1 Lecture 1 MGMT 650 Management Science and Decision Analysis

2Agenda nOperations Management nChapter 1 Management Science & relation to OM Quantitative Analysis and Decision Making Cost, Revenue, and Profit Models Management Science Techniques Introduction to Linear Programming

3 Operations Management The management of systems or processes that create goods and/or provide services Organization Finance Operations Marketing

4 Production of Goods vs. Delivery of Services nProduction of goods – tangible output nDelivery of services – an act nService job categories Government Wholesale/retail Financial services Healthcare Personal services Business services Education

5 nOperations Management includes: Forecasting Capacity planning Scheduling Managing inventories Assuring quality Deciding where to locate facilities And more... nThe operations function Consists of all activities directly related to producing goods or providing services Scope of Operations Management

6 Management Science nThe body of knowledge involving quantitative approaches to decision making is referred to as Management Science Operations research Decision science nIt had its early roots in World War II and is flourishing in business and industry with the aid of computers

7 nSteps of Problem Solving (First 5 steps are the process of decision making) Define the problem. Identify the set of alternative solutions. Determine the criteria for evaluating alternatives. Evaluate the alternatives. Choose an alternative (make a decision) Implement the chosen alternative. Evaluate the results. Problem Solving and Decision Making

8 Quantitative Analysis and Decision Making nPotential Reasons for a Quantitative Analysis Approach to Decision Making The problem is complex The problem is very important The problem is new The problem is repetitive

9 Models A model is an abstraction of reality. – Physical – Schematic – Mathematical What are the pros and cons of models? Tradeoffs

10 A Simulation Model

11 Models Are Beneficial  Easy to use, less expensive  Require users to organize  Systematic approach to problem solving  Increase understanding of the problem  Enable “what if” questions: simulation models  Specific objectives  Power of mathematics  Standardized format

12 Quantitative Approaches Linear programming: optimal allocation of resources Queuing Techniques: analyze waiting lines Inventory models: management of inventory Project models: planning, coordinating and controlling large scale projects Statistical models: forecasting

13 Analysis of Trade-offs  How many more jeans would Levi need to sell to justify the cost of additional robotic tailors?  Cost of additional robotic tailors vs Inventory Holding Cost

14 Product Mix Example Type 1Type 2 Profit per unit $60$50 Assembly time per unit 4 hrs10 hrs Inspection time per unit 2 hrs1 hr Storage space per unit 3 cubic ft ResourceAmount available Assembly time100 hours Inspection time22 hours Storage space39 cubic feet

15  Objective – profit maximization Maximize 60X X 2  Subject to Assembly 4X X 2 <= 100 hours Inspection 2X 1 + 1X 2 <= 22 hours Storage3X 1 + 3X 2 <= 39 cubic feet X 1, X 2 >= 0 A Linear Programming Model

16 Cost, Revenue and Profit Models

17 Cost Classification of Owning and Operating a Passenger Car

18 Cost-Volume Relationship

19 Cost-Volume Relationship

20 Cost-Volume Relationships Amount ($) 0 Q (volume in units) Total cost = VC + FC Total variable cost (VC) Fixed cost (FC) Amount ($) Q (volume in units) 0 Total revenue

21 Cost-Volume Relationships Amount ($) Q (volume in units) 0 BEP units Profit Total revenue Total cost

22 Example: Ponderosa Development Corp. nPonderosa Development Corporation (PDC) is a small real estate developer that builds only one style house. nThe selling price of the house is $115,000. nLand for each house costs $55,000 and lumber, supplies, and other materials run another $28,000 per house. Total labor costs are approximately $20,000 per house. nPonderosa leases office space for $2,000 per month. The cost of supplies, utilities, and leased equipment runs another $3,000 per month. nThe one salesperson of PDC is paid a commission of $2,000 on the sale of each house. PDC has seven permanent office employees whose monthly salaries are given on the next slide.

23 Employee Monthly Salary President $10,000 VP, Development 6,000 VP, Marketing 4,500 Project Manager 5,500 Controller 4,000 Office Manager 3,000 Receptionist 2,000 Example: Ponderosa Development Corp.

24 nIdentify all costs and denote the marginal cost and marginal revenue for each house. nWrite the monthly cost function c (x), revenue function r (x), and profit function p (x). nWhat is the breakeven point for monthly sales of the houses? nWhat is the monthly profit if 12 houses per month are built and sold? nDetermine the BEP for monthly sale of houses graphically. Example: Ponderosa Development Corp.

25 Graph of costs and Break-even point

26 Example: Step Fixed Costs nA manager has the option of purchasing 1, 2 or 3 machines nFixed costs and potential volumes are as follows: nVariable cost = $10/unit and revenue = $40/unit nIf the projected annual demand is between 580 and 630 units, how many machines should the manager purchase? # of machinesTotal annual FC ($)Range of output – – – 900

27 Break-Even Problem with Step Fixed Costs Quantity FC + VC = TC Step fixed costs and variable costs. 1 machine 2 machines 3 machines Total Revenue BEVs Total Cost

28 1. One product is involved 2. Everything produced can be sold 3. Variable cost per unit is the same regardless of volume 4. Fixed costs do not change with volume 5. Revenue per unit constant with volume 6. Revenue per unit exceeds variable cost per unit Assumptions of Cost-Volume Analysis

29 Three locations: Akron$30,000$75$180,000 Bowling Green$60,000$45$150,000 Chicago$110,000$25$160,000 FixedVariableTotal CityCostCostCost Total Cost = Fixed Cost + Variable Cost x Volume Cost-Volume & Locational Break-Even Analysis

30 – $180,000 $180,000 – – $160,000 $160,000 – $150,000 $150,000 – – $130,000 $130,000 – – $110,000 $110,000 – – $80,000 $80,000 – – $60,000 $60,000 – – $30,000 $30,000 – – $10,000 $10,000 – – Annual cost ||||||| 05001,0001,5002,0002,5003,000 Volume Akron lowest cost Bowling Green lowest cost Chicago lowest cost Chicago cost curve Akron cost curve Bowling Green cost curve Locational Break-Even Analysis Graph of Break-Even Points

31 Linear Programming – Chapter 2 George Dantzig –

32  Concerned with optimal allocation of limited resources such as  Materials  Budgets  Labor  Machine time  among competitive activities  under a set of constraints Linear Programming

33 Maximize 60X X 2 Subject to 4X X 2 <= 100 2X 1 + 1X 2 <= 22 3X 1 + 3X 2 <= 39 X 1, X 2 >= 0 Linear Programming Example Variables Objective function Constraints What is a Linear Program? A LP is an optimization model that has continuous variables a single linear objective function, and (almost always) several constraints (linear equalities or inequalities) Non-negativity Constraints

34  Decision variables  unknowns, which is what model seeks to determine  for example, amounts of either inputs or outputs  Objective Function  goal, determines value of best (optimum) solution among all feasible (satisfy constraints) values of the variables  either maximization or minimization  Constraints  restrictions, which limit variables of the model  limitations that restrict the available alternatives  Parameters: numerical values (for example, RHS of constraints)  Feasible solution: is one particular set of values of the decision variables that satisfies the constraints  Feasible solution space: the set of all feasible solutions  Optimal solution: is a feasible solution that maximizes or minimizes the objective function  There could be multiple optimal solutions Linear Programming Model

35 Another Example of LP: Diet Problem  Energy requirement : 2000 kcal  Protein requirement : 55 g  Calcium requirement : 800 mg FoodEnergy (kcal)Protein(g)Calcium(mg) Price per serving($) Oatmeal Chicken Eggs Milk Pie Pork

36 Example of LP : Diet Problem  oatmeal: at most 4 servings/day  chicken: at most 3 servings/day  eggs: at most 2 servings/day  milk: at most 8 servings/day  pie:at most 2 servings/day  pork: at most 2 servings/day Design an optimal diet plan which minimizes the cost per day

37 Step 1: define decision variables  x 1 = # of oatmeal servings  x 2 = # of chicken servings  x 3 = # of eggs servings  x 4 = # of milk servings  x 5 = # of pie servings  x 6 = # of pork servings Step 2: formulate objective function In this case, minimize total cost minimize z = 3x x x 3 + 9x x x 6

38 Step 3: Constraints  Meet energy requirement 110x x x x x x 6  2000  Meet protein requirement 4x x x 3 + 8x 4 + 4x x 6  55  Meet calcium requirement 2x x x x x x 6  800  Restriction on number of servings 0  x 1  4, 0  x 2  3, 0  x 3  2, 0  x 4  8, 0  x 5  2, 0  x 6  2

39 So, how does a LP look like? minimize 3x x x 3 + 9x x x 6 subject to 110x x x x x x 6  x x x 3 + 8x 4 + 4x x 6  55 2x x x x x x 6   x 1  4 0  x 2  3 0  x 3  2 0  x 4  8 0  x 5  2 0  x 6  2

40 Optimal Solution – Diet Problem Using LINDO 6.1  Cost of diet = $96.50 per day Food# of servings Oatmeal4 Chicken0 Eggs0 Milk6.5 Pie0 Pork2

41 Optimal Solution – Diet Problem Using Management Scientist  Cost of diet = $96.50 per day Food# of servings Oatmeal4 Chicken0 Eggs0 Milk6.5 Pie0 Pork2

42 Guidelines for Model Formulation  Understand the problem thoroughly.  Describe the objective.  Describe each constraint.  Define the decision variables.  Write the objective in terms of the decision variables.  Write the constraints in terms of the decision variables  Do not forget non-negativity constraints

43 Graphical Solution of LPs  Consider a Maximization Problem Max 5x 1 + 7x 2 s.t. x 1 < 6 2x 1 + 3x 2 < 19 x 1 + x 2 < 8 x 1, x 2 > 0

44 Slide © 2005 Thomson/South-Western Graphical Solution Example n Constraint #1 Graphed x 2 x 2 x1x1x1x1 x 1 < 6 (6, 0)

45 Slide © 2005 Thomson/South-Western Graphical Solution Example n Constraint #2 Graphed 2 x x 2 < 19 x 2 x 2 x1x1x1x1 (0, 6 1/3 ) (9 1/2, 0)

46 Slide © 2005 Thomson/South-Western Graphical Solution Example n Constraint #3 Graphed x 2 x 2 x1x1x1x1 x 1 + x 2 < 8 (0, 8) (8, 0)

47 Slide © 2005 Thomson/South-Western Graphical Solution Example n Combined-Constraint Graph 2 x x 2 < 19 x 2 x 2 x1x1x1x1 x 1 + x 2 < 8 x 1 <

48 Slide © 2005 Thomson/South-Western Graphical Solution Example n Feasible Solution Region x1x1x1x1 FeasibleRegion x 2 x 2

49 Slide © 2005 Thomson/South-Western Graphical Solution Example n Objective Function Line x1x1x1x1 x 2 x 2 (7, 0) (0, 5) Objective Function 5 x 1 + 7x 2 = 35 Objective Function 5 x 1 + 7x 2 = 35

50 Slide © 2005 Thomson/South-Western Graphical Solution Example n Optimal Solution x1x1x1x1 x 2 x 2 Objective Function 5 x 1 + 7x 2 = 46 Objective Function 5 x 1 + 7x 2 = 46 Optimal Solution ( x 1 = 5, x 2 = 3) Optimal Solution ( x 1 = 5, x 2 = 3)

51 1. Set up objective function and constraints in mathematical format 2. Plot the constraints 3. Identify the feasible solution space 4. Plot the objective function 5. Determine the optimum solution Graphical Linear Programming

52 Product Mix Problem Floataway Tours has $420,000 that can be used to purchase new rental boats for hire during the summer. The boats can be purchased from two different manufacturers. Floataway Tours would like to purchase at least 50 boats. They would also like to purchase the same number from Sleekboat as from Racer to maintain goodwill. At the same time, Floataway Tours wishes to have a total seating capacity of at least 200. Formulate this problem as a linear program

53 Maximum Expected Daily Boat Builder Cost Seating Profit Speedhawk Sleekboat $ $ 70 Silverbird Sleekboat $ $ 80 Catman Racer $ $ 50 Classy Racer $ $110 Product Mix Problem

54  Define the decision variables x 1 = number of Speedhawks ordered x 2 = number of Silverbirds ordered x 3 = number of Catmans ordered x 4 = number of Classys ordered  Define the objective function Maximize total expected daily profit: Max: (Expected daily profit per unit) x (Number of units) Max: 70x x x x 4 Product Mix Problem

55  Define the constraints (1) Spend no more than $420,000: 6000x x x x 4 < 420,000 (2) Purchase at least 50 boats: x 1 + x 2 + x 3 + x 4 > 50 (3) Number of boats from Sleekboat equals number of boats from Racer: x 1 + x 2 = x 3 + x 4 or x 1 + x 2 - x 3 - x 4 = 0 (4) Capacity at least 200: 3x 1 + 5x 2 + 2x 3 + 6x 4 > 200 Nonnegativity of variables: x j > 0, for j = 1,2,3,4 Product Mix Problem

56 Max 70x x x x 4 s.t. 6000x x x x 4 < 420,000 x 1 + x 2 + x 3 + x 4 > 50 x 1 + x 2 - x 3 - x 4 = 0 3x 1 + 5x 2 + 2x 3 + 6x 4 > 200 x 1, x 2, x 3, x 4 > 0 Product Mix Problem - Complete Formulation   Daily profit = $5040 Boat# purchased Speedhawk28 Silverbird0 Catman0 Classy28

57 Applications of LP  Product mix planning  Distribution networks  Truck routing  Staff scheduling  Financial portfolios  Capacity planning  Media selection: marketing