Graduate Program in Business Information Systems Integer and Goal Programming Aslı Sencer

BIS 517-Aslı Sencer2 Integer Program Structurally like the LP models where the solutions must be integers! Ex: Production order quantities Number of items sent from plants to warehouses Binary decisions produce or do not produce Assign staff i to job j or not Locate facility i in location j or not

BIS 517-Aslı Sencer3 Ex:Redwood Furniture Model (revisited) Optimal solution:

BIS 517-Aslı Sencer4 The Branch and Bound Method An effective search procedure that involves solving a succession of carefully formulated LP’s. If the optimal solution of the parent program includes fractions, then two descendent LP’s are developed and solved to force the noninteger variable to be an integer. The optimal objective function of the parent program constitues a bound on the optimal objective function value of the descendent LP’s.

BIS 517-Aslı Sencer5 Graphical Solution Problem1 Problem XtXt XcXc 45 Problem 3

BIS 517-Aslı Sencer 6 Branch and Bound Solution Problem 1 Xt=4.2 Xc=9.2 P=98.8 Problem 2 Xt=4 Xc=9.3 P=98.4 Problem 3 Xt=5 Xc=8 P=94 Problem 4 Xt=4 Xc=9 P=96 Problem 5 Xt=2.6 Xc=10 P=95.6 OPTIMAL SOLUTION

BIS 517-Aslı Sencer7 When do we round off the LP? For problems having small number of variables, IP is solved in a short time. Bigger problems require a lot of time. If no integer solution has been found, a rounded LP may be used. Which solution to be used is judgemental.

BIS 517-Aslı Sencer8 Solving Integer Programs with a Spreadsheet When using Excel Spreadsheet, it is sufficient to add a new constraint and define the decision variables as integers. You can also use WinQSB or QuickQuant to solve LP and IP problems.

BIS 517-Aslı Sencer9 Ex: Maui Miser Car Rentals Budget $500,000. Three types of cars: Economy vans, compact cars, large cars Income/vehicle: $15,000, $7,600, $10,600 Costs/vehicle: $25,000, $15,000, $21,000 At least 25 vehicles At least 5 vans, at least 5 large cars, at least 12 passenger cars # large cars can not exceed the #compacts Average daily rental rate should be less than $39 where Daily rental prices/vehicle: $50, $30, $40

BIS 517-Aslı Sencer10 Ex: Formulation

BIS 517-Aslı Sencer11 Optimal Solution

BIS 517-Aslı Sencer12 Goal Programming Maximize profit OR Minimize cost Single Goal Sometimes we have Multiple Goals. Minimize Cost AND Maximize Customer Satisfaction Maximize Return AND Minimize Risk What is the optimal balance in attaining the conflicting goals?

BIS 517-Aslı Sencer13 Basic Idea It is impossible to meet all goals simultaneously. The focus aims at achieving certain targets for each goal. The overall objective is to find the solution that collectively minimizes deviations from these targets.

BIS 517-Aslı Sencer14 Ex:Goal Programming in Redwood Furniture Co. Objectives Maximize Profit Maximize Revenue to maintain marketshare growth Maximize training time to increase work-force productivity TableChair Profit/unit$6$8 Revenue/unit$50$25 Train. time/unit1hr.3hr. Each LP has a different optimal solution. An idea is to set targets for the goals. Target $90 $450 30hr.

BIS 517-Aslı Sencer15 LP with Goal Targets Warning: When these constraints are added to the Redwood Product Mix Formulation there may be no feasible solution satisfying the minimum requirements of all goals! What should we do then? Any comments? Answer: Try to lower the targets Idea 1: We can treat targets as the lower bounds on the goals

BIS 517-Aslı Sencer16 Goal Deviations Idea 2: We can allow some goals to be partially met! Compared to LP, goal programming is flexible. Optimal balance between goals can be achieved in several ways. Goal Deviation: Amount of deviation from the goal target is a variable to be minimized.

BIS 517-Aslı Sencer17 Defining Goal Deviations Y p : The amount by which profit deviates from the target level When Yp is positive, When Yp is negative

BIS 517-Aslı Sencer18 Goal Deviation Constraints Regular Constrains:

BIS 517-Aslı Sencer19 Goal Programming Objective It costs $1 to fall a dollar below the profit target. However exceeding the profit target is desirable. Falling below revenue target lowers market share, lost earnings is $2 for a dollar below revenue target. Training time upto 30hrs enhances future efficiency. It costs $0.50 per hour obtained. Beyond 30hrs. training effect is negligable.

BIS 517-Aslı Sencer20 Ex: Optimal Solution Profit and Training levels are $6 and 6hrs. below their targets. Revenue target is met.

BIS 517-Aslı Sencer21 Ex: Maui Miser Case (revisited) Achieve annual rental income of at least $300,000 from fleet addition Spend as close to as possible to $500,000 Do not exceed average daily rental of $39 for the vehicles in the fleet addition. Penalty Costs: $1 for each dollar violation of goals 1 and 2. $10,000 for each dollar violation of goal 3.

BIS 517-Aslı Sencer22 Ex: Maui Miser Case GP Formulation

BIS 517-Aslı Sencer23 Ex: Maui Miser Case GP Optimal Solution