Mathematics Operations Research Lecture: Tuesday 1200 – 1400 Recitation: Tuesday1400 – 1500 Instructor: David Strimling Office: Mathematics 301 Office Hours: Tuesday, or after class by appointment Teaching Assistant: Romi Magori-Cohen

Mathematics Operations Research Syllabus

Mathematics Operations Research Course Rules You are expected to attend and participate in all lectures and recitation sessions Homework: –Due 2 weeks after assigned – no late HW will be accepted!! –Turned in at the recitation on the week it is due –Discussed at the recitation at which it is due Grades: –Comprehensive final exam – 90% –Homework – 10%

Mathematics Operations Research Lecture 1 – Definition of OR and Applications Areas Definitions History Procedure Modeling Techniques

Mathematics Operations Research Definitions 1.OR is an activity or process - something we do, rather than know - which by its very nature is applied. 2.The motivation of OR is to aid decision makers in dealing with complex real-world problems. 3.The real meaning of the OR activity is the construction and use of models.

Mathematics Operations Research History Pre-WWI ’s 1.Taylor is the “father of scientific management”. He studied manufacturing in the 1910’s. 2.Gantt worked in production scheduling in the 1910’s. 3.Frank & Lilian Gilbreth were concerned with motion study. 4.Fayol studied the organizational levels of the firm. His work complements Taylor’s. 5.Erlang studied variation of demand for telephone facilities in WWI thru Edison studied maneuvers of merchant ships to minimize shipping losses in WWI. 2.Lanchester published “Aircraft in Warfare: the Dawn of the fourth Arm” in WWII thru The RAF studied how to best use radar to locate enemy aircraft in The US studied military operations in The first mathematical technique of OR was developed by George B. Dantzig in 1946.

Five Phases of an OR Study: 1.Problem Definition - a.Description of goals or objectives. b.Identification of decision alternatives. c.Recognition of requirements and limitations. 2.Model Construction 3.Model Solution 4.Model Validation 5.Solution Implementation Mathematics Operations Research Procedure

A model is “good” if it is useful for the purpose for which it was intended. Mathematics Operations Research Models A model is an idealized representation of a real-life system. If the system exists, we use the mode to analyze its behavior to improving its performance. If the system does not exist, we use the model to define the ideal structure of the future system. Three types of models: 1.Iconic - scale model; globe, toy airplane, prototype, etc. 2.Analog - substitution of one property for another; relief maps, graphs, etc. 3.Symbolic - mathematical models, most widely used. מודל נציגות מואדרת של מערכת חיים אמיתיות. אם המערכת קיימת, אנחנו משתמשים במודוס לנתח את ההתנהגות שלו לשפר את ההצגה שלו. אם המערכת לא קיימת, אנחנו משתמשים במודל להגדיר את המבנה האידיאלי של מערכת העתיד

Mathematics Operations Research Modeling Determine the values of x i, i = 1, 2, - - -, n, that will Optimize f (x i ) Subject to the Constraints: h j (x j ) = 0, j = 1, 2, - - -, m g k (x k ) ≤ b k, k = 1, 2, - - -, p OR math models have three elements: 1.Decision Variable (DVs) — things you can control 2.Constraints — limit of control 3.Objective Function — measure of effectiveness

Mathematics Operations Research Applications Areas Resource Usage Inventory Distribution Service Conflict

Resource Usage A resource is an asset that can be used to accomplish a goal. Resources are distributed selectively between competing demands according to the each demand’s contribution to goal achievement. Decision Variables: –Resources Constraints: –Resource availability –Resource applicability Objective: –Maximize goal accomplishment

Inventory is an idle stock of items for future use. An inventory may have independent demand or dependent demand. In an independent demand inventory, the demand for an item is independent of the demands for other items in inventory. In a dependent demand inventory, the demand for an item is dependent upon the demands for other items in inventory. Decision Variables: –Quantity (how much to order) –Timing (when to order) Inventory Independent demand inventory – end-products (finished goods) Dependent demand inventories – assembly-products (components) Constraints: –Customer demand –Holding space –Budget Objective: –Minimize total inventory cost

Distribution A distribution system has “m” sources, each of which has available “a” units of a product, and “n” destinations, each of which requires “r” units of this product. There is a “cost” for transporting one unit of the product from each source to each destination. Sources Destinations Transshipment Points “Costs” Decision Variables: –Source capacities –Number of sources –TP capacity –Number of TPs Constraints: –Product availability –Destination demands –Budget Objective: –Minimize total distribution costs

A service involves customers arriving at a service facility, waiting in a line (queue) for service, receiving service, and finally leaving the facility. Service Input Source Queue Service Mechanism Service Facility Served Customers Arrival PatternFacility Capacity Number of Servers Service Pattern Service Time Decision Variables: –Facility capacity –Number of servers –Service pattern Constraints: –Arrival pattern –Service time –Budget Objectives: –Minimize total service cost –Maximize customer satisfaction

Conflict A conflict is a competition with a set of rules and payoffs. The rules define the actions available to the parties of the conflict. Different parties may have different rules, but each party has knowledge of the rules of the others. A pure strategy is a predefined sequence of actions a party will take during a conflict. A mixed strategy is a probability distribution over the set of possible pure strategies. Decision Variables: –Pure strategies –Probability distribution Constraints: –Rules of the conflict –Knowledge of others rules of the conflict Objective: –Maximize total payoff

Mathematics Operations Research Techniques Mathematical Programming Inventory Theory Network Theory Stochastic Processes Game Theory Simulation

Mathematics Operations Research Assignment 1 Identify 3 situations in you daily life that you would like to improve. Write a 2-4 sentence description of each situation. Define the decision variables, constraints, and objective(s).

Mathematical Programming  General Mathematical Programming Model  OPTIMIZEfxx S    X is an N - Dimensional Vector of DESICION VARIABLES  f is an OBJECTIVE FUNCTION  ()() {} S=: gx 0,= 1... n,hx= 0, = 1... m E Xand n i j ij  is a CONSTRAINT SET or FEASIBLE REGION in N- Dimensional Euclidean Space  ()() gx 0, hx= and i j i j  are the CONSTRAINT EQUATIONS that Define the Constraint Set  A Feasible Solution Vector, x S * , Which Optimizes the Objective Function, f, is the OPTIMUM FEASIBLE SOLUTION or Simply the OPTIMUM

DEFINITIONS  () ( ) {} N= S : d XX X, X   is an  - NEIGHBORHOOD ofX S   X A S  is a LIMIT POINT of A if ()     A N> 0 XXXX    A S  is CLOSED if it Contains Each of its Limit Points  A S  is BOUNDED if ()   > 0 A N X    A S  is COMPACT if it is Both Closed and Bounded EXAMPLES:1)X  0UNBOUNDED 2)0  X < 1NOT CLOSED 3)0  X  10COMPACT

Fundamental Theorem of Mathematical Programming Weierstrass Theorem Mathematical Statement A FUNCTION () f C X  DEFINED ON A COMPACT SET, S, HAS AN OPTIMUM, x * S  PROOF (MINIMUM CASE) () () {} Z 1 XZX fS= E : S = f    2) Every Compact Set of Real Numbers Contains its Greatest Lower Bound (GLB) ( ) () 1) i.e., Has A Compact Image on the Set of Real Numbers, 1 fS E X is Compact and X 1 S f C f : S E   () () 3) Z * GLB X * X * Z * = fS S f =    ( ) ( ) 4) ZXX * Z * X * is the Minimum x x* S, = f f = S      

A FUNCTION () f C X  DEFINED ON A COMPACT SET, S, HAS AN OPTIMUM, x * S  Feasible Region, S f(S) x x* z* Compact Image, f(S) Fundamental Theorem of Mathematical Programming Weierstrass Theorem Geometric Interpretation

 {} MAX x 2 x S : x 0    has no Solution Because S is unbounded and Therefore not Compact  {} MAX 10* x x S : 0 x< 1    has no Solution Because S is not closed And Therefore Not Compact  {} MAX xx S : 0< x 1 3    has a Solution at x = 1 Even Though S is not Compact S The Last Example Shows that the Conditions of the Weierstrass Theorem are SUFFICIENT but not NECESSARY Fundamental Theorem of Mathematical Programming Weierstrass Theorem Examples