GS BUSA 424 CH 2 Introduction to Optimization and Linear Programming
Mathematical programming (MP)
a field of management science that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual or a business.
Applications of Mathematical Optimization
• Determining Product Mix • Manufacturing • Routing and Logistics • Financial Planning
constraint The limited resources
1. A limit to a design process. Constraints may be such things as appearance, funding, space, materials, and human capabilities. 2. A limitation or restriction. The constraints in an optimization problem can be represented in a mathematical model in a number of ways. Three general ways of expressing the possible constraint relationships in an optimization problem are: A less than or equal to constraint: f(X1, X2, . . . , Xn) <= b A greater than or equal to constraint: f(X1, X2, . . . , Xn) >= b An equal to constraint: f(X1, X2, . . . , Xn) = b
Characteristics of Optimization Problems
1. Decisions 2. Constraints 3. Objectives
Steps in formulating an LP Model
1. Understand the problem. 2. Identify the decision variable. That is, what are the fundamental decisions that must be made in order to solve the problem? The answers to this question often will help you identify appropriate decision variables for your model. 3. State the objective function as a linear combination of decision variables. 4. State the constraints as linear combination of decision variables. These restrictions must be identified and stated in the form of constraints. 5. Identify any upper or lower bounds on the decision variables. You can view upper and lower bounds as additional constraints in the problem. In our example, there are simple lower bounds of zero on the variables X1 and X2 because it is impossible to produce a negative number of hot tubs.
Summary of Graphical Solution to LP Problems To summarize this section, a two-variable LP problem is solved graphically by performing these steps: 1. Plot the boundary line of each constraint in the model. 2. Identify the feasible region, that is, the set of points on the graph that simultaneously satisfies all the constraints.
3. Locate the optimal solution by one of the following methods: a. Plot one or more level curves for the objective function, and determine the direction in which parallel shifts in this line produce improved objective function values. Shift the level curve in a parallel manner in the improving direction until it intersects the feasible region at a single point. Then find the coordinates for this point. This is the optimal solution. b. Identify the coordinates of all the extreme points of the feasible region, and calculate the objective function values associated with each point. If the feasible region is bounded, the point with the best objective function value is the optimal solution.
non-linear
A relationship which does not create a straight line. curved lines or curved surfaces if the relationship between x factors is either a multiplication or a division this a non-linear relationship
Routing and Logistics Many retail companies have warehouses around the country that are responsible for keeping stores supplied with merchandise to sell. The amount of merchandise available at the warehouses and the amount needed at each store tends to fluctuate, as does the cost of shipping or delivering merchandise from the warehouses to the retail locations.
Determining the least costly method of transferring merchandise from the warehouses to the stores can save large amounts of money.
Finding the Optimal Solution by Enumerating the Corner Points So, another way of solving an LP problem is to identify all the corner points, or extreme points, of the feasible region and calculate the value of the objective function at each of these points. The corner point with the largest objective function value is the optimal solution to the problem. This approach is illustrated in Figure 2.7, where the X1 and X2 coordinates for each of the extreme points are identified along with the associated objective function values.
Enumerating the corner points to identify the optimal solution is often more difficult than the level curve approach because it requires that you identify the coordinates for all the extreme points of the feasible region. If there are many intersecting constraints, the number of extreme points can become rather large, making this procedure very tedious. Also, a special condition exists for which this procedure will not work. This condition, known as an unbounded solution, is described shortly.
Finding the Optimal Solution Using Level Curves (part 1) The very last level curve we can draw that still intersects the feasible region will determine the maximum profit we can achieve. This point of intersection, shown in Figure 2.6, represents the optimal feasible solution to the problem. If you compare Figure 2.6 to Figure 2.3, you see that the optimal solution occurs where the boundary lines of the pump and labor constraints intersect (or are equal). Thus, the optimal solution is defined by the point (X1, X2) that simultaneously satisfies equations 2.26 and 2.27, which are repeated here: X1 + X2 = 200 9X1 + 6X2 = 1,566
From the first equation, we easily conclude that X2 = 200 - X1. If we substitute this definition of X2 into the second equation we obtain: 9X1 + 6(200 - X1) = 1,566 Using simple algebra, we can solve this equation to find that X1 = 122. And because X2 = 200 - X1, we can conclude that X2 = 78. Therefore, we have determined that the optimal solution to our example problem occurs at the point (X1, X2) (122, 78). This point satisfies all the constraints in our model and corresponds to the point in Figure 2.6 identified as the optimal solution.
UNDERSTANDING HOW THINGS CHANGE It is important to realize that if changes occur in any of the coefficients in the objective function or constraints of this problem, then the level curve, feasible region, and optimal solution to this problem might also change. To be an effective LP modeler, it is important for you to develop some intuition about how changes in various coefficients in the model will impact the solution to the problem.
However, the spreadsheet shown in Figure 2.8 (and the file named Fig2-8.xlsm that accompanies this book) allows you to change any of the coefficients in this problem and instantly see its effect. You are encouraged to experiment with this file to make sure you understand the relationships between various model coefficients and their impact on this LP problem.
Plotting the second constraint The boundary line for the second constraint in our model is given by: 9X1 + 6X2 = 1,566
If X1 = 0 in equation 2.27, then X2 = 1,566/6 = 261. So, the point (0, 261) must fall on the line defined by equation 2.27. Similarly, if X2 = 0 in equation 2.27, then X1 = 1,566/9 = 174. So, the point (174, 0) must also fall on this line. These two points are plotted on the graph and connected with a straight line representing equation 2.27, as shown in Figure 2.2
Plotting the third constraint This constraint requires that no more than 2,880 feet of tubing be used in producing the hot tubs The boundary line for the third constraint in our model is: 12X1 + 16X2 = 2,880
If X1 = 0 in equation 2.28, then X2 = 2,880/16 = 180. So, the point (0, 180) must fall on the line defined by equation 2.28. Similarly, if X2 = 0 in equation 2.28, then X1 = 2,880/12 = 240. So, the point (240, 0) also must fall on this line. These two points are plotted on the graph and connected with a straight line representing equation 2.28, as shown in Figure 2.3.
Plotting the first constraint The boundary of the first constraint in our model, which specifies that no more than 200 pumps can be used, is represented by the straight line defined by the equation: X1 + X2 = 200
If X2 = 0, we can see from equation 2.26 that X1 = 200. Thus, the point (X1, X2) (200, 0) must fall on this line. If we let X1=0, from equation 2.26, it is easy to see that X2 = 200. So, the point (X1, X2) (0, 200) must also fall on this line.
Summary of the LP Model for the Example Problem The complete LP model for Howie's decision problem can be stated as: MAX: 350X1 + 300X2 2.5 Profit/Unit Subject to: 1X1 + 1X2 <= 200 2.6 Pump Constraint 9X1 + 6X2 <= 1,566 2.7 Hours Constraint 12X1 + 16X2 <= 2,880 2.8 Tubing Constraint 1X1 >= 0 2.9 non-negativity 1X2 >= 0 2.10 non-negativity
In this model, the decision variables X1 and X2 represent the number of Aqua-Spas and Hydro-Luxes to produce, respectively. Our goal is to determine the values for X1 and X2 that maximize the objective in equation 2.5 while simultaneously satisfying all the constraints in equations 2.6 through 2.10.
The objective in an optimization problem is represented mathematically by an objective function in the general format:
MAX (or MIN): f(X1, X2, . . . , Xn) The objective function identifies some function of the decision variables that the decision maker wants to either MAXimize or MINimize.
Alternate Optimal Solutions A situation in which more than one optimal solution is possible. It arises when the angle or slope of the objective function is the same as the slope of the constraint. For example, suppose Howie can increase the price of Aqua-Spas to the point at which each unit sold generates a profit of $450 rather than $350. The revised LP model for this problem is: MAX: 450X1 + 300X2 Because none of the constraints changed, the feasible region for this model is the same as for the earlier example. The only difference in this model is the objective function. Therefore, the level curves for the objective function are different from what we observed earlier.
Notice that the final level curve in Figure 2.9 intersects the feasible region along an edge of the feasible region rather than at a single point. All the points on the line segment joining the corner point at (122, 78) to the corner point at (174, 0) produce the same optimal objective function value of $78,300 for this problem. Thus, all these points are alternate optimal solutions to the problem. If we used a computer to solve this problem, it would identify only one of the corner points of this edge as the optimal solution.
Linear Programming (LP)
Optimization technique used to maximize an objective function (for example, contribution margin of a mix of products), when there are multiple constraints the functions in a model are linear in nature (that is, form straight lines or flat surfaces); If the relationship between the decision variables which are the x factors are either addition or subtraction then this relationship is called a linear relationship
When completing an optimization analysis the steps are:
Step 1:Establish the objective variable and the decision variables Step 2: Create the objective function Step 3: Find out all constraints
Unbounded Solutions When attempting to solve some LP problems, you might encounter situations in which the objective function can be made infinitely large (in the case of a maximization problem) or infinitely small (in the case of a minimization problem). As an example, consider this LP problem: MAX: X1 + X2 Subject to: X1 + X2 >= 400 -X1 + 2X2 <= 400 X1 >= 0 X2 >= 0 Although it is not unusual to encounter an unbounded solution when solving an LP model, such a solution indicates that there is something wrong with the formulation—for example, one or more constraints were omitted from the formulation, or a less than constraint was erroneously entered as a greater than constraint.
The feasible region and some level curves for this problem are shown in Figure 2.11. From this graph, you can see that as the level curves shift farther and farther away from the origin, the objective function increases. Because the feasible region is not bounded in this direction, you can continue shifting the level curve by an infinite amount and make the objective function infinitely large However, the goal here is to maximize the objective function value, which, as we have seen, can be done without limit. So, when trying to solve an LP problem by enumerating the extreme points of an unbounded feasible region, you must also check whether or not the objective function is unbounded.
Determining Product Mix Most manufacturing companies can make a variety of products. However, each product usually requires different amounts of raw materials and labor. Similarly, the amount of profit generated by the products varies.
The manager of such a company must decide how many of each product to produce in order to maximize profits or to satisfy demand at minimum cost.
Optimization
The process of reaching the optimal values
Infeasibility As an example, consider the LP model: MAX: X1 + X2 Subject to: X1 + X2 <= 150 X1 + X2 >= 200 X1 >= 0 X2 >= 0 Infeasibility can occur in LP problems, perhaps due to an error in the formulation of the model—such as unintentionally making a less than or equal to constraint a greater than or equal to constraint. Or there just might not be a way to satisfy all the constraints in the model. In this case, constraints will have to be eliminated or loosened in order to obtain a feasible region (and feasible solution) for the problem
The situation in which no solution to the linear programming problem satisfies all the constraints. The feasible solutions for the first two constraints in this model are shown in Figure 2.12. Notice that the feasible solutions to the first constraint fall on the left side of its boundary line, whereas the feasible solutions to the second constraint fall on the right side of its boundary line. Therefore, no possible values for X1 and X2 exist that simultaneously satisfy both constraints in the model.
Financial Planning The federal government requires individuals to begin withdrawing money from individual retirement accounts (IRAs) and other tax-sheltered retirement programs no later than age 70.5.
There are various rules that must be followed to avoid paying penalty taxes on these withdrawals. Most individuals want to withdraw their money in a manner that minimizes the amount of taxes they must pay while still obeying the tax laws.
redundant constraint A constraint that does not affect the feasible region. If a constraint is redundant, it can be removed from the problem without affecting the feasible region. plays no role in determining the feasible region of the problem. For example, in the hot tub example, suppose that 225 hot tub pumps are available instead of 200. The earlier LP model can be modified as follows to reflect this change: MAX: 350X1 + 300X2 Subject to: 1X1 + 1X2 <= 225 The constraints and feasible region for this revised model are shown in Figure 2.10. Notice that the pump constraint in this model no longer plays any role in defining the feasible region of the problem. That is, as long as the tubing constraint and labor constraints are satisfied (which is always the case for any feasible solution), then the pump constraint will also be satisfied.
Therefore, we can remove the pump constraint from the model without changing the feasible region of the problem—the constraint is simply redundant. The fact that the pump constraint does not play a role in defining the feasible region in Figure 2.10 implies that there will always be an excess number of pumps available. Because none of the feasible solutions identified in Figure 2.10 fall on the boundary line of the pump constraint, this constraint will always be satisfied as a strict inequality (1X1 + 1X2 = 225) and never as a strict equality (1X1 + 1X2 < 225). Again, redundant constraints are not really a problem. They do not prevent us (or the computer) from finding the optimal solution to an LP problem. However, they do represent "excess baggage" for the computer; so if you know that a constraint is redundant, eliminating it saves the computer this excess work. On the other hand, if the model you are working with will be modified and used repeatedly, it might be best to leave any redundant constraints in the model because they might not be redundant in the future. For example, from Figure 2.3, we know that if the availability of pumps is returned to 200, then the pump constraint again plays an important role in defining the feasible region (and optimal solution) of the problem.
Plotting the objective function (part 1) That is, we must determine which point in the feasible region will maximize the value of the objective function in our model. It can be shown that if an LP problem has an optimal solution with a finite objective function value, this solution will always occur at a point in the feasible region where two or more of the boundary lines of the constraints intersect.
These points of intersection are sometimes called corner points or extreme points of the feasible region.
Plotting the objective function (part 3) Now, suppose we are interested in finding the values of X1 and X2 that produce some higher level of profit, such as $52,500. Then, mathematically, we are interested in finding the points (X1, X2) for which our objective function equals $52,500, or where: $350X1 + $300X2 = $52,500
This equation also defines a straight line, which we could plot on our graph. If we do this, we'll find that the points (X1, X2) (0, 175) and (X1, X2) (150, 0) both fall on this line, as shown in Figure 2.5. The lines in Figure 2.5 representing the two objective function values are sometimes referred to as level curves because they represent different levels or values of the objective. Note that the two level curves in Figure 2.5 are parallel to one another. If we repeat this process of drawing lines associated with larger and larger values of our objective function, we will continue to observe a series of parallel lines shifting away from the origin, that is, away from the point (0, 0).
Plotting the objective function (part 2) To see why the finite optimal solution to an LP problem occurs at an extreme point of the feasible region, consider the relationship between the objective function and the feasible region of our example LP model. Suppose we are interested in finding the values of X1 and X2 associated with a given level of profit, such as $35,000. Then, mathematically, we are interested in finding the points (X1, X2) for which our objective function equals $35,000, or where: $350X1 + $300X2 = $35,000
This equation defines a straight line, which we can plot on our graph. Specifically, if X1 = 0 then, from equation 2.29, X2 = 116.67. Similarly, if X2 = 0 in equation 2.29, then X1 = 100. So, the points (X1, X2) (0, 116.67) and (X1, X2) (100, 0) both fall on the line defining a profit level of $35,000. (Note that all the points on this line produce a profit level of $35,000.) This line is shown in Figure 2.4.
Manufacturing Printed circuit boards, like those used in most computers, often have hundreds or thousands of holes drilled in them to accommodate the different electrical components that must be plugged into them.
This process is repeated hundreds or thousands of times to complete all the holes on a circuit board. Manufacturers of these boards would benefit from determining the drilling order that minimizes the total distance the drill bit must be moved.
The mathematical formulation of an optimization problem can be described in the general format: MAX (or MIN): f0(X1, X2, . . . , Xn) 2.1 Subject to: f1(X1, X2, . . . , Xn) <= b1 2.2 fk(X1, X2, . . . , Xn) >=bk 2.3 fm(X1, X2, . . . , Xn) = bm 2.4
This representation identifies the objective function (equation 2.1) that will be maximized (or minimized) and the constraints that must be satisfied (equations 2.2 through 2.4). Subscripts added to the f and b in each equation emphasize that the functions describing the objective and constraints can all be different. The goal in optimization is to find the values of the decision variables that maximize (or minimize) the objective function without violating any of the constraints.
Optimal Value
business factor's value that reaches its extreme (either minimum or maximum). Examples: Minimum costs, Maximum profits
Optimization Analysis
finds the optimum value for a target variable by repeatedly changing other variables, subject to specified constraints it is a process of analyzing how to allocate the limited amount of resources we have to the influencing factors so that we can reach the best ultimate outcomes in our business The modeling process of reaching the optimal values by changing other influential factors.
Loosening constraints involves
increasing the upper limits (or reducing the lower limits) to expand the range of feasible solutions. For example, if we loosen the first constraint in the previous model by changing the upper limit from 150 to 250, there is a feasible region for the problem. Of course, loosening constraints should not be done arbitrarily. In a real model, the value 150 would represent some actual characteristic of the decision problem (such as the number of pumps available to make hot tubs). We obviously cannot change this value to 250 unless it is appropriate to do so—that is, unless we know another 100 pumps can be obtained.
objective function coefficients
represent the marginal profits (or costs) associated with the decision variables The symbols c1, c2, . . . , cn in equation 2.11 are called objective function coefficients and might represent the marginal profits (or costs) associated with the decision variables X1, X2, . . . , Xn, respectively. The symbol aij found throughout equations 2.12 through 2.14 represents the numeric coefficient in the ith constraint for variable Xj.
The objective variable is dependent on
the influencing factors we call the influencing factors the decision variables and optimize this analysis
It is now easy to see which points satisfy all the constraints in our model. These points correspond to the shaded area in Figure 2.3, labeled "Feasible Region." The feasible region is
the set of points or values that the decision variables can assume and simultaneously satisfy all the constraints in the problem. the intersection of the graphs in a system of constraints The feasible region does not include: points which violate at least one of the functional or non-negativity constraints. The feasible region of a LP problem with two unknowns may be bounded or unbounded.
decision variables
within the context of optimization modeling, variables that will be manipulated to find the best solution These variables might represent the quantities of different products the production manager can choose to produce. They might represent the amount of different pieces of merchandise to ship from a warehouse to a certain store. They might represent the amount of money to be withdrawn from different retirement accounts.
2.11 Special Conditions in LP Models Several special conditions can arise in LP modeling:
• alternate optimal solutions, • redundant constraints, • unbounded solutions, and • infeasibility. The first two conditions do not prevent you from solving an LP model and are not really problems—they are just anomalies that sometimes occur. On the other hand, the last two conditions represent real problems that prevent us from solving an LP model.