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!