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DM Optimization
CHAPTER 6
DM Optimization 6.1 INTRODUCTION The conversion from logical decision tree (LDT) to binary decision diagram (BDD) was presented in Chapters 3 and 4. This method allows for obtaining the analytical expression of the top event occurrence probability in function of the probabilities of basic events. This expression defines quantitatively the behavior of a system, or process, but it does not take into account the exogenous variables that, in certain cases, could be more influential than the endogenous ones. This chapter presents a novel approach to consider external factors that may be important for optimizing the DM processes.82 It is based on considering the analytical expression obtained from the BDD as the objective function of a NLPP, where the exogenous variables are modeled by a set of constraints.83 An optimal investment, subject to the limitation of resources, is the main objective of the DM process. This book proposes two strategies to support the resource allocation when a MP is given. Fig. 6.1 shows the steps to be done before using the novel methods.
1. Obtaining LDT This step requires a qualitative analysis 2. Conversion from LDT into BDD This step requires a software able to execute control sentences 3. Obtaining cut-sets and analytical expression of probability Anytical expression of the main problem probability, depending on probabilities of BCs 4. Mathematical optimization approach or Birnbaum-cost measure method
Fig. 6.1 Proposed optimization method process.
Decision-Making Management http://dx.doi.org/10.1016/B978-0-12-811540-4.00006-5
© 2017 Elsevier Inc. All rights reserved.
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6.2 DM APPROACH The LDT design is not complicated, but it requires an adequate knowledge of the problem. Fig. 6.2 is composed of 32 basic causes and 26 nonbasic causes. One of the branches of this LDT has been zoomed in Fig. 6.2 to show some causes in detail. The MP is “delay in the orders,” therefore, the business is seeking to minimize these delays.
Delay in the orders
Wrong Projects Interaction
Wrong Stock Project
Mixture of information between transacctions
Lack of a global planning
Employees steal material
Lack of Employees Training
Shortage of statistic resources
Sampling mistakes
Parameters’ selection mistakes
Wrong Simulations
Unreal Warehouse Data
Poor production management
Forecast Mistakes
Out-dated analysis techniques
Delay in Repair Orders
Lack of notification
Employees with low qualification
No coordination between workers
Poor comunication between workers
Human Resources
Lack of simulations
Lack of daily notification
Errors in Paperwork Creation
Tools Shortage
Forecast Mistakes Wrong planning of job position
Out-dated analysis techniques
Shortage of statistic resources
Wrong Tools Stock
Limited number of tools
Employees with limited qualification
Manufacturin g Closed Late
Lack of resources in some departments
Limited connections between departments
Manufacturing order not scheduled
Low tools’ reliability
Wrong detailed needs in Organization Chart
Lack of inner training courses
Renege on inner Procedures
Lack of Employees Training
Employees with low qualification
Lack of inner training courses
Wrong Departmental Personnel Planning
Lack of incentives
Poor Organisation Between Departments
Mismanagement of Human Resources department
Limited salary
Mismatch between Needs
Overcharge of capacity
Bad work atmosphere Flow Absence of information between Departments
Unreal capacity work
Wrong work planning
Poor relationships between department’s leaders
Lack of Software implemented does not work properly
notification
Unpleasant relationships between employees
Wrong simulations
Unreal warehouse data
Employees steal material
Poor comunication between workers
No coordination between workers
Forecast mistakes
Lack of simulations
Lack of daily notification
Out-dated analysis techniques
Fig. 6.2 Example of a LDT.
• • • •
The following procedure is suggested to create a LDT: Define properly the MP and its scope. Detect the BCs related with the MP. Study the interrelation between mentioned BCs and the MP. Obtain the probability data for each BC.84
Shortage of statistic resources
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6.3 Mathematical NLPP Background The problem to address in this section is: minimize f ðxÞ subject to hðxÞ ¼ 0 gðxÞ 0 where in case that any of the functions involved are nonlinear, the problem will be a Non-Linear Programming Problem (NLPP).85 NLPP is defined by a function f : Rn ! R, and two vector functions that represent the constraints. h : Rn ! Rm g : Rn ! Rd x 2 Rn f : Rn ! R being m the number of (nonlinear) equality constraint, and d is the number of (nonlinear) inequality constraints. As a result, the following set of vectors will correspond to the feasible set: A ¼ fx 2 Rn : hðxÞ ¼ 0, gðxÞ 0g Rn The conditions of optimality are set by the Lagrange criterion. The Lagrange multipliers for a NLPP with both equality and inequality constraints are defined as: Lðx, λ, μÞ ¼ f ðxÞ + λ hðxÞ + μ gðxÞ In a NLPP, the necessary conditions of optimality are defined by KarushKhun-Tucker (KKT) conditions.86 If vector x0 is the optimal solution for the problem, then there exist two multipliers vectors λ 2 Rm and μ 2 Rd such that: rf ðx0 Þ + λ rhðxo Þ + μ rgðxo Þ ¼ 0, hðxo Þ ¼ 0 μ gðxo Þ ¼ 0 μ 0, gðxo Þ 0 where μ 0 and gðxo Þ 0 when the minimum is being found. These conditions are necessary but not sufficient.
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There are some important disadvantages about KKT conditions as: • The system is a nonlinear system. • A priori, the number of solutions are unknown. • Every single feasible point is evaluated in the main function f(x). Sufficient conditions to the problem are: The feasible set k ¼ fx 2 Rn : gðxÞ 0, hðxÞ ¼ 0g is limited by lim jxj!∞ ¼ + ∞ • If f ’ (x) is monotonically increasing, then f ’’ðxÞ > 0 and f(x) is convex. The sufficient conditions of optimality require the study of convexity for the function f, g, and the linearity for h constraints. Every optimal solution of: minimize f ðxÞ under hðxÞ ¼ 0 gðxÞ 0 must be a solution of the system of necessary conditions of optimality (KKT) rf ðxÞ + λ rhðxÞ + μ rgðxÞ ¼ 0 μ gðxÞ ¼ 0 hðxÞ ¼ 0 μ 0, gðxÞ 0 It is possible to demonstrate that the above conditions are equivalent to : 86
μ 0, gðxÞ 0 μi gðxÞi ¼ 0, i ¼ 1,2, …, m It will be required to consider equations and unknown variables to find all the solutions for the KKT conditions: d : inequalities, μ 2 Rd m : equalities, λ 2 Rm n : variables, x 2 Rn The solutions of the d + m + n equations with d + m + n unknown variables must be found (there are 2d cases).
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6.4 APPROACH TO THE OPTIMIZATION PROBLEM The case study presented in Fig. 5.5, has been considered in this section to describe the approach. The occurrence probability associated to each basic event is needed to obtain the Qsys. Therefore, a vector (q) collects all the events probabilities together: q ¼ ½0:02 0:03 0:4 0:08 0:2 0:4 0:09. Thus, the Qsys is provided by the BDD is: Qsys ¼ q1 + q6 ð1 q1 Þ + q7 ð1 q6 Þ ð1 q1 Þ + q5 q4 ð1 q7 Þ ð1 q6 Þ ð1 q1 Þ + q3 q2 ð1 q5 Þ ð1 q7 Þ ð1 q6 Þ ð1 q1 Þ + q3 q2 ð1 q4 Þð1 q7 Þ ð1 q6 Þ ð1 q1 Þ where, by setting qn for the corresponding probability, is equal to: Qsys ¼ 47:98 % Once the occurrence probability of the top is achieved, the main objective is to reduce it to a certain desired level. The reduction of the Qsys will be done directly by the different events. A new vector Imp is defined for that propose as: Imp ¼ ½Impðe1 Þ Impðe2 Þ Impðe3 Þ Impðe4 Þ Impðe5 Þ Impðe6 Þ Impðe7 Þ Therefore, the reduction of Qsys will be done modifying qi by: f ðImpðen ÞÞ ¼ q1 Impðe1 Þ + ðq6 Impðe6 ÞÞ 1 q1 Impðe1 Þ + q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ + q5 Impðe5 Þ q4 Impðe4 Þ 1 q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ + q3 Impðe3 Þ q2 Impðe2 Þ 1 ðq5 Impðe5 ÞÞ 1 q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ + q3 Impðe3 Þ q2 Impðe2 Þ 1 ðq4 Impðe4 ÞÞ 1 q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ The event ei has an associated cost related to the improvement Imp(ei). Therefore, a new vector Cost associated with each event improvement is obtained: Cost ¼ ½10000 4000 8000 12000 7000 5000 2000
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There is at last an associated budget that is not possible to be exceeded, defined as a scalar. The optimization problem will be: minimize f ðImpðen ÞÞ under
n X Ci qi Budget i¼1
0 Impðei Þ qi a where f(Imp(en)) is the optimization function; Ci is the cost associated with each event; qi is the probability assigned to each event; Budget is the available budget; a is the parameter for the optimization software that represents the minimum allowable value. The simulations presented are solved by the fmincon function of Matlab.87 A NLPP fmincon in a standard form is: min f ðxÞ x
under c ðxÞ 0 ceq ðxÞ ¼ 0 Axb Aeq x ¼ beq lb x ub where x, b, beq, lb, ub are vectors; A, Aeq are matrices; f(x) is a function that returns a scalar. fmincon finds a minimum of a constrained nonlinear multivariable function. The syntax used in this simulation is given by: ½x, fval, exitflag, output ¼ fmicomð fun, x0 , A, b, lb, up, optionsÞ
(6.1)
The function to be optimized need to be set, i.e., to define “fun” function. The CSs and Qsys are defined as in former sections. Fig. 6.3 shows as f(Imp(ei)) is obtained.
LDT
Conversion to BDD
Fig. 6.3 Flowchart for f(Imp(ei)).
BDD associated with LDT
CSs related with the variable ordering
Rewrite Qsys with the optimization variables
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The parameters for each simulation must be defined when f(Imp(en)) is set. Initial conditions and boundary conditions must be properly defined. In accordance to the standard definition of a NLPP defined by fmincon • Lower (lb) and upper (ub) bounds and b must be vectors. • A must be a matrix. The boundaries vectors lb and ub are written in the main algorithm. Nonetheless, two algorithms have been developed to define automatically the constraints to the optimization problem. Finally, an algorithm creates the matrix related with the inequalities considering the number of variables (n) implicated, i.e., the number of events. Fig. 6.4 shows the flowchart of the algorithm: Introduce a row and locate the events (variables) in the correct position
Introduce cost and number of Events (n)
Generate A matrix
No
Assign the cost in the matrix associated to e(i)
Introduce the same number of rows as previously for each variable
Is it the last event?
(2n +1) x n matrix completed
Yes Is it the last variable?
No
Yes
Fig. 6.4 Flowchart for A matrix.
The constraints on the left side of the inequality are defined with the information of each variable and its associated cost. b has the same size as A. b is defined by Fig. 6.5. It is achievable regardless of the number of events involved with this information and an explanatory flowchart to simulate a NLPP is provided. Fig. 6.6 shows the procedure to obtain x, fval, exitflag, and the output exit variables, where: • X: new probability associated to each event (Imp(qi)) • Fval: a scalar given by f(Imp(qi)). • Exitflag: will yield the information about the solution accuracy. • Output: solution to the optimization problem.
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Assign properly the probability associated with each event
Introduce q(i), number of variables (n) and Budget
Generate b vector
No
Need to adapt properly to matrix A dimension
Events probabilities gathered correctly
Yes
Is i < n?
No Yes (2n + 1) vector completed
Is i < n ?
Fig. 6.5 How to generate b vector.
Optimization with fmincon
Save x values in“P” matrix
Calculate the new values for optimization and save them in “Improvement” matrix
Assign the proper probability to each event
Define minimization function f(I(bn))
Save fval in “f(Imp)” vector
Ready to simulate
Save exitflag and output
Define cost vector
Define the available budget
Optimization concluded
Fig. 6.6 Optimization with fmincon.
6.5 EXAMPLE Let “Tools shortage” be the MP (Top Event) at a certain business, being “Poor work schedule” (e1), “Limited tools” (e2), and “Low tools’ reliability” (e3) the events to this problem. Fig. 6.7 shows the LDT. The CSs from Fig. 6.7 are: CS1 ¼ e1 CS2 ¼ e3 e2 e1
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Top event
g1
e1
e2
e3
Fig. 6.7 Tools shortage as MP.
Let the probability vector associated be defined as q ¼ ½0:6 0:8 0:9. Thus, Qsys ¼ 0:6 + 0:9 0:8 ð1 0:6Þ ¼ 0:888 The minimization function needed to fmincon will be: f ðI ðbn ÞÞ ¼ ½ð0:6 Imp1 Þ + ðð0:9 Imp3 Þ ð0:8 Imp2 Þ ð1 ð0:6 Imp1 ÞÞÞ The constraints are given by the budget and the cost vector is as follows: Budget ¼ 200 Cost ¼ ½1000 400 800 The optimization problem is defined as: minimize ð0:6 Ib1 Þ + ðð0:9 Ib3 Þ ð0:8 Ib2 Þ ð1 ð0:6 Ib1 ÞÞÞ such that 1000 Ib1 + 400 Ib2 + 800 Ib3 20000 ð4:9Þ 0 Ib1 0:6 0:001 0 Ib2 0:8 0:001 0 Ib3 0:9 0:001 The Lagrangian to this problem is given by: L ðImp1 , Imp2 , Imp3 , μ1 , μ2 , μ3 , μ4 ,μ5 ,μ6 ,μ7 Þ ¼ ð0:6 Imp1 Þ + ðð0:9 Imp3 Þ ð0:8 Imp2 Þ ð1 ð0:6 Imp1 ÞÞÞ + μ1 ðImp1 Þ + μ2 ðImp2 Þ + μ3 ðImp3 Þ + μ4 ðImp1 0:6Þ + μ5 ðImp2 0:8Þ + μ6 ðImp3 0:9Þ + μ7 ð1000 Imp1 + 400 Imp2 + 800 Imp3 200Þ
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Considering the KKT optimal conditions: rf ðxÞ + d X
d X μi rgðxÞi ¼ 0 i¼0
μi rgðxÞi ¼ 0
i¼0
μ 0,gðxÞ 0 writing down the KKT conditions: ð0:9 Imp3 Þ ð0:8 Imp2 Þ 2 1 m1 ¼ 0 ð0:6 Imp3 Þ ð1 ð0:6 Imp1 ÞÞ ð1Þ m2 ¼ 0 ð0:8 Imp2 Þ ð1 ð0:6 Imp1 ÞÞ ð1Þ m3 ¼ 0 m1 Imp1 ¼ 0 m2 Imp2 ¼ 0 m3 Imp3 ¼ 0 m4 ðImp1 0:6Þ ¼ 0 m5 ðImp2 0:8Þ ¼ 0 m6 ðImp3 0:9Þ ¼ 0 m7 ð1000 Imp1 + 400 Imp2 + 800 Imp3 200Þ ¼ 0 where there are ten equations and ten unknown variables, considering that the constraints of the multipliers must be μi 0. Therefore, 2d cases, being d the number of inequalities, must be studied and analyzed to find the optimal solution. Every feasible value should be evaluated in the main function to obtain the minimum. There are different methods to solve this NLPP. This case has no difficulty, being possible to solve it by hand-calculations. Nonetheless, larger LDTs with a great number of events could be found in a real case scenario where the computational cost is high. The feasible solutions found by fmincon are: Imp1 ¼ 0 Imp2 ¼ 0:5 Imp3 ¼ 0 i.e., the business should invest 0:5 400 ¼ 200 € for reducing the probability of the second event from 0.8 to 0.3. Table 6.1 presents the improvement in the occurrence probabilities of each event according to the investment and the associated cost.
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Table 6.1 Data obtained after optimization Initial qn qn after optimization
Cost associated
Investment
e1 e2 e3
1000 € 400 € 800 €
0€ 200 € 0€
60% 80% 90%
60% 30% 90%
If an investment of 200 € is done for reducing the occurrence probability of the second event (e2), the MP probability of occurrence will be reduced from 88% to 70%. According to the solution found, the business might improve its “Tools Shortage” problem and to reduce its occurrence probability when they invest 200 € in the basic cause “Limited tools.” A higher budget should be assigned to each event if the business requires a major improvement in the reduction of the occurrence probability of the top event.
6.6 DYNAMIC ANALYSIS AND OPTIMIZATION Fig. 6.8 presents the flowchart of the optimization algorithm. The algorithm evaluates the main function regarding the initial conditions. When the feasible solutions are obtained, then they are recorded as: • P is the feasible solutions of the NLPP, i.e., x values given after the optimization. • Improvement matrix contains the updated probabilities of each BC. • f(Imp) vector is the main function evaluated in the feasible points given by the NLPP. fmincon is employed to find the optimal solution, or at least, a good solution. The algorithm set the simulations when the feasible points are obtained and recorded in the aforementioned variables. The main purpose of the dynamic analysis is to study the gradual reduction of the occurrence probability of the top event with a certain degree of confidence. The algorithm is capable to stop the simulation when a certain threshold is reached. It provides the resources that should be invested to obtain a certain occurrence probability of the top event in a period. Fig. 6.9 shows the flowchart including the threshold limit.
6.7 EXAMPLE The case study presented in Fig. 6.7 is solved considering a period of four months. The first month approach is the same. The solutions of the iteration “t” are used as initial conditions for the iteration “t + 1.”
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Dynamic analysis
Define cost
Define budget
Define starting point
Define the initial probabilities associated to each Event
Define lower and upper bounds
Save initial probabilities
Start simulating
Calculate the updated probability
Save optimization exit variables
Define time desired to forecast
Save updated values and refresh initial probabilities
Is it the last simulation?
Save the results in “Improvement” matrix
No
Yes End
Fig. 6.8 Optimization algorithm flowchart.
The initial values/conditions are: f ðI ðbn ÞÞ ¼ ½ð0:6 Impe1 Þ +ðð0:9 Impe3 Þ ð0:8 Impe2 Þ ð1 ð0:6 Impe1 ÞÞÞ W ¼ ½0:6 0:8 0:9 Cost ¼ ½1000 400 800 Budget ¼ 200 The initial occurrence probability of the Top Event is Qsys ¼ 88:8%. A simulation over the first month is achieved. Then the variables defined in this algorithm are given by: Improvement ¼ ½0:6 0:3 0:9 P ¼ ½0 0:5 0 f ðImpÞ ¼ 70:8%
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Dynamic analysis
Define cost
Define starting point
Define the initial probabilities associated to each event
Define budget
Define lower and upper bounds
Save the initial probabilities
Save the results in “Improvement” matrix
Is fval < Threshold?
Calculate the updated probability
No
Save optimization exit variables
Define the threshold limit to be achieved
Start simulating
Save updated values and refresh the initial probabilities
Yes
Save updated values
End
Fig. 6.9 Threshold limit simulations flowchart.
The variables will be updated in each iteration over the time. The feasible points will form the new vector of probabilities to be optimized. The approach that will provide a reduction of the occurrence probability of the Top Event requires: • The cost required. • The months to reduce the probability to a certain level. • The resources allocation of the business. There will be some cases where the MP cannot be completely removed, i.e., only a reasonable level of reduction can be achieved. Table 6.2 shows the optimized values of each event, i.e., P matrix (the probability reduction values).
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Table 6.2 Probability reduction 1st month
2nd month
3rd month
4th month
Imp1 Imp2 Imp3
0.08 0.30 0
0.20 0 0
0.20 0 0
0 0.50 0
Fig. 6.10 shows the probability reduction of the events over the time. The second event (e2) has the most important reduction in the first and the second months. It is for the first event (e1) in the third month. It can be also observed that the approach raises the value of the first event, being close to 20% during the third and fourth months.
0.6 0.5 0.4 0.3 0.2 0.1 0 3 2 Events 1
1
4
3
2 Months
Fig. 6.10 Feasible points after optimization.
The new probability values are presented in Table 6.3. Table 6.3 Probabilities for the optimization of each event Initial 1st month 2nd month 3rd month
4th month
Probability e1 Probability e2 Probability e3
0.12 0 0.90
0.60 0.80 0.90
0.60 0.30 0.90
0.52 0 0.90
0.32 0 0.90
Fig. 6.11 shows the updated probability of the events over the time. In the first month, the probability of e2 is drastically reduced. Then, the probability of e1 is reduced and the probability of e3 does not change over the period in the following months.
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1 0.8 0.6 0.4 0.2 0 3 2 1
4
3
2
1
Fig. 6.11 Events improvement.
Table 6.4 shows the values obtained by f(Imp), i.e., the probability of the MP over the period considered. Table 6.4 MP probability of occurrence Qsys Initial 1st month 2nd month
3rd month
4th month
f(Imp)
32.16%
12.17%
88.8%
70.92%
52.14%
The initial probability of occurrence for “Tools Shortage” was around 88%. After a few iterations, the algorithm states that a reduction of the occurrence probability from 88% to 12% can be achieved if the resources of the business are allocated as Tables 6.3 and 6.5. The distribution of the budget would be: Table 6.5 Costs considering the optimization solution 1st month 2nd month
Investment e1 Investment e2 Investment e3
• • • • •
0€ 200 € 0€
80 € 120 € 0€
3rd month
4th month
200 € 0€ 0€
200 € 0€ 0€
The outcomes of fmincon are: Optimization terminated: “first-order optimality measure less than options. TolFun and maximum constraint violation is less than options.TolCon” Exitflag ¼ 1 Iterations ¼ 2 FuncCount ¼ 12 Firstorderoptimality ¼ 2.2204 1016
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The algorithm developed can minimize the probability of occurrence of a particular LDT by allocating the business resources optimally. With this approach, the business will be able to: • Set the best allocation of resources each month to reduce the occurrence probability of the Top Event. • Have under control the events which lead to a certain problem (Top Event) in the business. • Have a certain degree of confidence in the investment done in each event.
6.8 Real Case Study Fig. 5.1 shows a LDT related to a real case study of a confidential firm. Table 6.6 details the initial conditions. Fig. 5.2 BDD obtained from Fig. 5.1 presents the BDD obtained from this LDT. The probability of occurrence of the MP is 0.4825, calculated by using the analytical expression from Fig. 5.2 and the probabilities of Table 6.6. The maximum investment (MI) is: MI ¼ 0:4 0:9 500 + 0:4 0:5 150 + 0:2 0:4 400 + 0:3 0:9 400 + 0:1 0:8 150 + 0:2 0:5 200 + 0:25 0:6 500 + 0:2 0:3 300 + 0:1 0:95 900 ¼ 560:5 Fig. 6.12 shows the maximum investment for each BC. If this maximum budget were available, then the minimum probabilities of occurrence (Pmin) of the BCs will be: Pmin ¼ ½0:04 0:20 0:12 0:03 0:020 0:10 0:10 0:14 0:005 and, consequently, the minimum QMP is: QMP ¼ 0:1329 Therefore, when the available budget is less than 560.5 monetary units (mu), it will be necessary to choose the BCs that are the best to be improved in order to minimize the QMP. The budget considered in the following example is 350 mu. For this purpose, the problem to optimize will be: minimize QMP ðImpðBC ÞÞ
DM Optimization
Table 6.6 Case study: initial conditions Probability Description Notation of occurrence
BUDGET 350 Customer complaints Errors from the quality control department Lack of training No internal training is provided Lack of necessary equipment Failures in protocols Review protocols are not well defined Absenteeism training courses Errors from the logistics department Problems in the distribution Poorly optimized distribution Poorly established routes Forecast problems Driving inappropriate for fragile products Problems in stores Improperly stored products Obsolete warehouse
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IC (Monetary units)
Improvement limit (a)
MP Cause 1
– –
– –
– –
Cause 2 BC1
– 40%
– 500
– 90%
BC2
40%
150
50%
Cause 3 BC3
– 20%
– 400
– 40%
BC4
30%
400
90%
Cause 4
–
–
–
Cause 5
–
–
–
Cause 6
–
–
–
BC5
10%
150
80%
BC6 BC7
20% 25%
200 500
50% 60%
Cause 7 BC8
– 20%
– 300
– 30%
BC9
10%
900
95%
subject to ð500 ImpðBC1 Þ + 150 ImpðBC2 Þ + 400 ImpðBC3 Þ + 400 ImpðBC4 Þ + 150 ImpðBC5 Þ + 200 ImpðBC6 Þ + 500 ImpðBC7 Þ + 300 ImpðBC8 Þ + 900 ImpðBC9 ÞÞ 10 350 ImpðBC1 Þ 0:36 0; ImpðBC1 Þ 0
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Maximum investment
Layout (monetary units)
200
150
100
50
0
1
2
3
4
5
6
7
8
9
Basic causes
Fig. 6.12 Maximum investment allowed.
ImpðBC2 Þ 0:2 0; ImpðBC2 Þ 0 ImpðBC3 Þ 0:12 0; ImpðBC3 Þ 0 ImpðBC4 Þ 0:27 0; ImpðBC4 Þ 0 ImpðBC5 Þ 0:08 0; ImpðBC5 Þ 0 ImpðBC6 Þ 0:1 0; ImpðBC6 Þ 0 ImpðBC7 Þ 0:2 0; ImpðBC4 Þ 0 ImpðBC8 Þ 0:06 0; ImpðBC5 Þ 0 ImpðBC9 Þ 0:095 0; ImpðBC6 Þ 0 Fig. 6.13 shows the optimal investment allocation subject to a budget of 350 mu to minimize QMP. Optimal investment allocation
Outlays (monetary units)
200
150
100
50
0
1
2
Fig. 6.13 Optimal investment.
3
4
5 Basic causes
6
7
8
9
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There are 2 BCs that cannot be improved due to the fact that budget is limited, and there are 3 BCs whose maximum investments are not completed. The missing amounts to invest are: BC1 ¼ 105mu, BC6 ¼ 20mu, BC9 ¼ 85mu Fig. 6.14 shows the behavior of the QMP reduction according to the available budget. The QMP presents a nonlinear behavior. 0.5 0.45 0.4 QMP
0.35 0.3 0.25 0.2 0.15 0.1 50
100
150
200
250
300
350
400
450
500
550
Available budget
Fig. 6.14 Optimal investment allocation.
Fig. 6.15 shows the improvement of each investment compared to the previous one. The investment considered rises with a step of 50 mu. Improvement compared to the previous investment 10
Improvement (%)
8
6
4
2
0
100
150
200
250
300
350
400
450
500
550
Available budget
Fig. 6.15 Percentage improvement.
The slope is more pronounced at the beginning (Fig. 6.14). The first 250 mu are more useful than the rest of the investment (Fig. 6.15). This is useful information when the availability of budget is limited.
DM Optimization 6.1 INTRODUCTION The conversion from logical decision tree (LDT) to binary decision diagram (BDD) was presented in Chapters 3 and 4. This method allows for obtaining the analytical expression of the top event occurrence probability in function of the probabilities of basic events. This expression defines quantitatively the behavior of a system, or process, but it does not take into account the exogenous variables that, in certain cases, could be more influential than the endogenous ones. This chapter presents a novel approach to consider external factors that may be important for optimizing the DM processes.82 It is based on considering the analytical expression obtained from the BDD as the objective function of a NLPP, where the exogenous variables are modeled by a set of constraints.83 An optimal investment, subject to the limitation of resources, is the main objective of the DM process. This book proposes two strategies to support the resource allocation when a MP is given. Fig. 6.1 shows the steps to be done before using the novel methods.
1. Obtaining LDT This step requires a qualitative analysis 2. Conversion from LDT into BDD This step requires a software able to execute control sentences 3. Obtaining cut-sets and analytical expression of probability Anytical expression of the main problem probability, depending on probabilities of BCs 4. Mathematical optimization approach or Birnbaum-cost measure method
Fig. 6.1 Proposed optimization method process.
Decision-Making Management http://dx.doi.org/10.1016/B978-0-12-811540-4.00006-5
© 2017 Elsevier Inc. All rights reserved.
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6.2 DM APPROACH The LDT design is not complicated, but it requires an adequate knowledge of the problem. Fig. 6.2 is composed of 32 basic causes and 26 nonbasic causes. One of the branches of this LDT has been zoomed in Fig. 6.2 to show some causes in detail. The MP is “delay in the orders,” therefore, the business is seeking to minimize these delays.
Delay in the orders
Wrong Projects Interaction
Wrong Stock Project
Mixture of information between transacctions
Lack of a global planning
Employees steal material
Lack of Employees Training
Shortage of statistic resources
Sampling mistakes
Parameters’ selection mistakes
Wrong Simulations
Unreal Warehouse Data
Poor production management
Forecast Mistakes
Out-dated analysis techniques
Delay in Repair Orders
Lack of notification
Employees with low qualification
No coordination between workers
Poor comunication between workers
Human Resources
Lack of simulations
Lack of daily notification
Errors in Paperwork Creation
Tools Shortage
Forecast Mistakes Wrong planning of job position
Out-dated analysis techniques
Shortage of statistic resources
Wrong Tools Stock
Limited number of tools
Employees with limited qualification
Manufacturin g Closed Late
Lack of resources in some departments
Limited connections between departments
Manufacturing order not scheduled
Low tools’ reliability
Wrong detailed needs in Organization Chart
Lack of inner training courses
Renege on inner Procedures
Lack of Employees Training
Employees with low qualification
Lack of inner training courses
Wrong Departmental Personnel Planning
Lack of incentives
Poor Organisation Between Departments
Mismanagement of Human Resources department
Limited salary
Mismatch between Needs
Overcharge of capacity
Bad work atmosphere Flow Absence of information between Departments
Unreal capacity work
Wrong work planning
Poor relationships between department’s leaders
Lack of Software implemented does not work properly
notification
Unpleasant relationships between employees
Wrong simulations
Unreal warehouse data
Employees steal material
Poor comunication between workers
No coordination between workers
Forecast mistakes
Lack of simulations
Lack of daily notification
Out-dated analysis techniques
Fig. 6.2 Example of a LDT.
• • • •
The following procedure is suggested to create a LDT: Define properly the MP and its scope. Detect the BCs related with the MP. Study the interrelation between mentioned BCs and the MP. Obtain the probability data for each BC.84
Shortage of statistic resources
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6.3 Mathematical NLPP Background The problem to address in this section is: minimize f ðxÞ subject to hðxÞ ¼ 0 gðxÞ 0 where in case that any of the functions involved are nonlinear, the problem will be a Non-Linear Programming Problem (NLPP).85 NLPP is defined by a function f : Rn ! R, and two vector functions that represent the constraints. h : Rn ! Rm g : Rn ! Rd x 2 Rn f : Rn ! R being m the number of (nonlinear) equality constraint, and d is the number of (nonlinear) inequality constraints. As a result, the following set of vectors will correspond to the feasible set: A ¼ fx 2 Rn : hðxÞ ¼ 0, gðxÞ 0g Rn The conditions of optimality are set by the Lagrange criterion. The Lagrange multipliers for a NLPP with both equality and inequality constraints are defined as: Lðx, λ, μÞ ¼ f ðxÞ + λ hðxÞ + μ gðxÞ In a NLPP, the necessary conditions of optimality are defined by KarushKhun-Tucker (KKT) conditions.86 If vector x0 is the optimal solution for the problem, then there exist two multipliers vectors λ 2 Rm and μ 2 Rd such that: rf ðx0 Þ + λ rhðxo Þ + μ rgðxo Þ ¼ 0, hðxo Þ ¼ 0 μ gðxo Þ ¼ 0 μ 0, gðxo Þ 0 where μ 0 and gðxo Þ 0 when the minimum is being found. These conditions are necessary but not sufficient.
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There are some important disadvantages about KKT conditions as: • The system is a nonlinear system. • A priori, the number of solutions are unknown. • Every single feasible point is evaluated in the main function f(x). Sufficient conditions to the problem are: The feasible set k ¼ fx 2 Rn : gðxÞ 0, hðxÞ ¼ 0g is limited by lim jxj!∞ ¼ + ∞ • If f ’ (x) is monotonically increasing, then f ’’ðxÞ > 0 and f(x) is convex. The sufficient conditions of optimality require the study of convexity for the function f, g, and the linearity for h constraints. Every optimal solution of: minimize f ðxÞ under hðxÞ ¼ 0 gðxÞ 0 must be a solution of the system of necessary conditions of optimality (KKT) rf ðxÞ + λ rhðxÞ + μ rgðxÞ ¼ 0 μ gðxÞ ¼ 0 hðxÞ ¼ 0 μ 0, gðxÞ 0 It is possible to demonstrate that the above conditions are equivalent to : 86
μ 0, gðxÞ 0 μi gðxÞi ¼ 0, i ¼ 1,2, …, m It will be required to consider equations and unknown variables to find all the solutions for the KKT conditions: d : inequalities, μ 2 Rd m : equalities, λ 2 Rm n : variables, x 2 Rn The solutions of the d + m + n equations with d + m + n unknown variables must be found (there are 2d cases).
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6.4 APPROACH TO THE OPTIMIZATION PROBLEM The case study presented in Fig. 5.5, has been considered in this section to describe the approach. The occurrence probability associated to each basic event is needed to obtain the Qsys. Therefore, a vector (q) collects all the events probabilities together: q ¼ ½0:02 0:03 0:4 0:08 0:2 0:4 0:09. Thus, the Qsys is provided by the BDD is: Qsys ¼ q1 + q6 ð1 q1 Þ + q7 ð1 q6 Þ ð1 q1 Þ + q5 q4 ð1 q7 Þ ð1 q6 Þ ð1 q1 Þ + q3 q2 ð1 q5 Þ ð1 q7 Þ ð1 q6 Þ ð1 q1 Þ + q3 q2 ð1 q4 Þð1 q7 Þ ð1 q6 Þ ð1 q1 Þ where, by setting qn for the corresponding probability, is equal to: Qsys ¼ 47:98 % Once the occurrence probability of the top is achieved, the main objective is to reduce it to a certain desired level. The reduction of the Qsys will be done directly by the different events. A new vector Imp is defined for that propose as: Imp ¼ ½Impðe1 Þ Impðe2 Þ Impðe3 Þ Impðe4 Þ Impðe5 Þ Impðe6 Þ Impðe7 Þ Therefore, the reduction of Qsys will be done modifying qi by: f ðImpðen ÞÞ ¼ q1 Impðe1 Þ + ðq6 Impðe6 ÞÞ 1 q1 Impðe1 Þ + q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ + q5 Impðe5 Þ q4 Impðe4 Þ 1 q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ + q3 Impðe3 Þ q2 Impðe2 Þ 1 ðq5 Impðe5 ÞÞ 1 q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ + q3 Impðe3 Þ q2 Impðe2 Þ 1 ðq4 Impðe4 ÞÞ 1 q7 Impðe7 Þ 1 q6 Impðe6 Þ 1 q1 Impðe1 Þ The event ei has an associated cost related to the improvement Imp(ei). Therefore, a new vector Cost associated with each event improvement is obtained: Cost ¼ ½10000 4000 8000 12000 7000 5000 2000
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There is at last an associated budget that is not possible to be exceeded, defined as a scalar. The optimization problem will be: minimize f ðImpðen ÞÞ under
n X Ci qi Budget i¼1
0 Impðei Þ qi a where f(Imp(en)) is the optimization function; Ci is the cost associated with each event; qi is the probability assigned to each event; Budget is the available budget; a is the parameter for the optimization software that represents the minimum allowable value. The simulations presented are solved by the fmincon function of Matlab.87 A NLPP fmincon in a standard form is: min f ðxÞ x
under c ðxÞ 0 ceq ðxÞ ¼ 0 Axb Aeq x ¼ beq lb x ub where x, b, beq, lb, ub are vectors; A, Aeq are matrices; f(x) is a function that returns a scalar. fmincon finds a minimum of a constrained nonlinear multivariable function. The syntax used in this simulation is given by: ½x, fval, exitflag, output ¼ fmicomð fun, x0 , A, b, lb, up, optionsÞ
(6.1)
The function to be optimized need to be set, i.e., to define “fun” function. The CSs and Qsys are defined as in former sections. Fig. 6.3 shows as f(Imp(ei)) is obtained.
LDT
Conversion to BDD
Fig. 6.3 Flowchart for f(Imp(ei)).
BDD associated with LDT
CSs related with the variable ordering
Rewrite Qsys with the optimization variables
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The parameters for each simulation must be defined when f(Imp(en)) is set. Initial conditions and boundary conditions must be properly defined. In accordance to the standard definition of a NLPP defined by fmincon • Lower (lb) and upper (ub) bounds and b must be vectors. • A must be a matrix. The boundaries vectors lb and ub are written in the main algorithm. Nonetheless, two algorithms have been developed to define automatically the constraints to the optimization problem. Finally, an algorithm creates the matrix related with the inequalities considering the number of variables (n) implicated, i.e., the number of events. Fig. 6.4 shows the flowchart of the algorithm: Introduce a row and locate the events (variables) in the correct position
Introduce cost and number of Events (n)
Generate A matrix
No
Assign the cost in the matrix associated to e(i)
Introduce the same number of rows as previously for each variable
Is it the last event?
(2n +1) x n matrix completed
Yes Is it the last variable?
No
Yes
Fig. 6.4 Flowchart for A matrix.
The constraints on the left side of the inequality are defined with the information of each variable and its associated cost. b has the same size as A. b is defined by Fig. 6.5. It is achievable regardless of the number of events involved with this information and an explanatory flowchart to simulate a NLPP is provided. Fig. 6.6 shows the procedure to obtain x, fval, exitflag, and the output exit variables, where: • X: new probability associated to each event (Imp(qi)) • Fval: a scalar given by f(Imp(qi)). • Exitflag: will yield the information about the solution accuracy. • Output: solution to the optimization problem.
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Assign properly the probability associated with each event
Introduce q(i), number of variables (n) and Budget
Generate b vector
No
Need to adapt properly to matrix A dimension
Events probabilities gathered correctly
Yes
Is i < n?
No Yes (2n + 1) vector completed
Is i < n ?
Fig. 6.5 How to generate b vector.
Optimization with fmincon
Save x values in“P” matrix
Calculate the new values for optimization and save them in “Improvement” matrix
Assign the proper probability to each event
Define minimization function f(I(bn))
Save fval in “f(Imp)” vector
Ready to simulate
Save exitflag and output
Define cost vector
Define the available budget
Optimization concluded
Fig. 6.6 Optimization with fmincon.
6.5 EXAMPLE Let “Tools shortage” be the MP (Top Event) at a certain business, being “Poor work schedule” (e1), “Limited tools” (e2), and “Low tools’ reliability” (e3) the events to this problem. Fig. 6.7 shows the LDT. The CSs from Fig. 6.7 are: CS1 ¼ e1 CS2 ¼ e3 e2 e1
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Top event
g1
e1
e2
e3
Fig. 6.7 Tools shortage as MP.
Let the probability vector associated be defined as q ¼ ½0:6 0:8 0:9. Thus, Qsys ¼ 0:6 + 0:9 0:8 ð1 0:6Þ ¼ 0:888 The minimization function needed to fmincon will be: f ðI ðbn ÞÞ ¼ ½ð0:6 Imp1 Þ + ðð0:9 Imp3 Þ ð0:8 Imp2 Þ ð1 ð0:6 Imp1 ÞÞÞ The constraints are given by the budget and the cost vector is as follows: Budget ¼ 200 Cost ¼ ½1000 400 800 The optimization problem is defined as: minimize ð0:6 Ib1 Þ + ðð0:9 Ib3 Þ ð0:8 Ib2 Þ ð1 ð0:6 Ib1 ÞÞÞ such that 1000 Ib1 + 400 Ib2 + 800 Ib3 20000 ð4:9Þ 0 Ib1 0:6 0:001 0 Ib2 0:8 0:001 0 Ib3 0:9 0:001 The Lagrangian to this problem is given by: L ðImp1 , Imp2 , Imp3 , μ1 , μ2 , μ3 , μ4 ,μ5 ,μ6 ,μ7 Þ ¼ ð0:6 Imp1 Þ + ðð0:9 Imp3 Þ ð0:8 Imp2 Þ ð1 ð0:6 Imp1 ÞÞÞ + μ1 ðImp1 Þ + μ2 ðImp2 Þ + μ3 ðImp3 Þ + μ4 ðImp1 0:6Þ + μ5 ðImp2 0:8Þ + μ6 ðImp3 0:9Þ + μ7 ð1000 Imp1 + 400 Imp2 + 800 Imp3 200Þ
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Considering the KKT optimal conditions: rf ðxÞ + d X
d X μi rgðxÞi ¼ 0 i¼0
μi rgðxÞi ¼ 0
i¼0
μ 0,gðxÞ 0 writing down the KKT conditions: ð0:9 Imp3 Þ ð0:8 Imp2 Þ 2 1 m1 ¼ 0 ð0:6 Imp3 Þ ð1 ð0:6 Imp1 ÞÞ ð1Þ m2 ¼ 0 ð0:8 Imp2 Þ ð1 ð0:6 Imp1 ÞÞ ð1Þ m3 ¼ 0 m1 Imp1 ¼ 0 m2 Imp2 ¼ 0 m3 Imp3 ¼ 0 m4 ðImp1 0:6Þ ¼ 0 m5 ðImp2 0:8Þ ¼ 0 m6 ðImp3 0:9Þ ¼ 0 m7 ð1000 Imp1 + 400 Imp2 + 800 Imp3 200Þ ¼ 0 where there are ten equations and ten unknown variables, considering that the constraints of the multipliers must be μi 0. Therefore, 2d cases, being d the number of inequalities, must be studied and analyzed to find the optimal solution. Every feasible value should be evaluated in the main function to obtain the minimum. There are different methods to solve this NLPP. This case has no difficulty, being possible to solve it by hand-calculations. Nonetheless, larger LDTs with a great number of events could be found in a real case scenario where the computational cost is high. The feasible solutions found by fmincon are: Imp1 ¼ 0 Imp2 ¼ 0:5 Imp3 ¼ 0 i.e., the business should invest 0:5 400 ¼ 200 € for reducing the probability of the second event from 0.8 to 0.3. Table 6.1 presents the improvement in the occurrence probabilities of each event according to the investment and the associated cost.
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Table 6.1 Data obtained after optimization Initial qn qn after optimization
Cost associated
Investment
e1 e2 e3
1000 € 400 € 800 €
0€ 200 € 0€
60% 80% 90%
60% 30% 90%
If an investment of 200 € is done for reducing the occurrence probability of the second event (e2), the MP probability of occurrence will be reduced from 88% to 70%. According to the solution found, the business might improve its “Tools Shortage” problem and to reduce its occurrence probability when they invest 200 € in the basic cause “Limited tools.” A higher budget should be assigned to each event if the business requires a major improvement in the reduction of the occurrence probability of the top event.
6.6 DYNAMIC ANALYSIS AND OPTIMIZATION Fig. 6.8 presents the flowchart of the optimization algorithm. The algorithm evaluates the main function regarding the initial conditions. When the feasible solutions are obtained, then they are recorded as: • P is the feasible solutions of the NLPP, i.e., x values given after the optimization. • Improvement matrix contains the updated probabilities of each BC. • f(Imp) vector is the main function evaluated in the feasible points given by the NLPP. fmincon is employed to find the optimal solution, or at least, a good solution. The algorithm set the simulations when the feasible points are obtained and recorded in the aforementioned variables. The main purpose of the dynamic analysis is to study the gradual reduction of the occurrence probability of the top event with a certain degree of confidence. The algorithm is capable to stop the simulation when a certain threshold is reached. It provides the resources that should be invested to obtain a certain occurrence probability of the top event in a period. Fig. 6.9 shows the flowchart including the threshold limit.
6.7 EXAMPLE The case study presented in Fig. 6.7 is solved considering a period of four months. The first month approach is the same. The solutions of the iteration “t” are used as initial conditions for the iteration “t + 1.”
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Dynamic analysis
Define cost
Define budget
Define starting point
Define the initial probabilities associated to each Event
Define lower and upper bounds
Save initial probabilities
Start simulating
Calculate the updated probability
Save optimization exit variables
Define time desired to forecast
Save updated values and refresh initial probabilities
Is it the last simulation?
Save the results in “Improvement” matrix
No
Yes End
Fig. 6.8 Optimization algorithm flowchart.
The initial values/conditions are: f ðI ðbn ÞÞ ¼ ½ð0:6 Impe1 Þ +ðð0:9 Impe3 Þ ð0:8 Impe2 Þ ð1 ð0:6 Impe1 ÞÞÞ W ¼ ½0:6 0:8 0:9 Cost ¼ ½1000 400 800 Budget ¼ 200 The initial occurrence probability of the Top Event is Qsys ¼ 88:8%. A simulation over the first month is achieved. Then the variables defined in this algorithm are given by: Improvement ¼ ½0:6 0:3 0:9 P ¼ ½0 0:5 0 f ðImpÞ ¼ 70:8%
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Dynamic analysis
Define cost
Define starting point
Define the initial probabilities associated to each event
Define budget
Define lower and upper bounds
Save the initial probabilities
Save the results in “Improvement” matrix
Is fval < Threshold?
Calculate the updated probability
No
Save optimization exit variables
Define the threshold limit to be achieved
Start simulating
Save updated values and refresh the initial probabilities
Yes
Save updated values
End
Fig. 6.9 Threshold limit simulations flowchart.
The variables will be updated in each iteration over the time. The feasible points will form the new vector of probabilities to be optimized. The approach that will provide a reduction of the occurrence probability of the Top Event requires: • The cost required. • The months to reduce the probability to a certain level. • The resources allocation of the business. There will be some cases where the MP cannot be completely removed, i.e., only a reasonable level of reduction can be achieved. Table 6.2 shows the optimized values of each event, i.e., P matrix (the probability reduction values).
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Table 6.2 Probability reduction 1st month
2nd month
3rd month
4th month
Imp1 Imp2 Imp3
0.08 0.30 0
0.20 0 0
0.20 0 0
0 0.50 0
Fig. 6.10 shows the probability reduction of the events over the time. The second event (e2) has the most important reduction in the first and the second months. It is for the first event (e1) in the third month. It can be also observed that the approach raises the value of the first event, being close to 20% during the third and fourth months.
0.6 0.5 0.4 0.3 0.2 0.1 0 3 2 Events 1
1
4
3
2 Months
Fig. 6.10 Feasible points after optimization.
The new probability values are presented in Table 6.3. Table 6.3 Probabilities for the optimization of each event Initial 1st month 2nd month 3rd month
4th month
Probability e1 Probability e2 Probability e3
0.12 0 0.90
0.60 0.80 0.90
0.60 0.30 0.90
0.52 0 0.90
0.32 0 0.90
Fig. 6.11 shows the updated probability of the events over the time. In the first month, the probability of e2 is drastically reduced. Then, the probability of e1 is reduced and the probability of e3 does not change over the period in the following months.
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1 0.8 0.6 0.4 0.2 0 3 2 1
4
3
2
1
Fig. 6.11 Events improvement.
Table 6.4 shows the values obtained by f(Imp), i.e., the probability of the MP over the period considered. Table 6.4 MP probability of occurrence Qsys Initial 1st month 2nd month
3rd month
4th month
f(Imp)
32.16%
12.17%
88.8%
70.92%
52.14%
The initial probability of occurrence for “Tools Shortage” was around 88%. After a few iterations, the algorithm states that a reduction of the occurrence probability from 88% to 12% can be achieved if the resources of the business are allocated as Tables 6.3 and 6.5. The distribution of the budget would be: Table 6.5 Costs considering the optimization solution 1st month 2nd month
Investment e1 Investment e2 Investment e3
• • • • •
0€ 200 € 0€
80 € 120 € 0€
3rd month
4th month
200 € 0€ 0€
200 € 0€ 0€
The outcomes of fmincon are: Optimization terminated: “first-order optimality measure less than options. TolFun and maximum constraint violation is less than options.TolCon” Exitflag ¼ 1 Iterations ¼ 2 FuncCount ¼ 12 Firstorderoptimality ¼ 2.2204 1016
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The algorithm developed can minimize the probability of occurrence of a particular LDT by allocating the business resources optimally. With this approach, the business will be able to: • Set the best allocation of resources each month to reduce the occurrence probability of the Top Event. • Have under control the events which lead to a certain problem (Top Event) in the business. • Have a certain degree of confidence in the investment done in each event.
6.8 Real Case Study Fig. 5.1 shows a LDT related to a real case study of a confidential firm. Table 6.6 details the initial conditions. Fig. 5.2 BDD obtained from Fig. 5.1 presents the BDD obtained from this LDT. The probability of occurrence of the MP is 0.4825, calculated by using the analytical expression from Fig. 5.2 and the probabilities of Table 6.6. The maximum investment (MI) is: MI ¼ 0:4 0:9 500 + 0:4 0:5 150 + 0:2 0:4 400 + 0:3 0:9 400 + 0:1 0:8 150 + 0:2 0:5 200 + 0:25 0:6 500 + 0:2 0:3 300 + 0:1 0:95 900 ¼ 560:5 Fig. 6.12 shows the maximum investment for each BC. If this maximum budget were available, then the minimum probabilities of occurrence (Pmin) of the BCs will be: Pmin ¼ ½0:04 0:20 0:12 0:03 0:020 0:10 0:10 0:14 0:005 and, consequently, the minimum QMP is: QMP ¼ 0:1329 Therefore, when the available budget is less than 560.5 monetary units (mu), it will be necessary to choose the BCs that are the best to be improved in order to minimize the QMP. The budget considered in the following example is 350 mu. For this purpose, the problem to optimize will be: minimize QMP ðImpðBC ÞÞ
DM Optimization
Table 6.6 Case study: initial conditions Probability Description Notation of occurrence
BUDGET 350 Customer complaints Errors from the quality control department Lack of training No internal training is provided Lack of necessary equipment Failures in protocols Review protocols are not well defined Absenteeism training courses Errors from the logistics department Problems in the distribution Poorly optimized distribution Poorly established routes Forecast problems Driving inappropriate for fragile products Problems in stores Improperly stored products Obsolete warehouse
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IC (Monetary units)
Improvement limit (a)
MP Cause 1
– –
– –
– –
Cause 2 BC1
– 40%
– 500
– 90%
BC2
40%
150
50%
Cause 3 BC3
– 20%
– 400
– 40%
BC4
30%
400
90%
Cause 4
–
–
–
Cause 5
–
–
–
Cause 6
–
–
–
BC5
10%
150
80%
BC6 BC7
20% 25%
200 500
50% 60%
Cause 7 BC8
– 20%
– 300
– 30%
BC9
10%
900
95%
subject to ð500 ImpðBC1 Þ + 150 ImpðBC2 Þ + 400 ImpðBC3 Þ + 400 ImpðBC4 Þ + 150 ImpðBC5 Þ + 200 ImpðBC6 Þ + 500 ImpðBC7 Þ + 300 ImpðBC8 Þ + 900 ImpðBC9 ÞÞ 10 350 ImpðBC1 Þ 0:36 0; ImpðBC1 Þ 0
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Maximum investment
Layout (monetary units)
200
150
100
50
0
1
2
3
4
5
6
7
8
9
Basic causes
Fig. 6.12 Maximum investment allowed.
ImpðBC2 Þ 0:2 0; ImpðBC2 Þ 0 ImpðBC3 Þ 0:12 0; ImpðBC3 Þ 0 ImpðBC4 Þ 0:27 0; ImpðBC4 Þ 0 ImpðBC5 Þ 0:08 0; ImpðBC5 Þ 0 ImpðBC6 Þ 0:1 0; ImpðBC6 Þ 0 ImpðBC7 Þ 0:2 0; ImpðBC4 Þ 0 ImpðBC8 Þ 0:06 0; ImpðBC5 Þ 0 ImpðBC9 Þ 0:095 0; ImpðBC6 Þ 0 Fig. 6.13 shows the optimal investment allocation subject to a budget of 350 mu to minimize QMP. Optimal investment allocation
Outlays (monetary units)
200
150
100
50
0
1
2
Fig. 6.13 Optimal investment.
3
4
5 Basic causes
6
7
8
9
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There are 2 BCs that cannot be improved due to the fact that budget is limited, and there are 3 BCs whose maximum investments are not completed. The missing amounts to invest are: BC1 ¼ 105mu, BC6 ¼ 20mu, BC9 ¼ 85mu Fig. 6.14 shows the behavior of the QMP reduction according to the available budget. The QMP presents a nonlinear behavior. 0.5 0.45 0.4 QMP
0.35 0.3 0.25 0.2 0.15 0.1 50
100
150
200
250
300
350
400
450
500
550
Available budget
Fig. 6.14 Optimal investment allocation.
Fig. 6.15 shows the improvement of each investment compared to the previous one. The investment considered rises with a step of 50 mu. Improvement compared to the previous investment 10
Improvement (%)
8
6
4
2
0
100
150
200
250
300
350
400
450
500
550
Available budget
Fig. 6.15 Percentage improvement.
The slope is more pronounced at the beginning (Fig. 6.14). The first 250 mu are more useful than the rest of the investment (Fig. 6.15). This is useful information when the availability of budget is limited.