Quantitative Analysis Test 1

For any inefficient DMU, what other DMU's represent the virtual DMU to which the inefficient DMU is compared?

- In the excel output for the inefficient DMU, the DMU's whose input costs = output values are the DMU's upon which the virtual DMU is based - an inefficient DMU could become efficient by an input reduction (cost savings) for each input, calculated by multiplying all input values by the difference between its present efficiency and 100%

Date Envelopment Analysis

- LP-based technique for measuring relative performance of organizational units, sometimes referred to as decision making units (DMU's) - an increasingly popular and practical management tool - used to identify the 'best' performer or practice (benchmarking)

which DMU's are efficient and which ones are inefficient?

- a DMU is determined to be efficient if maximized output value = 1. If the efficiency = 1, the DMU is 100% efficient relative to the other DMU's - a DMU is determined to be inefficient if maximized output value < 1 (say X). If the efficiency is < 1, the DMU is X% as efficient as a virtual DMU created by the efficient DMU's

Analytical hierarcy process (AHP)

- a multiple criteria approach to decision making which does not lose sight of the fact that many decisions are ultimately dependent on the creative (and subjective) process by which the decision problem is formulated

advantages of SolverTable

- allows you to vary any of the outputs - requires a bit more work and some experimentation to find the appropriate input ranges - much more flexible in this respect - very straightforward - outputs have the same interpretation for any type of optimization model - separate add-in not included with Excel

weaknesses of DEA

- an extreme point technique, noise such as measurement error can cause significant problems - estimating the "relative" efficiency of a DMU, but it converges very slowly to "absolute" efficiency - is a nonparametric technique, statistical hypothesis tests are difficult and often are the focus of ongoing research - since a standard formulation of DEA creates a separate linear program for each DMU, large problems can be computationally intensive

strengths of DEA

- can handle multiple input and multiple output models - doesn't require an assumption of a functional form relating inputs to outputs - are directly compared against a peer or combination of peers - inputs and outputs can have very different units

Applications areas in DEA

- education - health care - banking - manufacturing

issues with sensitivity report

- focuses only on the coefficients of the objective function and the right hand sides of the constraints - provides very useful information through its reduced costs, shadow prices, and allowable increases and decreases - based on changing only one objective function coefficient or one RHS value at a time. This one-at-a-time restriction prevents us from answering more complex questions directly. - based on a well-established mathematical theory of sensitivity analysis in linear programming - is not available for models with integer or binary based variables, and its interpretation for nonlinear models is more difficult than for linear models - comes with Excel

relative productivity

- idea is to limit the relative productivity of any DMU by placing an upper bound on the relative productivity of all units - then it is easy to measure your DMU's productivity against the best

most common varieties of mathematical programming

- linear programming - integer programming - nonlinear programming - network analysis - dynamic programming

failures of traditional measurement of productivity

- not realistic for most firms, only one product - more practical to use some common measures, but difficult to find - different types of outputs and inputs - differing levels of productivity are possible

examples of binary integer program

- should we undertake a particular project? - should we make a certain fixed investment? - should we locate a facility in South Dakota? - any decision where the choice is yes or no

issues with the choice of weights of DEA

- the weights (decision variables) in DEA are dependent on the scale of the data, rendering them meaningless in terms of interpretation - one alternative is to scale the data so that the weights are truly relative to each other across DMU's inputs, and outputs (scaling example to be completed in class) - a unit having the highest ratio of one of the outputs to one of the inputs will put its weights there, and thus be evaluated as efficient. This means that there is at least the potential for a number of efficient units equal to the number of inputs times the number of outputs. Therefore a unit can appear efficient simply because of its patter of inputs and not because of any inherent efficiency

managing performance using DEA

- utilizing the identification of peer groups for future improvement - benchmarks - identifying efficient operating practices - possibly multiple best practices - evaluation of managerial decision making

consistency vector can be calculated by:

1) calculating the row averages for the normalized matrix to get weights 2) multiplying the original matrix by the weights to get the product 3) dividing the product by the weights to get the consistency vector

six steps in decision making

1) clearly define the problem at hand 2) list the possible decision alternatives 3) identify the possible outcomes (state of nature) that might occur regardless of the decision 4) list the payoff (profit) of each combination of alternatives and outcomes 5) select one of the decisions theory models 6) apply the model and make your decision

maximum regret steps

1) converting the payoff table into a regret table 2) subtract the payoff for every alternative for a given state of nature from the maximum payoff for that state of nature 3) repeat the process for each state of nature to complete the table 4) find the maximum regret for each decision alternative 5) choose the minimum of the maximum regrets

How to use AHP to solve a problem

1) get an idea of the structure of the problem 2) identify a set of weights to reflect the importance of each criteria and how each decision alternative is judged on each of the criteria and how each decision alternative is judged on each of the criteria 3) by combining all above information, we can financially get composite priorities for each of the decision alternatives

DEA modeling in excel

1) set up matrix of DMU's vs. Inputs/outputs 2) set up a vector of weights for inputs/outputs 3) use SUMPRODUCT to get weighted inputs and weighted outputs for each DMU 4) constrain all DMU's so that weighted inputs are > weighted outputs 5) constrain the DMU of interest to have weighted inputs = 1 6) maximize the weighted outputs of the DMU of interest 7) repeat for each DMU

guidelines for mathematical programs

1) understand the problem thoroughly 2) verbally and concisely state the following - the objective - the decision variables - the constraints 3) develop mathematical expressions using the decision variables as the unknown 4) implement the expressions in a spreadsheet and run

consistency index (CI)

= x-n/n-1 - x is the average value of the consistency vector - n is the number of potential decisions being compared

Professor Thomas L Saaty

AHP was first developed by _____________________ in the 1970s and since that time has received wide application in the variety of areas

P(A/B) =

P(A^B)/P(B) ** bottom is given**

conditional probability

P(PS/FM) or anything with the given line in it

expected monetary value (EMV)

a long run weighted average value for a decision (as long as the sum of the probabilities is equal to one)

when the problem solution is very sensitive to changes in the input data and model specification,

additional testing should be performed to make sure that the model and input data are accurate and valid

of exactly producing at required demand

after solver, binding constraints are _________

adjusted market probabilities

all answers should add up to 1

equally likely

also called Laplace, decision assumes equal likelihood that any state of nature will occur

surplus (for a > constraint)

amount to be added to the right-hand side to make the constraint binding

slack (for a > constraint)

amount to be subtracted from the right-hand side to make the constraint binding

cell references

are the only way for SolverTable to work

decision variables

aspects of the problem you can control that will help achieve the stated objective

banking

bank branch operating efficiency, technical and scale efficiency in the banking sector

binary variable

can take on only two values -1 or 0

reduced cost

change in an objective function coefficient necessary to have a positive optimal value for that variable

maximax

choose the maximum payoff for each decision alternative and then choose the maximum of the maximums

maximin

choose the minimum payoff for each decision alternative and then choose the maximum of the minimums

decision making under certainty outcome

choose the option that you are "certain" will yield the best return

product mix

classical type of linear programming model

constraints

condition that must be satisfied for the solution to be feasible

constraints

conditions on the decision variables that put restrictions on the possible values of the variables

binding constraint

constraint that holds as an equality in the optimal solution

are based on allocating limited resources, satisfying requirements, or achieving desired ratios of products produced or activities utilized

constraints in a product mix problem

y = 0 if

decision is no

y = 1 if

decision is yes

decision making under uncertainty

decision makers have no idea which of several outcomes will result from choosing a particular decision alternative, but still knows the payoff of each outcome and decision alternative

decision making under certainty

decision makers know with certainty what the consequences will be from a given set of decisions

sensitivity analysis

determines how sensitive the solution, objective function, or other problem conditions are to changes in problem data

mathematical programming is typically applied to

deterministic problems where probability theory is not needed

portfolio selection model

formulated using data for the Morningstar Principia mutual fund database for the purpose of determining the optimal allocations of investor funds amount six categories of mutual funds for the bull market of the late 1990s

decision node

has a "branch" for each decision alternative

state of nature node

has a "branch" for each state of nature that is possible for a given decision

up and down and then across, diagonally

in excel, have to go _____________, can't go __________

scheduling models

integer linear programming applied to human resource and operations management

mathematical programming

involves constructing a mathematical model to represent a problem of interest and applying a programmable process to find the solution to the problem

"perfect" information

is almost impossible to acquire legally and ethically

value of CR < 0.1

is typically considered acceptable; larger values require the decision-maker to reduce the inconsistencies by revising judgments

binary integer program

linear program where all decision variables must be binary

algorithm

mathematical model to represent a problem of interest

linear program

mathematical program with a linear objective function and linear constraints

MMULT function

matrix multiplier - have to highlight all cells you need in advance - ctrl, shift, enter for it to work

health care

measuring nursing service efficiency, reimbursement rate setting for prescription drugs, evaluating the efficiency of nonprofit organizations, resource allocation, frontier estimation for health economics, measuring hospital efficiency, quality targets perinatal care

minimax regret

minimizing the maximum regret (like the concept of opportunity cost).

asking the "what-if" questions makes management

more nimble with respect to decision making efficiency and effectiveness

maximin criterion

most pessimistic way to make the decision

infeasible solution

no answer satisfies all the constraints

select a mix of products or activities that will optimize the objective function

objective of a product mix problem

minimize labor cost or number of employees while meeting demand for employees of other human resource issues

objective of scheduling models

cutting stock problems

occur where a "product" is produced in a standard size, which then must be cut into several smaller sizes to satisfy customer orders

manufacturing

operational performance ratings, resource use efficiency, product quality and cost leadership, determinants of manufacturing performance, factor substitution, capacity utilization, and total factor productivity growth, technology selection, alternative machine component grouping, selection of a flexible manufacturing system

maximax criterion

optimistic way to make the decision

decision making under uncertainty outcome

payoff of each outcome and decision alternative

education

performance assessment, comparing university departments, improving pupil transportation, guiding schools to improved performance

traditional measure of productivity

productivity = output/input

F4

puts $ in formula in excel

random index (RI)

random matrices were generated

dual/shadow price

rate of improvement in the objective function if a constraint's right-hand side increases by 1 (price of extra resources or savings on extra production)

newer measure of productivity (relative)

relative productivity = weighted sum of outputs/weighted sum of inputs

even if the input data is accurate,

sensitivity analysis will allow a manager to identify how changing conditions will affect the objective and decisions without actual implementation

prior probabilities

something that has happened before

feasible region

the domain of the set of decision variables (example: all possible combinations of the decision variables that satisfy all of the constraints)

objective function

the function (of the decision variables) to be minimized or maximized

objective

the goal of the problem (maximizing profit, sales or productivity, or minimizing the cost)

unbounded solution

the objective function has no limit

consistency ratio

the ratio of the CI to the RI - the measure of how a given matrix compared to a purely random matrix in terms of the CIs

multiple optimal solutions

two or more solutions provide the same best objective function value

production planning models

used for determining production of service provision planning over time

integer linear programs (IP)

used when all decision variables must be integers - production of airplanes - number of employees - anything that can't be a fraction in reality

mixed integer linear program (MIP)

used when some of the decision variables must be integers, while others may be continuous

decision variables

variables included in the math program that represent the decisions to be made

blending models

various inputs must be blended together to produce the desired outputs

given

what does the line in the probabilities mean

decision making under risk

when more information is available about the probabilities of the different states of nature for a decision, it is wise to use that information

non-linear function

where mathematical modifications other than multiplication by a constant exist

linear function

where the only mathematical modification of a variable is multiplication by a constant