Constraint management

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

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

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. Identifying decision variables (e.g. quantity of a product) Identifying constraints 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

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.

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.

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

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: 0,1*P2 < 10 m1, m2: 10 < P1 < 100 m3, m4: 10 < P2 < 100 ofCM: 200 P1+200P2=MAX 200 200

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)

Solution with linear programming 33 jugs and 33 plaits a per week Contribution margin: 13 200 HUF / week T1 e1 100 ofF e3 e1: 1*P1+0,5*P2 < 50 e2: 0,5*P1+1*P2 < 50 e3: 0,1*P2 < 10 m1,m2: 10 < P1 < 100 m3, m4: 10 < P2 < 100 ofCM: 200 P1+200P2=MAX 33,3 e2 100 T2 33,3

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

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

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

Thank you for your attention!