming or binary or binary integer programming (BIP) is the special case of integer programming where variables are required to be 0 or 1 (rather than arbitrary integers). This problem is also classified as NP-hard, and in fact the decision version was one of Karp's 21 NP-complete problems.
Problem No.1
Current
Airtel
TATA
(x1)
Motors
Industries (x5)
(x2)
(x4)
276.3
688
DLF(x3) Reliance
ICICI
HU L (x6)
290.15
981.85
819
237.55
0.0247 0.15
.053
0.048
0.0642
0.12
2.0494 17.25
1.6638
.6331
.48
0.33
.75
1.54
1.11
1.53
.37
price per share (Rs.) Projected annual growth rate Projected annual dividend per share (Rs.) Projected
1.21
risk in return
The total am ount available for investm ent is Rs. 10.92 lakhs and the followin g conditions are required to be satisfied. satisfied. . ( i)
The m aximum rupee amount to be invested invested in alternative F is Rs. 1,00,000.
(ii) No m ore than Rs. 2,00,000 should be invested in alternativ alternatives es A and B com bined
(iii) (iii) Total w eighted risk should not be greater than 1.2, -. (iv) (iv) For the sake o f diversity, at least 40 shares of each stock should be purchased. (v) A t least 10 percent of the total investment should be in alternativ alternativee A and B combined. (vi) Dividends for the year should b e at least Rs. 5000 Rup ees return per share of stock is defined defined a s price per share one year hence less current price per share PLUS d ividend per share. If the the objective is to m axim ize total rupee return, formulate the Linear Programming model for determining the optimal optimal number number of shares to b e purch ased in each of the shares under consideration. Y ou m ay assume that the time t ime horizon/or the investmen t is is one year
Subject to a) 276.3(x1) + 688(x2) + 290.15(x3) + 981.85(x4) + 819(x 5) + 237.55(x6) 237.55(x6 ) <= 1092000 b) 237.55(x6) <= 100000 c) 276.3(x1) + 688(x2) <=200000 d) -124.335x1 +6.88x2 +98.651x3 -88.3665x4 +270.27x5 197.1665x6<=0
e) x1, x2, x3, x4,x5,x6 >=40 f) 207.5x1 +619.2x2 -29.015x3 -98.185x4-819x5-23.755x6 >=0 g) .0494x1 +17.25x2 +1.6638x3 +.6331x4 +.48x5 +.33x6 >+5000
Maximi Maximi e the functio function n 8.87x1 +120.45x2 +17.042x3 + 47.76x4 +53.0598x5 +28.836x6
Problem No. 2
Investment HDFC Bank is in the process of devising a loan policy that involves a maximum of 10 million . The following table provides the pertinent data about available types of loans. Type of Loan
Interest Rate
Bad-debt Ratio
PERSONAL
0.175
0.08
Car
0.0875
0.05
Home
0.0867
0.08
Farm
0.115
0.20
Commercial
0.11
0.01
Bad debts are unrecoverable and produce no interest revenue. Competion with other financial institution requires that the bank allocate at least 40% of the fund to farm and commercial loans. To assist the housing industry in the region, home loans must equal atleast 50% of the personal, car, and home loans. The bank also has a stated policy of not allowing the overall ratio of bad debts on all loans to exceed 4%.
Subject to
(a) (b) (c) (d)
P+C+H+F +CO<=10000000 .6F+.6Co-.4P-.4H-.4C>=0 .5H-.5P-.5C .5H-.5P-.5C >=0 0.04P+.01C+.04H+.16F-.03CO<=0
MAXIMIZE .081P + .033125C -.000236H -.000236 H - .108F + .0989CO
