1. Constraint management

2. 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

3. 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

4. Linear programming

5. 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.

6. Typical areas of problems
Determining optimal schedules
Establishing locations
Identifying optimal worker-job assignments
Determining optimal diet plans
Identifying optimal mix of products in a factory (!!!)
etc.

7. Linear programming models
…are mathematical representations of constrained optimization problems.
BASIC CHARACTERISTICS:
Components
Assumptions

8. 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

9. 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

10. Model formulation
The procesess of assembling information about a problem into a model.
This way the problem became solved mathematically.
Identifying decision variables (e.g. quantity of a product)
Identifying constraints
Solve the problem.

11. 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

12. Graphical linear programming
Set up the objective function and the constraints into mathematical format.
Plot the constraints.
Identify the feasible solution space.
Plot the objective function.
Determine the optimum solution.
Sliding the line of the objective function away from the origin to the farthes/closest point of the feasible solution space.
Enumeration approach.

13. Corporate system-matrix
1.) 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.

14. 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

15. Resource-Product Relation types
Non-convertible relations R1
a11 R2
a22 R3
a32
Partially convertible relations R4
a43
a44
a45 R5
a56
a57 R6
a66
a67

16. Product-mix in a pottery – corporate system matrix
Jug
Plate
Capacity
50 kg/week
100 HUF/kg
50 hrs/week
800 HUF/hr
10 kg/week
Clay (kg/pcs)
1,0
0,5
Weel time (hrs/pcs)
0,5
1,0
Paint (kg/pcs)
0,1
Minimum (pcs/week) 10
Maximum (pcs/week)
100
Price (HUF/pcs)
700
1060
Contribution margin (HUF/pcs)
e1: 1*P1+0,5*P2 < 50
e2: 0,5*P1+1*P2 < 50
e3: ,1*P2 < 10
m1, m2: < P1 < 100
m3, m4: < P2 < 100
ofCM: P1+200P2=MAX
200
200

17. variables (amount of produced goods)
Objective function
refers to choosing the best element from some set of available alternatives.
X*P1 + Y*P2 = max
weights (depends on what we want to maximize: price, contribution margin)
variables (amount of produced goods)

18. Solution with linear programming
33 jugs and 33 plaits a per week
Contribution margin: HUF / week T1 e1
100
ofF e3
e1: 1*P1+0,5*P2 < 50
e2: 0,5*P1+1*P2 < 50
e3: ,1*P2 < 10
m1,m2: < P1 < 100
m3, m4: < P2 < 100
ofCM: P1+200P2=MAX
33,3 e2
100 T2
33,3

19. 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
5 000
MIN (pcs/y) 50
100
MAX (pcs/y)
1 500
2000
p (HUF/pcs)
150
f (HUF/pcs) 30 20

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*T *T6 = max
ofCM: 30*T5 + 20*T6 = max

21. T1 r1
Contr. max: P5=1500, P6=1000
Rev. max: P5=50, P6=1966
3000 r2
2500
ofCM
ofTR
2000
2500 T2

22. Thank you for your attention!