Graduate Program in Business Information Systems LNEAR PROGRAMMNG

Graduate Program in Business Information Systems LİNEAR PROGRAMMİNG AND APPLİCATİONS Aslı Sencer

OPERATIONS RESEARCH 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. BIS 517 -Aslı Sencer

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

Some news about OR: 4 A Wall Street Journal Article lists the use of LP as one of the greatest technological innovations of the past 1000 years. 1975 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. BIS 517 -Aslı Sencer

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

LINEAR PROGRAMMING 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! BIS 517 -Aslı Sencer

Basic LP Models: Product Mix 7 Redwood Furniture Co. Unit Requirements Table 30 Chair 20 Amount Available in a Period 300 Labor (hrs) 5 10 110 Unit profit $6 $8 Resource Wood (ft) BIS 517 -Aslı Sencer

What is the optimal plan to max. Profit? 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 BIS 517 -Aslı Sencer

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

How is an LP solved? 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. BIS 517 -Aslı Sencer

Graphical Solution 11 Xc 15 Optimal Solution: Xt=4 tables, Xc=9 Chairs Profit*=$96 Constraint 1 11 (4, 9) Constraint 2 Xt 10 BIS 517 -Aslı Sencer 22

Basic LP Models: Feed Mix 12 Two types of seeds are mixed to formulate the wheat of wild birdseed. Proportional Content Buckwheat Sunflower wheat Total Requirement Fat . 04 . 06 ≥ 480 lb Protein . 12 . 10 ≥ 1200 lb Roughage . 10 . 15 ≤ 1500 Cost per lb $. 18 $. 10 Nutritional Item BIS 517 -Aslı Sencer

LP Formulation 13 BIS 517 -Aslı Sencer

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

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

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

Cases in an LP: Multiple Optima 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 Infinite number of optimal solutions exist in the form (1) (2) BIS 517 -Aslı Sencer

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

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

Solution in the Excel Solver 20 BIS 517 -Aslı Sencer

Applications of LP: Transportation Models 21 Sporting goods company Capacity Plants Warehouses Demand Frankfurt 150 Seoul NY 100 Tel Aviv Phoenx 200 Yokohama 150 100 Juarez 300 200 BIS 517 -Aslı Sencer

LP: Transportation Models (cont’d. ) 22 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? Shipping Costs per pair of skis Destination Frankfurt NY Phoenix Yokohama Juarez $19 $7 $3 $21 Seoul 15 21 18 6 Tel Aviv 11 14 15 22 From Plant BIS 517 -Aslı Sencer

LP: Transportation Models (cont’d. ) 23 Xij: Number of units shipped from plant i to warehouse j. i=1, 2, 3 and j=1, 2, 3, 4. Minimize shipping costs=19 X 11+7 X 12+3 X 13+21 X 14 +15 X 21+21 X 22+18 X 23+6 X 24 +11 X 31+14 X 32+15 X 33+22 X 34 From Plant Destination Frankfurt NY Phoenix Yokohama Capacity Juarez X 11 X 12 X 13 X 14 100 Seoul X 21 X 22 X 23 X 24 300 Tel Aviv X 31 X 32 X 33 X 34 200 Demand 150 100 200 150 600 BIS 517 -Aslı Sencer

LP: Transportation Models (cont’d. ) 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 Xij ≥ 0 for all i, j. BIS 517 -Aslı Sencer

LP: Transportation Models (cont’d. ) 25 Optimal Solution: Optimal cost=$6, 250 Capacity Warehouses Plants 100 Juarez 50 Demand Frankfurt 150 NY 100 Phoenx 200 Yokohama 150 100 300 Seoul 100 200 Tel Aviv 100 150 BIS 517 -Aslı Sencer

LP: Marketing Applications 26 How to allocate advertising budget between mediums such as TV, radio, billboard or magazines? Ex: Real Reels Co. Allocated ad. Budget=$100, 000 Playboy True Esquire 10 million 6 million 4 million Significant Buyers 10% 15% 7% Cost per ad $10, 000 $5, 000 $6, 000 1, 000 900, 000 280, 000 Readers Exposures per ad • No more than 5 ads in True and at least two ads in Playboy and Esquire BIS 517 -Aslı Sencer

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

LP: Assignment Models 28 Assignment of a set of workers to a set of jobs Time required to complete one job Drilling 5 min Grinding 10 min Lathe 10 min Bud 10 5 15 Chuck 15 15 10 Individual Ann BIS 517 -Aslı Sencer

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

LP: Diet Problem 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 BIS 517 -Aslı Sencer

LP: Labor Planning 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 11 a. m. and 1 p. 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. BIS 517 -Aslı Sencer

LP: Labor Planning (Cont’d. ) 32 Time Period Minimum labor required 9 a. m. -10 a. m. 10 10 a. m. -11 a. m. 12 11 a. m. -noon 14 Noon-1 p. m. -2 p. m. -3 p. m. -4 p. m. -5 p. m. BIS 517 -Aslı Sencer 16 18 17 15 10

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