constraint management
Post on 03-Jan-2016
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Constraint Constraint managementmanagement
Constraint
Something that limits the performance of a process or system in achieving its goals.
Categories: Market (demand side) Resources (supply side)
Labour Equipment Space Material and energy Financial Supplier Competency and knowledge Policy and legal environment
Steps of managing constraints
Identify (the most pressing ones)Maximizing the benefit, given the
constraints (programming)Analyzing the other portions of the process
(if they supportive or not)Explore and evaluate how to overcome the
constraints (long term, strategic solution)Repeat the process
Linear programming
Linear programming…
…is a quantitative management tool to obtain optimal solutions to problems that involve restrictions and limitations (called constrained optimization problems).
…consists of a sequence of steps that lead to an optimal solution to linear-constrained problems, if an optimum exists.
Typical areas of problems
Determining optimal schedulesEstablishing locationsIdentifying optimal worker-job
assignmentsDetermining optimal diet plansIdentifying optimal mix of products in a
factory (!!!)etc.
Linear programming models
…are mathematical representations of constrained optimization problems.
BASIC CHARACTERISTICS:ComponentsAssumptions
Components of the structure of a linear programming model
Objective function: a mathematical expression of the goal e. g. maximization of profits
Decision variables: choices available in terms of amounts (quantities)
Constraints: limitations restricting the available alternatives; define the set of feasible combinations of decision variables (feasible solutions space). Greater than or equal to Less than or equal to Equal to
Parameters. Fixed values in the model
Assumptions of the linear programming model
Linearity: the impact of decision variables is linear in constraints and the objective functions
Divisibility: noninteger values are acceptable
Certainty: values of parameters are known and constant
Nonnegativity: negative values of decision variables are not accepted
Model formulation
The procesess of assembling information about a problem into a model.
This way the problem became solved mathematically.
1. Identifying decision variables (e.g. quantity of a product)
2. Identifying constraints
3. Solve the problem.
2. Identify constraints
Suppose that we have 250 labor hours in a week. Producing time of different product is the following: X1:2 hs, X2:4hs, X3:8 hs
The ratio of X1 must be at least 3 to 2.
X1 cannot be more than 20% of the mix. Suppose that the mix consist of a variables x1, x2 and x3
2
3
x
x
2
1 0x3x2 21
)xxx(2,0x 3211
0x2,0x2,0x8,0 321
250x8x4x2 321
Graphical linear programming
1. Set up the objective function and the constraints into mathematical format.
2. Plot the constraints.3. Identify the feasible solution space.4. Plot the objective function.5. Determine the optimum solution.
1. Sliding the line of the objective function away from the origin to the farthes/closest point of the feasible solution space.
2. Enumeration approach.
Corporate system-matrix1.) Resource-product matrix
Describes the connections between the company’s resources and products as linear and deterministic relations via coefficients of resource utilization and resource capacities.
2.) Environmental matrix (or market-matrix): Describes the minimum that we must, and maximum that we can sell on the market from each product. It also describes the conditions.
Contribution margin
Unit Price - Variable Costs Per Unit = Contribution Margin Per Unit
Contribution Margin Per Unit x Units Sold = Product’s Contribution to Profit
Contributions to Profit From All Products – Firm’s Fixed Costs = Total Firm Profit
Resource-Product Relation typesP1 P2 P3 P4 P5 P6 P7
R1 a11
R2 a22
R3 a32
R4 a43 a44 a45
R5 a56 a57
R6 a66 a67
Non-convertible relations Partially convertible relations
Product-mix in a pottery – corporate system matrix
Jug Plate
Clay (kg/pcs) 1,0 0,5
Weel time (hrs/pcs)
0,5 1,0
Paint (kg/pcs) 0 0,1
Capacity
50 kg/week 100 HUF/kg
50 hrs/week 800 HUF/hr
10 kg/week 100 HUF/kg
Minimum (pcs/week) 10 10
Maximum (pcs/week)
100 100
Price (HUF/pcs) 700 1060
Contribution margin (HUF/pcs)
e1: 1*P1+0,5*P2 < 50e2: 0,5*P1+1*P2 < 50e3: 0,1*P2 < 10m1, m2: 10 < P1 < 100m3, m4: 10 < P2 < 100ofCM: 200 P1+200P2=MAX200 200
Objective function
refers to choosing the best element from some set of available alternatives.
X*P1 + Y*P2 = max
variables (amount of produced
goods)
weights(depends on what we want to maximize:
price, contribution margin)
Solution with linear programming
T1
T2
33,3
33,3
33 jugs and 33 plaits a per week
Contribution margin: 13 200 HUF / week
e1: 1*P1+0,5*P2 < 50e2: 0,5*P1+1*P2 < 50e3: 0,1*P2 < 10m1,m2: 10 < P1 < 100m3, m4: 10 < P2 < 100ofCM: 200 P1+200P2=MAX
e1
e2
e3ofF
100
100
What is the product-mix, that maximizes the revenues and the contribution to profit!
P1 P2 b (hrs/y)
R1 2 3 6 000
R2 2 2 5 000MIN (pcs/y) 50 100
MAX (pcs/y) 1 500 2000
p (HUF/pcs) 50 150
f (HUF/pcs) 30 20
P1&P2: linear programming
r1: 2*T5 + 3*T6 ≤ 6000
r2: 2*T5 + 2*T6 ≤ 5000
m1, m2: 50 ≤ T5 ≤ 1500
p3, m4: 100 ≤ T6 ≤ 2000
ofTR: 50*T5 + 150*T6 = max
ofCM: 30*T5 + 20*T6 = max
r2
r1
ofCM
ofTR
Contr. max: P5=1500, P6=1000Rev. max: P5=50, P6=1966
T1
T22000
3000
2500
2500
Thank you for your attention!
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