COURSE INSTRUCTOR : PROF. DR. SHAHAB KHUSHNOOD Chapter#10Techniques for making better engineering management decisions Lecture No. 08 Course: Engineering Management MED DEPARTMENT, U.E.T TAXILA
INTRODUCTION • During middle of 1940s, a significant growth of quantitative management techniques occurred. • The increase in the use of computers and the modern problem complexities have enhanced the importance of many of these techniques. For Example: • Linear Programming. • Non-Linear Programming.
Today, the mostly management techniques which are in practice: • Linear Programming • Decision Trees • Exponential Smoothing • Discounted Cash Flow Analysis
Optimization Techniques There are two widely known optimization techniques known as: • Lagrangian Multiplier Technique (Technique of undetermined Multipliers) • Linear Programming Technique (Advanced form of simplex method technique)
1-LAGRANGIAN MULTIPLIER TECHNIQUE • This technique is demonstrated for a two variable function. • However on similar lines it can be extended fornvariables. • Assume that we have defined function f(y1,y2) Subject to the constraint function k(y1,y2)=0
1-LAGRANGIAN MULTIPLIER TECHNIQUE • With the aid of these two functions the Lagrangian function , L(y1,y2,λ) formulated as follows:
1-LAGRANGIAN MULTIPLIER TECHNIQUE • Subject to the following necessary conditions for estimating a relative maximum or minimum value: • Expressions for y1,y2 and λ can be obtained by solving the simultaneous equations.
Solution: The Lagrange function is formulated as follows Taking the partial derivative with respect to y1,y2 and λresults in
2y1+2λ = 0 2y2- λ=0 2y1-y2=0 So Y1=8/5 Y2=-8/10 λ =-8/5 Thus the critical point of f, subjected to specified condition, is (8/5,-8/10) .
2-LINEAR PROGRAMMING • In real life situations the objective and constraint functions will be rather complex. • However the simplest form of linear programming problem formulation may be expressed as follows
Example 2 For above equations following values for symbols are defined Write down the resulting equations by assuming that the objective function is to be maximized.
(1) (2) (3) (4) (5) (6)
From Plot of equations 1-6: • Company profit is optimum at point F. (F point satisfies all the constraints) • F point shows maximum profit. The optimum values of Z1 and Z2 at point F are 6.5 and 2. substitute these values in equation.1.
Discounted Cash Flow Analysis • In various engineering investment decisions, the time value of money plays an important role. • Therefore it is necessary for the engineers to have some knowledge of engineering economics.
Discounted cash flow analysis The basics of engineering economics are • Simple Interest • This is the interest which is calculated on the original sum of money, called the original principal, for the period in which the lent or borrowed sum is being utilized. • The simple interest,St,is given by: (1) Where M is the principal amount lent or borrowed i is the interest rate per period ( this is normally a year) k is the interest periods (these are usually years)
Simple Interest (Cont.…) The total amount of money, Mt, after the specified lent or borrowed period is given by (2)
Solution: (1) (2)
2. Compound Interest • In this case at the end of each equal specified period, the earned interest is added to the original principal or amount lent or borrowed at the beginning of that period • Thus, this new principal, or amount, acts as a principal for the next period and the process continues.
To calculate compound amount, Mck, the resulting formula is developed as follows:
3.Present Worth • The present worth of a single payment is given by: • In simple terms, this formula is used to obtained the present worth, M, of money,Mck, after k periods, discounted at the periodic interest rate of i. • Sometime above equation can be written as
Formula For Uniform Periodic Payments • For using formula it is assumed that at the end of each of the K period or years, the depositor adds D amount of money. The money is invested at the interest rate i, compounded annually or periodically. • Thus the total amount of money (1) where
It should be noted in this equation that the D amount of money is first time deposited at the end of first year or period.
By substituting k=3, D=2000 and i= 0.15 in equation (1): From above equation the total amount of money after three year period is:
Example # 8 Given Data: (From Example 7)
Solution: Substitute the data in given equation:
PRESENT VALUE OF UNIFORM PERIODIC PAYMENTS • In present value of uniform periodic payment we wish to find the present value of uniform periodic payments after K periods or years instead of total amount. • Thus the present worth, PW, of uniform periodic payments is given by • Multiply both side with (1+i)-1,The resulting present worth formula
Solution: In this example following values are given: Substitute these values in equation:
Example 9 Cont. • Thus decision to purchase the machine will be profitable investment.
Depreciation Techniques • The term “Depreciation” means a decline in value. • In order to take into consideration the change in the value of the product, the depreciation charges are made during the useful life of the engineering products. • The three depreciation techniques are: 1-Declining-Balance Depreciation Method 2-Straight-line Depreciation Method 3-Sum Of Digits Depreciation Method
Depreciation Techniques (Cont..) 1. Declining-Balance Depreciation Method • This method dictates the accelerated write-off of the product worth in its early productive years and corresponding lower write-off near the end of useful life years. • The depreciation rate αd is given by where
The declining balance techniques requires a positive value of s. The product book value, Vbv(M), at the end of year M is given by:
Solution: From example statement Substitute in equation