BUS 2420 Management Science Instructor: Vincent WS Chow Office: WLB 818 Ext: 7582 E-mail: vwschow@hkbu.edu.hk URL: http://ww.hkbu.edu.hk/~vwschow Office hours: (to p2)

Refer to my website (to p3)

Subject outline Subject outline (see handout) Textbook: Grading: Bernard W. Taylor III, Introduction to Management Science, 10th Edition, Prentice Hall, 2010 Grading: Topics: Refer to handout Tutorials Start from 3rd hr of 3rd week lecture Typically, we assign few questions in each lecture and then taken them up for discussion in the next week session. How you are being graded? (to p4) (to p5) (to p6) (lecture)

Grading: Assignments 15% Class Participation 15% Test 20% Most likely be1-3 assignments Group Memberships (refer to our web site) Class Participation 15% Tutorial performance Test 20% One mid-term exam Examination 50% One final exam (to p3)

How you are being graded? Students will award marks if they show their works (by submission!) in the tutorial sessions Students are thus strongly encouraged to bring their works to show in tutorials or prepare materials for presentation ..… Note: you may like to approach me later to see how we could improve this process of grading! (to p3)

Lecture 1 Introduction to Management Science What is Management Science? How to apply Management Science technique? Types of Management Science Models/techniques We start with the most popular Management Science technique: Linear Programming (to p7) (to p9) Have we seen or used then before? (to p11) (to p13)

Management Science Management science uses a scientific approach to solving management problems. It is used in a variety of organizations to solve many different types of problems. It encompasses a logical mathematical approach to problem solving. History of Management Science (to p8) (to p6)

History of Management Science It was originated from two sources: Operational Research Management Information Systems It is thus more emphasizing on the analysis of solution applications than learning their on how models were derived. Other names for management science: quantitative methods, quantitative analysis and decision sciences. (to p7)

Steps in applying Management Science teniques (1) (to p10) (2) (3) (4) In practice, this step is critical (5) (to p6)

Steps Observation — Identification of a problem that exists in the system or organisation. Definition of the Problem — Problem must be clearly and consistently defined showing its boundaries and interaction with the objectives of the organisation. Model Construction — Development of the functional mathematical relationships that describe the decision variables, objective function and constraints of the problem. Model Solution — Models solved using management science techniques. Model Implementation — Actual use of the model or its solution. (to p9)

Models to be considered in this subject * * * * * * * * * Their Characteristics (to p12) (to p6) * Topics that will cover in this subject!

Characteristics of Modeling Techniques Linear mathematical programming: clear objective; restrictions on resources and requirements; parameters known with certainty. Probabilistic techniques: results contain uncertainty. Network techniques: model often formulated as diagram; deterministic or probabilistic. Forecasting and inventory analysis techniques: probabilistic and deterministic methods in demand forecasting and inventory control. Other techniques: variety of deterministic and probabilistic methods for specific types of problems. (to p11)

Linear Programming Or denote as LP Overview of LP How does LP look like? Components of LP General LP format Example 1: Maximizing Z Example 2: Minimizing Z We will talk about more LP formulations and its solutions in next lecture (to p14) (to p20) (to p15) (to p21) (to p23) (to p25)

Linear Programming - An Overview Objectives of business firms frequently include maximizing profit or minimizing costs, or denote as Max Z or Min Z Linear programming is an analysis technique in which linear algebraic relationships represent a firm’s decisions given a business objective and resource constraints. Steps in application: 1- Identify problem as solvable by linear programming. 2- Formulate a mathematical model of managerial problems. 3- Solve the model. (to p13)

4 Components of LP Decision variables: mathematical symbols representing levels of activity of a firm. Objective function: a linear mathematical relationship describing an objective of the firm, in terms of decision variables, that is maximized or minimized Constraints: restrictions placed on the firm by the operating environment stated in linear relationships of the decision variables. Parameters: numerical coefficients and constants used in the objective function and constraint equations. Non-negativity (or necessary) constraints (to p16) (to p17) (to p18) (to p19) (to p13)

Example of Decision Variables It is used to represent decision problem to be solve Let, x1=number of bowls to produce/day x2= number of mugs to produce/day How of them are needed is depended on the nature of the problem! (to p15)

Objective Functions It is used to represent the type of problems we are to solve In this subject, we only emphasize to either Maximizing a profit margin or Minimizing a production cost Example: An Objective function maximize Z = $40x1 + 50x2 (to p15) Refer to how much we made for each x is produced

Constraints It is also referred to resource constraints They are to indicate how much resources made available in a firm Example: Resource Constraints: 1x1 + 2x2  40 hours of labor 4x1 + 3x2  120 pounds of clay (to p15)

Non-negativity constraints We assumed that all decision variables are carried out positive values (why?) Example: Non-negativity Constraints: x10; x2  0 (to p15)

Sample of LP or Maximize Z=$40x1 + 50x2 subject to Decision variables Let xi be denoted as xi product to be produced, and i = 1, 2 or Let x1 be numbers of product x1 to be produced and x2 be numbers of product 21 to be produced Maximize Z=$40x1 + 50x2 subject to 1x1 + 2x2  40 hours of labor 4x2 + 3x2  120 pounds of clay x1, x2  0 Cost Objective function Constraints (to p13)

General LP format subject to xij ≥ 0, for i=1,…,m, j=1,…,n Max/Min Z : Σ cixi subject to Σ aij xij (=, ≤, ≥) bj , j = 1,…., n xij ≥ 0, for i=1,…,m, j=1,…,n (to p22) General steps for LP formulation It means there are total of m decision variables n resource constraints (to p13)

Steps for LP formulation Step 1: define decision variables Step 2: define the objective function Step 3: state all the resource constraints Step 4: define non-negativity constraints (to p21)

Example 1: Max Problem A Maximisation Model Example The Beaver Creek Pottery Company produces bowls and mugs. The two primary resources used are special pottery clay and skilled labour. The two products have the following resource requirements for production and profit per item produced (that is, the model parameters). Resource available: 40 hours of labour per day and 120 pounds of clay per day. How many bowls and mugs should be produced to maximizing profits give these labour resources? LP formulation (to p24)

Max LP problem Step 1: define decision variables Let x1=number of bowls to produce/day x2= number of mugs to produce/day Step 2: define the objective function maximize Z = $40x1 + 50x2 where Z= profit per day Step 3: state all the resource constraints 1x1 + 2x2  40 hours of labor ( resource constraint 1) 4x1 + 3x2  120 pounds of clay (resource constraint 2) Step 4: define non-negativity constraints x10; x2  0 Complete Linear Programming Model: \ maximize Z=$40x1 + 50x2 subject to 1x1 + 2x2  40 4x2 + 3x2  120 x1, x2  0 (to p13)

Example 2: Min Z A farmer is preparing to plant a crop in the spring. There are two brands of fertilizer to choose from, Supper-gro and Crop-quick. Each brand yields a specific amount of nitrogen and phosphate, as follows: The farmer’s field requires at least 16 pounds of nitrogen and 24 pounds of phosphate. Super-gro costs $6 per bag and Crop-quick costs $3 per bag. The farmer wants to know how many bags of each brand to purchase in order to minimize the total cost of fertilizing. LP formulation (to p26)

Min Z x1  number of bags of Super-gro, Step 1: define their decision variables x1  number of bags of Super-gro, x2  number of bags of Crop-quick. Step 2: define the objective function Minimise Z  6x1  3x2 Step 3: state all the resource constraints 2x1  4x2  16, (resource 1) 4x1  3x2  24 (resource 2) Step 4: define the non-negativity constraints x1  0, x2  0 Overall LP: Minimise Z  6x1  3x2 subject to 2x1  4x2  16, 4x1  3x2  24, (to p13)