Chapter 1 Introduction n Introduction: Problem Solving and Decision Making n Quantitative Analysis and Decision Making n Quantitative Analysis n Model Development n Software Packages

Operations Research n n The body of knowledge involving quantitative approaches to decision making is referred to as Management Science Operations Research Decision Science n n It had its early roots in World War II and is flourishing in business and industry due, in part, to: numerous methodological developments (e.g. simplex method for solving linear programming problems) a virtual explosion in computing power

Quantitative Analysis and Decision Making Define the Problem Define the Problem Identify the Alternatives Identify the Alternatives Determine the Criteria Determine the Criteria Identify the Alternatives Identify the Alternatives Choose an Alternative Choose an Alternative Structuring the Problem Analyzing the Problem n n Decision-Making Process

Analysis Phase of Decision-Making Process n n Qualitative Analysis based largely on the manager’s judgment and experience includes the manager’s intuitive “ feel ” for the problem is more of an art than a science Quantitative Analysis and Decision Making

Analysis Phase of Decision-Making Process n n Quantitative Analysis analyst will concentrate on the quantitative facts or data associated with the problem analyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problem analyst will use one or more quantitative methods to make a recommendation Quantitative Analysis and Decision Making

n Potential 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.

Quantitative Analysis n Quantitative Analysis Process Model Development Data Preparation Model Solution Report Generation

Model Development n Models are representations of real objects or situations n Three forms of models are: Iconic models - physical replicas (scalar representations) of real objects Analog models - physical in form, but do not physically resemble the object being modeled Mathematical models - represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses

Advantages of Models n Generally, experimenting with models (compared to experimenting with the real situation): requires less time is less expensive involves less risk n The more closely the model represents the real situation, the accurate the conclusions and predictions will be.

Example: Production Problem (1) The total profit from the sale of a product can be determined by multiplying the profit per unit by the quantity sold. If we let x represent the number of units sold and P the total profit, then, with a profit of $10 per unit, the following mathematical model defines the total profit earned by selling x units The above mathematical expression describes the problem’s objective and the firm will attempt to maximize profit.

A production capacity constraint would be necessary if, for instance, 5 hours are required to produce each unit and only 40 hours of production time are available per week. The production time constraint is given by The value of 5x is the total time required to produce x units; the symbol ≤ indicates that the production time required must be less than or equal to the 40 hours available Example: Production Problem (2)

The decision problem or question is the following: How many units of the product should be scheduled each week to maximize profit? A complete mathematical model for this simple production problem is This model is an example of a linear programming model. Example: Production Problem (3)

Mathematical Models n Objective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost n Constraints – a set of restrictions or limitations, such as production capacities n Uncontrollable Inputs – environmental factors that are not under the control of the decision maker n Decision Variables – controllable inputs; decision alternatives specified by the decision maker, such as the number of units of Product X to produce

Mathematical Models n n Deterministic Model – if all uncontrollable inputs to the model are known and cannot vary n n Stochastic (or Probabilistic) Model – if any uncontrollable are uncertain and subject to variation n n Stochastic models are often more difficult to analyze. n n Frequently a less complicated (and perhaps less precise) model is more appropriate than a more complex and accurate one due to cost and ease of solution considerations.

Transforming Model Inputs into Output Uncontrollable Inputs (Environmental Factors) Uncontrollable Inputs (Environmental Factors) Controllable Inputs (Decision Variables) Controllable Inputs (Decision Variables) Output (Projected Results) Output (Projected Results) Mathematical Model Mathematical Model

Flowchart for the Production Model

Example: Iron Works, Inc. Iron Works, Inc. manufactures two products made from steel and just received this month's allocation of b pounds of steel. It takes a 1 pounds of steel to make a unit of product 1 and a 2 pounds of steel to make a unit of product 2. Let x 1 and x 2 denote this month's production level of product 1 and product 2, respectively. Denote by p 1 and p 2 the unit profits for products 1 and 2, respectively. Iron Works has a contract calling for at least m units of product 1 this month. The firm's facilities are such that at most u units of product 2 may be produced monthly.

Example: Iron Works, Inc. n Mathematical Model The total monthly profit = (profit per unit of product 1) x (monthly production of product 1) + (profit per unit of product 2) x (monthly production of product 2) = p 1 x 1 + p 2 x 2 We want to maximize total monthly profit: Max p 1 x 1 + p 2 x 2

Example: Iron Works, Inc. n Mathematical Model (continued) The total amount of steel used during monthly production equals: (steel required per unit of product 1) x (monthly production of product 1) + (steel required per unit of product 2) x (monthly production of product 2) = a 1 x 1 + a 2 x 2 This quantity must be less than or equal to the allocated b pounds of steel: a 1 x 1 + a 2 x 2 < b

Example: Iron Works, Inc. n Mathematical Model (continued) The monthly production level of product 1 must be greater than or equal to m : x 1 > m The monthly production level of product 2 must be less than or equal to u : x 2 < u However, the production level for product 2 cannot be negative: x 2 > 0

Example: Iron Works, Inc. n Mathematical Model Summary Max p 1 x 1 + p 2 x 2 s.t. a 1 x 1 + a 2 x 2 < b x 1 > m x 2 < u x 2 > 0 Objectiv e Function “Subject to” Constraint s

Example: Iron Works, Inc. n Question: Suppose b = 2000, a 1 = 2, a 2 = 3, m = 60, u = 720, p 1 = 100, p 2 = 200. Rewrite the model with these specific values for the uncontrollable inputs.

Example: Iron Works, Inc. n Answer: Substituting, the model is: Max 100x x 2 s.t. 2x 1 + 3x 2 < 2000 x 1 > 60 x 2 < 720 x 2 > 0

Example: Iron Works, Inc. n Question: The optimal solution to the current model is x 1 = 60 and x 2 = 626 2/3. If the product were engines, explain why this is not a true optimal solution for the "real-life" problem. n Answer: One cannot produce and sell 2/3 of an engine. Thus the problem is further restricted by the fact that both x 1 and x 2 must be integers. (They could remain fractions if it is assumed these fractions are work in progress to be completed the next month.)

Example: Iron Works, Inc. Uncontrollable Inputs $100 profit per unit Prod. 1 $200 profit per unit Prod. 2 2 lbs. steel per unit Prod. 1 3 lbs. Steel per unit Prod lbs. steel allocated 60 units minimum Prod units maximum Prod. 2 0 units minimum Prod. 2 $100 profit per unit Prod. 1 $200 profit per unit Prod. 2 2 lbs. steel per unit Prod. 1 3 lbs. Steel per unit Prod lbs. steel allocated 60 units minimum Prod units maximum Prod. 2 0 units minimum Prod units Prod units Prod units Prod units Prod units Prod units Prod. 2 Controllable Inputs Profit = $131, Steel Used = 2000 Profit = $131, Steel Used = 2000 OutputOutput Max 100(60) + 200(626.67) s.t. 2(60) + 3(626.67) < > > < < > > 0 Max 100(60) + 200(626.67) s.t. 2(60) + 3(626.67) < > > < < > > 0 Mathematical Model

Linear Programming (LP) Problems MAX (or MIN): c 1 X 1 + c 2 X 2 + … + c n X n Subject to:a 11 X 1 + a 12 X 2 + … + a 1n X n <= b 1 : a k1 X 1 + a k2 X 2 + … + a kn X n >=b k : a m1 X 1 + a m2 X 2 + … + a mn X n = b m

An Example LP Problem Blue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes. There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available. Aqua-SpaHydro-Lux Pumps11 Labor 9 hours 6 hours Labor 9 hours 6 hours Tubing12 feet16 feet Unit Profit$350$300

Steps In Formulating LP Models: 1. Understand the problem. 2. Identify the decision variables. X 1 =number of Aqua-Spas to produce X 2 =number of Hydro-Luxes to produce 3.State the objective function as a linear combination of the decision variables. MAX: 350X X 2

Steps In Formulating LP Models (continued) 4. State the constraints as linear combinations of the decision variables. 1X 1 + 1X 2 <= 200} pumps 9X 1 + 6X 2 <= 1566} labor 12X X 2 <= 2880} tubing 5. Identify any upper or lower bounds on the decision variables. X 1 >= 0 X 2 >= 0

LP Model for Blue Ridge Hot Tubs MAX: 350X X 2 S.T.:1X 1 + 1X 2 <= 200 9X 1 + 6X 2 <= X X 2 <= 2880 X 1 >= 0 X 2 >= 0 X 2 >= 0

Solving LP Problems: An Intuitive Approach n Idea: Each Aqua-Spa (X 1 ) generates the highest unit profit ($350), so let’s make as many of them as possible! n How many would that be? Let X 2 = 0 1st constraint:1X 1 <= 200 2nd constraint:9X 1 <=1566 or X 1 <=174 3rd constraint:12X 1 <= 2880 or X 1 <= 240 n If X 2 =0, the maximum value of X 1 is 174 and the total profit is $350*174 + $300*0 = $60,900 n This solution is feasible, but is it optimal? n No!