Presentation on theme: "Linear Programming Models & Case Studies"— Presentation transcript:
1 Linear Programming Models & Case Studies Adapted from lectures of Prof. Rifat Rustom
2 The Optimization Problem is: What is OptimizationThe Optimization Problem is:Find values of the variables that minimize or maximize the objective function while satisfying the constraints.
3 Components of Optimization Problem An objective function which we want to minimize or maximizeA set of unknowns or variables which affect the value of the objective functionA set of constraints that allow the unknowns to take on certain values but exclude othersOptimization problems are made up of three basic ingredients:
5 Linear ProgrammingA linear programming problem is one in which we are to find the maximum or minimum value of a linear expressionax + by + cz(called the objective function), subject to a number of linear constraints of the formAx + By + Cz NThe largest or smallest value of the objective function is called the optimal value, and a collection of values of x, y, z, that gives the optimal value constitutes an optimal solution. The variables x, y, z, are called the decision variables
6 LP Properties and Assumptions PROPERTIES OF LINEAR PROGRAMS1. One objective function2. One or more constraints3. Alternative courses of action4. Objective function and constraints are linearASSUMPTIONS OF LP1. Certainty2. Proportionality3. Additivity4. Divisibility5. Nonnegative variables
7 Basic Assumptions of LP We assume conditions of certainty exist and numbers in the objective and constraints are known with certainty and do not change during the period being studiedWe assume proportionality exists in the objective and constraintsconstancy between production increases and resource utilization – if 1 unit needs 3 hours then 10 require 30 hoursWe assume additivity in that the total of all activities equals the sum of the individual activitiesWe assume divisibility in that solutions need not be whole numbersAll answers or variables are nonnegative as we are dealing with real physical quantities
8 Example Find the maximum value of Subject to: p = 3x + 2y + 4z 4x + 3y + z >=3 x + 2y + z >=4 x >=0, y >=0, z >=0
9 Examples of LP Problems (1) 1. A Product Mix ProblemA manufacturer has fixed amounts of different resources such as raw material, labor, and equipment.These resources can be combined to produce any one of several different products.The quantity of the ith resource required to produce one unit of the jth product is known.The decision maker wishes to produce the combination of products that will maximize total income.
10 Examples of LP Problems (2) 2. A Blending ProblemBlending problems refer to situations in which a number of components (or commodities) are mixed together to yield one or more products.Typically, different commodities are to be purchased. Each commodity has known characteristics and costs.The problem is to determine how much of each commodity should be purchased and blended with the rest so that the characteristics of the mixture lie within specified bounds and the total cost is minimized.
11 Examples of LP Problems (3) 3. A Production Scheduling ProblemA manufacturer knows that he must supply a given number of items of a certain product each month for the next n months.They can be produced either in regular time, subject to a maximum each month, or in overtime. The cost of producing an item during overtime is greater than during regular time. A storage cost is associated with each item not sold at the end of the month.The problem is to determine the production schedule that minimizes the sum of production and storage costs.
12 Examples of LP Problems (4) 4. A Transportation ProblemA product is to be shipped in the amounts al, a2, ..., am from m shipping origins and received in amounts bl, b2, ..., bn at each of n shipping destinations.The cost of shipping a unit from the ith origin to the jth destination is known for all combinations of origins and destinations.The problem is to determine the amount to be shipped from each origin to each destination such that the total cost of transportation is a minimum.
13 Examples of LP Problems (5) 5. A Flow Capacity ProblemOne or more commodities (e.g., traffic, water, information, cash, etc.) are flowing from one point to another through a network whose branches have various constraints and flow capacities.The direction of flow in each branch and the capacity of each branch are known.The problem is to determine the maximum flow, or capacity of the network.
14 Formulating LP Problems Formulating a linear program involves developing a mathematical model to represent the managerial problemThe steps in formulating a linear program areCompletely understand the managerial problem being facedIdentify the objective and constraintsDefine the decision variablesUse the decision variables to write mathematical expressions for the objective function and the constraints
15 Example 1: Flair Furniture Company 1. A Product Mix ProblemExample 1: Flair Furniture CompanyThe Flair Furniture Company produces inexpensive tables and chairsProcesses are similar in that both require a certain amount of hours of carpentry work and in the painting and varnishing departmentEach table takes 4 hours of carpentry and 2 hours of painting and varnishingEach chair requires 3hours of carpentry and 1 hour of painting and varnishingThere are 240 hours of carpentry time available and 100 hours of painting and varnishingEach table yields a profit of $70 and each chair a profit of $50
16 Example 1: Flair Furniture Company The company wants to determine the best combination of tables and chairs to produce to reach the maximum profitHOURS REQUIRED TO PRODUCE 1 UNITDEPARTMENT(T) TABLES(C) CHAIRSAVAILABLE HOURS THIS WEEKCarpentry43240Painting and varnishing21100Profit per unit$70$50
17 Example 1: Flair Furniture Company The objective is toMaximize profitThe constraints areThe hours of carpentry time used cannot exceed 240 hours per weekThe hours of painting and varnishing time used cannot exceed 100 hours per weekThe decision variables representing the actual decisions we will make areT = number of tables to be produced per weekC = number of chairs to be produced per week
18 Example 1: Flair Furniture Company We create the LP objective function in terms of T and CMaximize profit = $70T + $50CDevelop mathematical relationships for the two constraintsFor carpentry, total time used is(4 hours per table)(Number of tables produced) + (3 hours per chair)(Number of chairs produced)We know thatCarpentry time used ≤ Carpentry time available4T + 3C ≤ 240 (hours of carpentry time)
19 Example 1: Flair Furniture Company SimilarlyPainting and varnishing time used ≤ Painting and varnishing time available2 T + 1C ≤ 100 (hours of painting and varnishing time)This means that each table produced requires two hours of painting and varnishing timeBoth of these constraints restrict production capacity and affect total profit
20 Example 1: Flair Furniture Company The values for T and C must be nonnegativeT ≥ 0 (number of tables produced is greater than or equal to 0)C ≥ 0 (number of chairs produced is greater than or equal to 0)The complete problem stated mathematicallyMaximize profit = $70T + $50Csubject to4T + 3C ≤ 240 (carpentry constraint)2T + 1C ≤ 100 (painting and varnishing constraint)T, C ≥ 0 (nonnegativity constraint)
21 Max 70T + 50C Subject to 4T + 3C <=240. 2T + 1C <=100 T >= Max 70T + 50C Subject to 4T + 3C <=240 2T + 1C <=100 T >= 0 C >= 0Global optimal solution found.Objective value:Infeasibilities:Total solver iterations:Model Class: LPTotal variables:Nonlinear variables:Integer variables:Total constraints:Nonlinear constraints:Total nonzeros:Nonlinear nonzeros:Variable Value Reduced CostTCRow Slack or Surplus Dual PriceRustom
22 Example 2: Solution of the Two Quarrying Sites 2. A Blending ProblemExample 2: Solution of the Two Quarrying SitesA Quarrying Company owns two different rock sites that produce aggregate which, after being crushed, is graded into three classes: high, medium and low-grade. The company has contracted to provide a concrete batching plant with 12 tons of high-grade, 8 tons of medium-grade and 24 tons of low-grade aggregate per week. The two sites have different operating characteristics as detailed below.Site Cost per day ($'000) Production (tons/day)High Medium LowXYHow many days per week should each Site be operated to fulfill the batching plant contract?
23 Example 2: Solution of the Two Quarrying Sites Translate the verbal description into an equivalent mathematical description.Determine:- Variables- Constraints- ObjectiveFormulating the problem (mathematical representation of the problem).(1) VariablesThese represent the "decisions that have to be made" or the "unknowns".Letx = number of days per week Site X is operatedy = number of days per week Site Y is operatedNote here that x >= 0 and y >= 0.
24 Example 2: Solution of the Two Quarrying Sites (2) ConstraintsIt is best to first put each constraint into words and then express it in a mathematical form.Aggregate production constraintsbalance the amount produced with the quantity required under the batching plant contractAggregateHigh x + 1y >= 12Medium 3x + 1y >= 8Low x + 6y >= 24Days per week constraintwe cannot work more than a certain maximum number of days a week e.g. for a 5 day week we have:x <= y <= 5Constraints of this type are often called implicit constraints because they are implicit in the definition of the variables.
25 Example 2: Solution of the Two Quarrying Sites (3) ObjectiveAgain in words our objective is (presumably) to minimize cost which is given by 180x + 160yHence we have the complete mathematical representation of the problem as:Minimize 180x + 160ySubject to6x + y >= 123x + y >= 84x + 6y >= 24x <= 5y <= 5x >= 0Global optimal solution found.Objective value:Infeasibilities:Total solver iterations:Variable Value Reduced CostXYRow Slack or Surplus Dual Price
26 Example 3: Optimization of Well Treatment Problem Statement:Three wells in Gaza are used to pump water for domestic use. The maximum discharge of the wells as well as the properties of water pumped are shown in the table below. The water is required to be treated using chlorine before pumped into the system.Determine the minimum cost of chlorine treatment per cubic meter per day for the three wells given the cost in $/m3 for the three wells. The total discharge is 5000 m3/day.
27 Properties of Pumped Water ClQmaxCl treatmentMax. workingmg/lm3/hr$/m3hoursWell No. 12001000.0520Well No. 25001500.1218Well No. 33001200.0815
28 Mathematical Model Minimize Cost of Chlorine Treatment per Day Objective Function: C = 0.05Q Q Q3ConstraintsSubject to:Well 1: max discharge should be equal to or less than 2000 m3/day (100 m3/hr x 20hr/day)Q1 < = (1)Well 2: max discharge should be equal to or less than 2700 m3/day (150 m3/hr x 18hr/day)Q2 < = (2)Well 3: max discharge should be equal to or less than 1800 m3/day (120 m3/hr x 15hr/day)Q3 < = (3)
29 Wells 1, 2, 3:total discharge should be equal to 5000 m3/day.Q1 + Q2 + Q3 = (4)` discharge of each well should be greater than 0.Q1, Q2, Q3 > (5)