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How to Influence and Improve Decisions Through Optimization Models
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How to Influence and Improve Decisions Through Optimization Models
. Forest University, School of Business, Winston-Salem, North Carolina 27109 Contact: (JDC)
Industry’s recent increased focus on data-driven decision making and the use of analytics in all sectors from sports to financial services to technology and healthcare has led to a resurgence in the interest of traditional operations research tools such as optimization, simulation, and decision analysis. As organizations mature analytically, it seems likely that we will see a further increase in interest in prescriptive analytics, including optimization modeling, which is the focus of this tutorial. With massive amounts of data being routinely collected in real time and an increased awareness on the part of management of the value of data, the availability of data are typically no longer the bottleneck in the optimization modeling process. Increased computing speed, improved algorithms, parallel processing, and cloud computing have increased the size of optimization problems we can solve to optimality. Considering better data availability and the dramatic increase in our ability to solve problems, what are the impediments keeping us from having significant influence and impact on decision making? Going forward, it is possible that our inability to (1) structure a messy decision problem into a useful optimization framework, (2) properly use the model to deliver valuable insights for management, and (3) communicate to management the value prop- osition of our insights will become the new reasons we might fail to have the impact we know is possible. In this tutorial, we review several types of optimization models and the art of modeling—that is, the process of going from mess to model. We discuss how to use an optimization model to provide not just “the answer” but also insights that will be useful to managers and influence their decision making. We discuss the importance of communication in influencing decisions and provide examples and best practices rele- vant for optimization.
optimization • math programming • modeling • influence • data-driven decision making 1. Introduction to Optimization Modeling
Mathematical optimization is the process of maximizing or minimizing some function, called the objective function, usually subject to a set of constraints. In operations research, we often refer to mathematical optimization as mathematical programming, because the act of op- timizing results in an optimal program—that is, an optimal course of action to follow. In analytics, optimization falls into the category of prescriptive analytics, the set of analytical methodologies that prescribe a course of action.
An optimization model is a mathematical representation of a decision problem where the goal is to maximize (or minimize) the objective function, usually subject to constraints. Both the objective function and constraints involve mathematical functions of the decision vari- ables. We use optimization algorithms such as the simplex algorithm, branch and bound, and generalized reduced gradient to solve the optimization model. By “solve,” we mean find values of the decision variables that satisfy all the constraints of the problem and provide the
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Camm: How to Influence and Improve Decisions Through Optimization Models 2 Tutorials in Operations Research, © 2018 INFORMS
maximum (or minimum) value of the objective function. In the vernacular of mathematical programming, any setting of the decision variables that satisfies all the constraints is called a feasible solution. Optimization algorithms therefore find a best feasible solution where best is measured by the objective function. Although fast and effective algorithms are necessary for solving optimization models, the focus of this tutorial is on the process of building optimization models and how to effectively use such models for improved decision making.
The applications of optimization models in industry are plentiful. Capital budgeting, fi- nancial portfolio optimization, supply chain network design, operations planning and scheduling, inventory optimization, marketing mix models, product design and product line management, and vehicle routing and facility location are but a few of the types of applications of optimization that have had significant impact on how businesses are run every day.
For the purposes of this tutorial, we define an optimization model as follows: Maximize fðx1;x2;x3;:::;xnÞ
subject to
gjðx1;x2;x3;:::;xnÞ bj gjðx1;x2;x3;:::;xnÞ!bj gjðx1;x2;x3;:::;xnÞ1⁄4bj
j 1⁄4 1;2;:::;k;
j 1⁄4 k þ 1;k þ 2;:::;k þ l;
j 1⁄4kþlþ1;kþlþ2;:::;kþlþm:
Here, we have n decision variables and k + l + m constraints with objective function f, constraint functions gj, and constants bj.
In Section 2, we discuss the process of modeling a decision problem as an optimization problem. We use examples of actual optimization problems from industry (“real-world” problems). In Section 3, we discuss how to generate alternative optima, and in Section 4, we discuss how to generate a family of solutions for your client to consider (even suboptimal solutions). We have found the approaches in Sections 3 and 4 to be incredibly useful in consulting. The importance of communication and implementation issues are discussed in Section 5. Although not often taught in the classroom, these “soft skills” are, in our experience, often the biggest impediment to having an impact on decision making. Section 6 is a discussion about benchmarking and measuring potential impact for your client. In Section 7, we discuss implementation issues. In Section 8, we discuss how to deal with the resistance to change that can come from using optimization models over time. We end with a summary and other lessons learned from 20 years of consulting in optimization with the humble understanding that we are still learning after all these years.
2. Building Optimization Models
Camm et al. [8] list five steps of the decision-making process:
Step 1. Identify and define the problem.
Step 2. Determine the criteria that will be used to evaluate possible solutions. Step 3. Determine the set of possible solutions.
Step 4. Evaluate the possible solutions.
Step 5. Choose one of the possible solutions.
The first step is far from trivial and arguably the most difficult step. Managers typically do not describe problems; similar to a medical patient, they describe the symptoms of underlying problems. The job of the analyst is to listen, ask many questions, and diagnose/define the real problem to be solved. For more on the importance of defining the right problem, please see Cooper [10] and Weddell-Wedellsborg [21].
In the context of optimization modeling, the remaining steps involve developing the ob- jective function (Step 2), determining the constraint set (Step 3), and solving the optimization model (Steps 4 and 5). The modeling process is the process of moving from the mess to a model, going from Step 1 to Steps 2 and 3.
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Once a problem has been properly defined, there are several techniques that may be used to structure the modeling process, including influence diagrams (Powell and Baker [18]) and semantic networks (Evans and Camm [11]). If the problem is an optimization problem, we have found that answering the following list of simple questions is useful in model construction:
1. What am I trying to decide?
2. How am I restricted?
3. What is my metric of solution quality?
We use the Mill case (Camm et al. [6]) to demonstrate the modeling process as well as other key concepts in this tutorial.
The sales department at has confirmed orders for each of 15 fabrics. Man- agement is concerned that there is not enough production capacity to meet demand.
Calhoun has two types of looms: dobby and regular. Dobby looms can be used to make all fabrics and are the only looms that can weave certain fabrics, such as plaids. Given in Table 1 are the demand for each fabric, the production rate for each fabric on each type of loom, the variable production cost for each fabric, and the outsource cost for each fabric. Note that if a fabric can be woven on each type of loom, then the production rates are equal, and so the variable cost per yard of producing in the mill is the same regardless of which type of loom is utilized. Also note that fabrics 1–4 cannot be made on a regular loom.
Calhoun has 90 regular looms and 15 dobby looms. The mill operates continuously during the quarter: 13 weeks, seven days a week, and 24 hours a day. Assume that the changeover cost for the looms is negligible.
For now, management would like to know how to allocate the looms to the fabrics and which fabrics to outsource in order to minimize the cost of meeting demand. How much of each fabric should be produced on each type of loom, and how much of each fabric should be outsourced?
What Am I Trying to Decide? We are trying to decide how much of each fabric to produce on a regular loom or a dobby loom, or how much to outsource.
How Am I Restricted? We have the following restrictions: (i) demand must be satisfied for each fabric, (ii) we cannot make fabrics 1–4 on a regular machine, (iii) we have a limited
Table 1. Data for Mills problem. Dobby rate
Regular rate (yd/hr)
x 5.194 3.309 4.135 5.232 5.232 5.232 3.733 4.135 4.439 5.232 4.135
Mill cost ($/yd)
0.66 0.56 0.66 0.55 0.61 0.62 0.65 0.49 0.50 0.44 0.64 0.57 0.50 0.31 0.50
Outsource cost ($/yd)
0.80 0.70 0.85 0.70 0.75 0.75 0.80 0.60 0.70 0.60 0.80 0.75 0.65 0.45 0.70
1 2 3 4 5 6 7 8 9
Demand (yd)
16,500 52,000 45,000 22,000 76,500
110,000 122,000 62,000 7,500 69,000 70,000 82,000 10,000 380,000 62,000
4.653 4.653 4.653 4.653 5.194 3.809 4.185 5.232 5.232 5.232 3.733 4.135 4.439 5.232 4.185
Note. x means that the fabric cannot be made on a regular loom.
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Camm: How to Influence and Improve Decisions Through Optimization Models 4 Tutorials in Operations Research, © 2018 INFORMS
number of dobby looms and hence a limited number of dobby loom hours, and (iv) we have a limited number of regular looms and hence a limited number of regular loom hours.
What Is My Metric of Solution Quality? The measure of solution quality is the total cost of satisfying demand.
On the basis of the answer to the first question, we may define the following decision variables:
ri = the amount in yards of fabric i to produce on a regular loom, i = 1, 2, …, 15. di = the amount in yards of fabric i to produce on a dobby loom, i = 1, 2, …, 15. oi = the amount in yards of fabric i to outsource, i = 1, 2, …, 15.
For ease of exposition, we may also define the follow data:
ci = the cost per yard to produce fabric i, i = 1, 2, …, 15.
si = the cost per yard to outsource fabric i, i = 1, 2, …, 15.
pi = production rate for fabric i in yards per loom hour, i = 1, 2, . . ., 15.
ti = 1/pi, loom hours per yard required for producing fabric i, i = 1, 2, . . ., 15. demi = the demand for fabric i in yards, i = 1, 2, …, 15.
Finally, we note that, ignoring any changeover time or down time, the number of loom hours available are approximately 32,760 dobby loom hours (15 24 7 13 = 32,760) and 196,560 regular loom hours (90 24 7 13 = 196,560). We may then use the following linear pro- gram to help management answer the question at hand:
Minimize subject to
X15 i1⁄41 X15 i1⁄41
ri þdi þoi 1⁄4 demi
ri ; di ; oi ! 0
i 1⁄4 1;2;:::;15; (4)
i1⁄41;2;3;4; (5)
i 1⁄4 1; 2; : : : ; 15: (6)
ci ðri þ diÞ þ
The objective function (1) is the production cost plus the outsource cost. Constraint (2) ensures that we do not exceed regular loom capacity. Constraint (3) ensures we do not exceed the dobby hours available. The constraint set (4) (there are 15 such constraints, one for each fabric) ensures that demand for each fabric is satisfied. Constraint set (5) ensures that we do not allow production for fabrics 1–4 on a regular loom. Finally, we refer to constraint set (6) as the nonnegativity constraints, as we cannot produce or outsource negative amounts of fabric.
The model (1)–(6) is a linear program because all functions are linear in the decision variables, the decision variables are nonnegative, and the decision variables are continuous (i.e., we allow fractional solutions). We may solve (1)–(6) using the simplex algorithm or an interior point algorithm.
A few observations are in order. First, this model is designed to help management un- derstand how much of each fabric should be made on each type of loom or outsourced. The
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fundamental question really is, “Does anything need to be outsourced?” A closer look at the data in Table 1 shows that it is always cheaper to produce than to outsource. Hence, if we have enough capacity, we should produce everything. This tells us a lot about what to expect when we solve (1)–(6). If we solve our model and nothing is outsourced, it means we had enough capacity to do so. If we solve the model and some fabric is outsourced, we should see that all loom time is used.
Takeaway. Always understand the data and anticipate model behavior—that is, the general solution characteristics. Understanding your data and your model is critical to being able to communicate recommendations to your client.
The solution of (1)–(6) is shown in Table 2. All regular and dobby loom time is used. Only fabric 6 is outsourced, and it is split between regular looms and the outsourcing firm. Fabric 15 is also split, but between dobby and regular looms. The total cost is $599,108.16.
For completeness, we mention that inequality constraints that hold as equality are said to be binding. At optimality, the binding constraints form the bottleneck that prevents us from doing better. A by-product of solving a linear program is that we are provided a shadow price that indicates the change in the objective function for an increase of one on the right-hand side of a constraint. Each variable has a reduced cost, which is the shadow price for the non- negativity constraint for that variable. In the case, regular and dobby loom hours are both binding constraints.
Now let us discuss some extensions of the basic model we have constructed. If in addition to nonnegativity we require that the decision variables take on only integer values—that is, a model consisting of (1)–(5) and (7) from below—then we have a linear integer program:
ri;di;oi !0 and integer i 1⁄4 1;2;:::;15: (7)
Variables that are required to be integer but also between 0 and 1 inclusive are called binary variables. Binary variables allow us to model yes–no decisions. To illustrate the use of binary variables, let us consider another extension of (1)–(6). Suppose that we require that only one of the three options—regular loom, dobby loom, or outsource—may be used for each fabric.
We define the following binary variables:
Ri =1iffabriciisproducedonregularloomsand0ifnot;i=1,2,…,15. Di =1iffabriciisproducedondobbyloomsand0ifnot;i=1,2,…,15. Oi =1iffabriciisoutsourcedand0ifnot;i=1,2,…,15.
Table 2. An optimal solution to the case.
Fabric Dobby Regular Outsource
16,500 52,000 45,000 22,000 76,500
110,000 122,000 62,000 7,500 69,000 70,000 82,000 10,000 380,000 62,000
1 16,500.00 2 52,000.00 3 45,000.00 4 220,00.00 5 0.00 6 0.00 7 0.00 3 0.00 9 7,500.00
10 0.00 11 0.00 12 0.00 13 0.00 14 0.00 15 9,230.09
0.00 0.0 0.00 0.0 0.00 0.0 0.00 0.0
76,500.00 0.00 6,871.51 103,128.49 122,000.00 0.00 62,000.00 0.00 0.00 0.00 69,000.00 0.00 70,000.00 0.00 82,000.00 0.00 10,000.00 0.00 380,000.00 0.00 52,769.91 0.00
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The following constraints, when added to (1)–(6), will ensure that only one option is used for each fabric:
oi demiOi RiþDiþOi 1⁄4 1
Ri;Di;Oi 2f0;1g
i 1⁄4 1;2;:::;15; (8) i 1⁄4 1;2;:::;15; (9) i 1⁄4 1;2;:::;15; (10) i 1⁄4 1;2;:::;15; (11)
i 1⁄4 1;2;:::;15: (12)
Constraints (8)–(10) are sometimes referred to as setup constraints. Consider, for example, constraint (8). If Ri = 0, then ri = 0. If Ri = 1, then we have ri demi, but (11) implies Di = Oi = 0, and so by (4), (9), and (11), we have ri = demi. The previously mentioned constraint (11) ensures that only one option is used for each fabric, and (12) ensures that Ri, Di, and Oi are binary.
The solution of the extended model is shown in Table 3. Note that the solution is quite different from the solution in Table 2. Fabrics 11 and 12 are outsourced. There are unused dobby and regular hours (2,205.5 hours and 9,065 hours, respectively). Also, note that because we added constraints to the original model, it cannot be that cost decreases. The total cost of $6
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