LINEAR PROGRAMMING AND APPLICATIONS Graduate Program in Business Information Systems Aslı Sencer

OPERATIONS RESEARCH BIS 517-Aslı Sencer 2 What is Operations Research?  Collection of techniques used to  allocate the scarce resources  in the “best” –OPTIMAL – way! Best of what?  We need an “objective” to be minimized or maximized!  Profit, Cost, Utility, Delay, Distance, Flow, etc.

Some applications BIS 517-Aslı Sencer 3 Resource allocation Production and inventory planning Capacity Planning Workers and machine scheduling Investment planning Formulating marketing and military strategies

Some news about OR: BIS 517-Aslı Sencer 4 A Wall Street Journal Article lists the use of LP as one of the greatest technological innovations of the past 1000 years Nobel Prize for economics: T.C. Koopmans and L.V. Kantoprovich for their contributions in the field 1992 Nobel Prize: Harry Markowitz for his LP based research.

Basic Optimization Techniques BIS 517-Aslı Sencer 5 Linear Programming Integer and Goal Programming  Transportation, Assignment Models  Network Models Nonlinear Programming Stochastic Programming Also Simulation

LINEAR PROGRAMMING BIS 517-Aslı Sencer 6 Most successful of all modern quantitative methods. Program here is not a computer code! It is a plan that efficiently allocate limited resources to achive a goal. Involves linear relationships, i.e. relations are in the form of lines, planes!

Basic LP Models: Product Mix Redwood Furniture Co. Resource Unit Requirements Amount Available in a Period TableChair Wood (ft) Labor (hrs) Unit profit$6$8 BIS 517-Aslı Sencer 7

What is the optimal plan to max. Profit? BIS 517-Aslı Sencer 8 Option 1: Allocate all resources to the more profitable item. Total quantity, profit? Any resource left? Option 2: Is it more profitable to produce less chairs and more tables? Linear Programming

Formulating a Linear Problem Define variables: : number of tables produced in a period : number of chairs produced in a period Define constraints: Define Objective Function BIS 517-Aslı Sencer 9

How is an LP solved? BIS 517-Aslı Sencer 10 Graphical Method: Applicable to a maximum of two decision variables. Simplex Method: Applicable to all LP. Takes long to implement manually. Use softwares based on simplex and other techniques.

Graphical Solution BIS 517-Aslı Sencer 11 Constraint 1 Constraint Optimal Solution: Xt=4 tables, Xc=9 Chairs Profit*=$96 (4,9) XtXt XcXc

Basic LP Models: Feed Mix Two types of seeds are mixed to formulate the wheat of wild birdseed. Nutritional Item Proportional Content Total Requirement Buckwheat Sunflower wheat Fat.04.06≥480 lb Protein.12.10≥1200 lb Roughage.10.15≤1500 Cost per lb$.18$.10 BIS 517-Aslı Sencer 12

LP Formulation BIS 517-Aslı Sencer 13

Graphical Solution to Feed Mix Problem BIS 517-Aslı Sencer 14 Fat Roughage Protein ( 15000,0 ) (10000,0) (3750,7500) XbXb XsXs Optimal Solution: X b *=3750 lb, X s *=7500 lb Cost*=$1425

Types of Feasible Regions BIS 517-Aslı Sencer 15 Bounded Feasible Region Unbounded Feasible Region

Cases in an LP: Infeasible Solution BIS 517-Aslı Sencer 16 No feasible region If an LP has no feasible region, then the solution is INFEASIBLE! Do all LP has an optimal Solution?

Cases in an LP: Multiple Optima BIS 517-Aslı Sencer 17 Optimal Solutions: Point(1): X 1 *=4 2/7, X 2 *=6 3/7 Point(2): X 1 *=6 6/7, X 2 *=4 2/7 P*=$120 (1) (2) Infinite number of optimal solutions exist in the form

Cases in an LP: Unbounded Optimal Solution BIS 517-Aslı Sencer 18 Optimal Solution: X 1 *=, X 2 *= P*=

Solving Linear Programs with a Spreadsheet BIS 517-Aslı Sencer 19 Write out the formulation table Put the formulation table into a spreadsheet Use Excel’s Solver to obtain a solution

Solution in the Excel Solver BIS 517-Aslı Sencer 20

Applications of LP:Transportation Models BIS 517-Aslı Sencer 21 Sporting goods company Capacity Plants Warehouses Demand Juarez Seoul Tel Aviv Yokohama Phoenx NY Frankfurt

LP:Transportation Models (cont’d.) From Plant Destination FrankfurtNYPhoenixYokohama Juarez $19$7$3$21 Seoul Tel Aviv BIS 517-Aslı Sencer 22 Shipping Costs per pair of skis What are the optimal shipping quantities from the plants to the warehouses, if the demand has to be met by limited capacities while the shipping cost is minimized?

LP:Transportation Models (cont’d.) X ij : Number of units shipped from plant i to warehouse j. i=1,2,3 and j=1,2,3,4. Minimize shipping costs=19X 11 +7X 12 +3X X X X X 23 +6X X X X X 34 From Plant Destination Capacity FrankfurtNYPhoenixYokohama JuarezX11X12X13X14100 SeoulX21X22X23X24300 Tel AvivX31X32X33X34200 Demand BIS 517-Aslı Sencer 23

LP:Transportation Models (cont’d.) BIS 517-Aslı Sencer 24 subject to #shipped from a plant can not exceed the capacity: X 11 +X 12 +X 13 +X 14 ≤ 100 (Juarez Plant) X 21 +X 22 +X 23 +X 24 ≤ 300 (Seoul Plant) X 31 +X 32 +X 33 +X 34 ≤ 200 (Tel Aviv Plant) #shipped to a warehouse can not be less than the demand: X 11 +X 21 +X 31 ≥ 150 (Frankfurt) X 12 +X 22 +X 32 ≥ 100 (NY) X 13 +X 23 +X 33 ≥ 200 (Phoenix) X 14 +X 24 +X 34 ≥ 150 (Yokohama) Nonnegativity X ij ≥0 for all i,j.

LP:Transportation Models (cont’d.) BIS 517-Aslı Sencer 25 Capacity Plants Warehouses Demand Juarez Seoul Tel Aviv Yokohama Phoenx NY Frankfurt Optimal Solution: Optimal cost=$6,

LP: Marketing Applications How to allocate advertising budget between mediums such as TV, radio, billboard or magazines? Ex: Real Reels Co. Allocated ad. Budget=$100,000 PlayboyTrueEsquire Readers10 million6 million4 million Significant Buyers 10%15%7% Cost per ad$10,000$5,000$6,000 Exposures per ad 1,000,000900,000280,000 BIS 517-Aslı Sencer 26 No more than 5 ads in True and at least two ads in Playboy and Esquire

LP: Marketing Applications (cont’d.) BIS 517-Aslı Sencer 27 Not integer?

LP: Assignment Models Assignment of a set of workers to a set of jobs Individual Time required to complete one job DrillingGrindingLathe Ann5min10min Bud10515 Chuck15 10 BIS 517-Aslı Sencer 28

LP: Assignment Models (cont’d.) BIS 517-Aslı Sencer 29

LP: Diet Problem BIS 517-Aslı Sencer 30 How much to use of each ingredient so that the nutritional requirements are met in the cheapest way? Ex: Feed Mix problem given at the beginning of the lecture

LP:Labor Planning BIS 517-Aslı Sencer 31 Addresses staffing needs over a specific time period. Hong Kong Bank of Commerce:  12 Full time workers available, but may fire some.  Use part time workers who has to work for 4 consequtive hours in a day.  Luch time is one hour between 11a.m. and 1p.m. shared by full time workers.  Total part time hours is less than 50% of the day’s total requirement.  Part-timers earn $4/hr (=$16/day) and full timers earn $50/day.

LP:Labor Planning (Cont’d.) Time PeriodMinimum labor required 9a.m.-10a.m.10 10a.m.-11a.m.12 11a.m.-noon14 Noon-1p.m.16 1p.m.-2p.m.18 2p.m.-3p.m.17 3p.m.-4p.m.15 4p.m.-5p.m.10 BIS 517-Aslı Sencer 32

LP:Labor Planning (cont’d.) BIS 517-Aslı Sencer 33