Before developing and stating any conclusions from the article, it is essential to identify all assumptions that have been made.  In this particular article the flow of graduate students at different states in an academic career were examined.  Thus students can be enrolled in the master or doctoral program as either full or part-time (Nicholls, 2007, pp. 770-771).  In most cases, the flow between these programs and statuses are intertwined meaning that one could change from the master program to the doctoral program; note that the converse of these situations are also possible.  However in this article the flow between these programs is not permitted because at this particular university there have not been any cases in which this switch has occurred (Nichols, 2007, p 771).  This implies that once a candidate declares themselves as a member of the master or doctoral program then they will either complete or withdraw from that specified program.  On the other hand it is possible to transfer from part time to full time status and vice-versa.  Furthermore, academic durations will be generalized as follows; doctoral candidates who are full time typically complete their degree within two to five years and part time candidates within three to eight years, while master candidates who are full time finish within one to three years and part time candidates range from a year-and-a-half to six years (Nicholls, 2007, pp. 775; 783).  With the assumptions stated, the Markov chain models for doctoral and master students will be developed and analyzed.

The doctoral student flow model will be established by first identifying the possible states, which are summarized in table 1 (Nicholls, 2007, p. 776).

Table 1: Definition of transient and absorbing states for Doctoral Student Flow Model

Notice that there is a time restriction based on the assumptions stated above, and the absorbing states consist of either withdrawal from the program or an accepted thesis.  It should be noted that a candidate can also fail their required thesis which then creates another absorbing state, j^* = 16, or allows this situation to be considered a withdrawal (Nicholls, 2007, p. 775).  The latter is how this situation will be handled in the following findings.  Now that the states of the system have been defined, it is possible to calculate long-run probabilities, first passage times and expected completion rates.  In order to understand the process of calculating this information, one can either use the formulas presented in the article or the methods illustrated in the succeeding simple examples.

Consider the following problem to show how steady-state probabilities can be calculated.  If it is raining today, the odds of it raining tomorrow is forty percent, while the odds of it not raining is sixty percent.  If it is not raining today, there is a twenty percent chance of rain tomorrow, while there is an eighty percent chance of no rain tomorrow (Hillier & Lieberman, 2010, p. 724).  Thus state one will represent it raining and state two will represent it being clear.  The stochastic model could be represented by the following directed graph:

After the states of the system are defined, the transition matrix, P, can be formed such that .  In order to calculate the long-run (steady-state) probabilities of the system, let t = [x, y] and find the values of x and y that satisfy tP = t.  Note that t is a row matrix that is determined by the number of states in the system.  Thus if there are five states in a system then t will be a one-by-five row matrix.  In general, t is a one-by-n matrix where n is the number of states in the system.  Hence, tP = t implies [x, y] = [x, y]; [0.4x + 0.2y, 0.6x + 0.8y] = [x, y]; 0.4x + 0.2y = x and 0.6x + 0.8y = y.  By letting x = 1 it follows that y = 3.  Now [x, y] = [1, 3] must be normalized which implies that  (1/(1+3))*[1, 3] = [ 0.25, 0.75] = t.  Therefore for this example there is a twenty-five percent chance of rain and a seventy-five percent chance of a given day being clear in the long-run.

To demonstrate how to calculate the first passage time and probability of absorption consider the following example.  Suppose the following transition matrix is given such that  (McGovern, 2011).  Notice that states three and four are absorbing which means that P₂ can be rewritten as P₂  .  With transition matrix P₂ rearranged in the above form, the expected number of times the Markov process will be in a given state before being absorbed can be calculated as follows: (I – N)^-1 =  .  Therefore, if the system starts in state one then the first passage time until absorption in state three or four is 1.43 + 1.02 = 2.45 steps.  Furthermore,

(I – N)^-1 A = provides the probability of absorption by any given absorbing state.  Thus if starting in state two, there is a twenty-nine percent chance the process will be absorbed by state three, and a seventy-one percent chance that the process will be absorbed by state four.  By using these simple examples to illustrate the methodology of obtaining steady-state probabilities, first passage times, and probability of absorption we can now return to the calculations of the article with an understanding of how these results could be formed.

The steady-state absorption probabilities for a full time doctoral candidate indicated that sixty-five percent of first year students had their thesis accepted, and this number increased up to approximately seventy-five percent for third year candidates (Nicholls, 2007, p. 779).  These results make sense since the longer a candidate is in the program, the more likely that candidate will finish.  However due to the assumptions of this system, if a candidate is at the completion of his/her fifth year then their probability of withdrawal is one-hundred percent because of the limited timeframe.  Ultimately, the same relationship holds true for part time doctoral candidates.  A positive correlation exists between the probability of an accepted thesis and the length of time enrolled in the program, with respect to the specified restrictions.  Most notably, only forty percent of part time candidates in their first year attain an approved thesis where sixty-seven percent of candidates in their third year attain this goal.  Moreover, the first passage times indicate that the majority of full time candidates enter an absorbing state very close to the five year limit.  The one exception is students who are currently in their fifth year.  These students typically become absorbed within the next year-and-a-half.  Likewise, part time candidate absorption normally occurs between years four and six.  Lastly, the results indicate that the majority of full and part time candidates reach completion between years five and seven (Nicholls, 2007, pp. 780-781).  With the conclusion of the doctoral flow model, it follows that the master flow model can be calculated in a similar manner.

Using the same methods as the doctoral flow model, the master flow model will consist of fewer states since the timeframe until completion is much shorter compared to a doctoral candidate.  The results indicate that eighty-three percent of first year full time master candidates withdraw from the program compared to only thirty-eight percent of first year part time candidates (Nicholls, 2007, p. 784).  These large discrepancies between the probabilities of withdrawal verses thesis acceptance appears to be the reverse of the situation in the doctoral flow model.  Also, full time candidates are expected to be in the system for approximately three years, except when they are currently in their third year, and part time candidates are typically in the system for three to six years.  Finally, the expected completion rates are significantly lower when compared to the doctoral completion rates and only attains an acceptable completion percentage of sixty-three percent for part time candidates during their fifth year (Nicholls, 2007, pp. 784-785).

Thus, with the cumulative results, the article concludes that the highest completion rates are full time doctoral candidates and part time master candidates.  The most concerning area is the full time master candidate since they have an inadequate completion percentage of seventeen percent (Nicholls, 2007, pp. 788-789).  This study has created a more defined understanding of the flow of graduate students and emphasized problems within the system that need to be addressed in the future.

References

Hillier, F.S., & Lieberman, G.J. (2010). Introduction to Operations Research. 9th ed. New York, NY: McGraw-Hill Higher Education. 723-753.

McGovern, S. (2011). Probabilistic Operations Research: Lecture 4. [Presentation]. Boston, MA.

Nicholls, M.G. (2007). Assessing the Progress and the Underlying Nature of the Flows of Doctoral and Master Degree Candidates Using Absorbing Markov Chains. Higher Education, 53 (6). 769-790.

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The problem of assigning tasks to stations in such a way that some specific objectives, such as minimizing the number of stations needed for a given cycle time or maximizing the efficiency of the assembly line, are optimized subject to the precedence relationships among tasks is called the assembly line balancing (ALB) problem.  The main constraints of ALB are that each task must be assigned to exactly one station, all precedence relationships among tasks must be satisfied, and the total task times of all the tasks assigned to a station cannot exceed the cycle time (Ozcan & Toklu, 2009, p. 822).  Many studies on assembly lines include exact solution methods, heuristic approaches, and metaheuristic approaches.  However, more literature and solution methods exist for one-sided assembly lines verses two-sided assembly lines.  In the following paper metaheuristic approaches for the two-sided assembly line balancing (TALB) problem will be discussed, with a primary focus on the tabu search algorithm.

The tabu search algorithm (TSA), which was developed by Glover, is a metaheuristic designed to solve combinatorial optimization problems by using basic local search strategies.  Tabu search uses short- and long-term memory to forbid certain moves in a solution space.  The short-term memory forbids cycling around a local neighborhood in the solution space and helps to move away from a local optimal solution.  Long-term memory allows searches to be conducted in the most promising neighborhoods.  TSA consists of several elements such as move, neighborhood, initial solution, search strategy, memory structure, aspiration criterion, and stopping rules (Ozcan & Toklu, 2009, pp. 823-824).  In the article A tabu search algorithm for two-sided assembly line balancing a TSA is developed for the TALB problem with an objective to maximize the line efficiency and minimize the smoothness index.  Line efficiency (LE) is the ratio of total station time to the cycle time multiplied by the number of workstations.  It is expressed as LE =  , where ST is the station time of station i, W is the total number of workstations, and C is the cycle time.  The smoothness index (SI) is an index to indicate the relative smoothness of a given assembly line balance.  A SI of zero indicates a perfect balance.  SI =  where STmax is the maximum station time, ST_i is the station time of station i, and W is the total number of workstations (Elsayed & Boucher, 1985, pp. 260-261).  Now that the objective of the problem has been defined the proposed TSA can be established.

The proposed TSA proceeds as follows: The first step states that an initial solution is constructed as a priority list (PL).  The tasks are assigned to the stations sequentially by the priority value of tasks.  The position of a PL represents a task i, and the value of the position represents the priority value of task i (PR_i).  A random number with uniform (0, 1) distribution is generated for each task to obtain priority values.  To create a feasible line balance the tasks are assigned to stations such that the first assignable task with the highest priority value is assigned to the first mated station, two stations that face one another, according to its preferred operational direction.  The set of assignable tasks is updated and this process continues until all tasks have been assigned.  Thus, the initial solution (x₀) is stored as the current solution (x_k) and the best solution (x*).  The cost of the initial solution (f(x₀)) becomes the current value of the objective function (f(x_k)) and the best value of the objective function (f(x*)) (Ozcan & Toklu, 2009, p. 824).

The next step is to generate neighborhood solutions (m(x_k)) by applying a move (m) to a current solution x_k.  A swap operator obtains all permutations x’_k from x_k by swapping the priority value of tasks placed at the b^th position and randomly selected a^th position.  These neighborhood solutions of x_k are considered candidate solutions and are evaluated by the objective function.  A candidate solution (x’_k) which is the best not tabu or satisfies the aspiration criterion is selected as the new x_k.  This selection is called a move and added to the tabu list (TL); the oldest move is removed from the TL if it is overloaded.  The TL consists of a two-dimensional array that is used to check if a move from a solution to its neighborhood is forbidden or allowed.  Whenever a pair of tasks are declared tabu, TL[a][b] is set to the current iteration number plus the tabu size.  If TL[a][b] is empty then the priority values for tasks a and b are free to swap.  Otherwise TL[a][b] is T and the priority values of tasks a and b cannot be moved until iteration T.  The tabu restriction may be overridden if the move will produce a solution that is better than what has been found in the former iterations.  This rule is known as the aspiration criterion.  Therefore if the new x_k is better than x* and feasible, it is stored as the new x*, otherwise x* remains the same.  This searching process is repeated until the iteration counter (k) equals the given number of iterations (K) (Ozcan & Toklu, 2009, pp. 824-825).

Lastly, when the iteration number reaches its maximum limit the algorithm terminates and the performance measures of the system are calculated.  The two performance measures being calculated are the line efficiency (LE) and smoothness index (SI) using the following formulas:

The first two equations calculate the simple lower bound (LB) of the TALB problem for the number of stations.  Note that LTotal, RTotal, and ETotal are the total task time of the left, right, and either directional tasks, respectively.  The remaining equations calculate the desired performance measures.  Note that S_w and S_q are the station times of right-side station w and left-side station q, respectively, and S_max is the maximum station time.  Also, m_R and m_L represent the number of stations on the right- and left-side respectively.  Finally the objective function of the proposed algorithm is formulated as follows: , where f₁^max(LE) is set equal to 100 and f₂^min(SI) is set equal to zero since these values create a perfectly balanced line (Ozcan & Toklu, 2009, p. 825).  With the general outline of the TSA developed an example will be used to illustrate the procedure.

Consider the 12-task problem depicted in figure 1.  The numbers in the nodes represent the tasks, the labels (t_i,d_i) below the nodes represent the task times and preferred operation direction respectively.  L/R indicates that task i should be assigned to a left-side station/a right-side station while E indicates that the task can be performed on either side of the line.  The directed arrow between two task nodes implies the precedence relationships among tasks.

The cycle time is fixed to 8 and the parameters of the algorithm are selected as follows: K = 12 and tabu size = 4.  At each iteration of the solution process, the number of solutions in the neighborhood is 11.  The lower bound (LB) of the number of stations is 4 (Ozcan & Toklu, 2009, pp. 823, 826).

The initial solution (x₀) is randomly generated at the beginning of the solution process (k = 0).  Then a random number is generated for each task with uniform (0, 1) distribution to obtain PR_i values.  Figure 2 shows the PR_i values for this given example.

In order to obtain an initial assembly line balance the following steps must be applied to build a feasible solution.  Let P_i be the set of tasks that precede task i, SAT be the set of assignable tasks, ST_i be the starting time of task i, and FT_i be the finishing time of task i.  The first step is to set w =1, q = 1, S_w = 0, and S_q = 0.  Step two is to determine SAT, if SAT = Ø go to step five.  Step three sorts the tasks in SAT in decreasing order of PR_i.  Step four assigns the first task h in SAT for which: a) If d_h = R, t_h + S_w ≤ C, and t_h + FT_a ≤ C, then assign task h to station w, otherwise set w = w + 1, S_w = 0 and go to step two.  Basically step 4a states that if the desired direction of a task in SAT is the right-side of the assembly line and the task time can be processed within the cycle time, which is 8 in this example, then assign the task to the station, otherwise return to step two.  b) Identical to 4a except that it focuses on tasks that have a preferred direction of the left-side of the assembly line.  c) If there is no preferred direction for the task, that is d_h = E, then generate a random number rn for task h with uniform (0, 1) distribution.  If rn < 0.5 go to step 4a, otherwise go to step 4b.  Step five is to calculate f, the objective function value. When these steps are applied to the above example the initial two-sided assembly line balance is created, which is shown in figure 3 (Ozcan & Toklu, 2009, p. 826).

In order to fully understand the previously defined steps and how the initial feasible solution for figure 3 was obtained begin by referring to figure 1.  At time zero (t = 0) the precedence diagram indicates that tasks 1, 2, and 3 can be processed.  To determine the order at which these tasks will be processed refer to figure 2.  Since it is only possible to process tasks 1, 2, and 3 the PR_i values that correspond to these tasks are put in decreasing order thus producing the sequence 1, 3, and 2.  Since 1 has the highest PR_i it is assigned to the left side of the assembly line, based on directional preference, and will require a time length of two.  Next on the list is task 3 which has no preference of location which means that a random number rn will be generated, which happened to be greater than or equal to 0.5 so at t = 0 task 3 will begin on the right side of the assemble line and finish at t = 2.  Therefore the assembly line is busy until t = 2.  At t= 2 the process will be repeated with SAT = tasks 2, 4, and 6.  From the results displayed in figure 3 the objective function value of the initial line balance is equal to 2.00.  Next the first iteration is performed.

Set k = 1, x* = x₀, f* = f(x₀) = 2.00, and x₁ = x₀.  By using the swap move, all neighborhoods of x₁ are generated and the objective function values are calculated for each move.  Then a random task, which happened to be task 9, is selected and whole examination is applied between assignment orders.  Since there are a total of 12 tasks it follows that task 9 can be swapped with the remaining 11 tasks so the number of candidate solutions is 11.  After calculating the objective function value for all swaps it is determined that the move of task 9 with task 3 is the best.  The move 9-3 is accepted as the new solution and the move is added to the tabu list.  Figures 4 and 5 show the new priority values and two-sided assembly line balance for the first iteration.  The objective function value of the first iteration is equal to 1.34 which is an improvement from the old objective function value of 2.00 (Ozcan & Toklu, 2009, p. 826).

Even though there is little literature related to two-sided assembly line balancing problems the article A tabu search algorithm for two-sided assembly line balancing applied the proposed algorithm, which was developed in this paper, to several test problems with various cycle time lengths.  When compared with other procedures for solving TALB such as genetic algorithm (GA), group assignment procedure (GAPR), an ant-colony based (ACO) heuristic, and an enumerative algorithm (EA) the results showed that TSA performed well throughout the test data.  The proposed TSA obtained the LB values for 24 of the 41 test problems, and for some test problems found the optimal solution (Ozcan & Toklu, 2009, pp. 827-829).  There are many areas in which research for TALB problems can be further developed.  As research efforts continue in this field more literature will be available to compare similar problem types.  Analyzing multi-objective optimization problems that take into consideration several criteria such as load balancing and smoothing is just one possible example for further research.

References

Bartholdi, J. (1993). Balancing Two-sided Assembly Lines: A Case Study. International Journal of Production Research, 31 (10). 2447-2461.

Baykasoglu, A., & Dereli. T. (2008). Two-sided Assembly Line Balancing Using an Ant-colony-based Heuristic. The International Journal of Advanced Manufacturing Technology, 36 (5). 582-588.

Elsayed, E., & Boucher, T. (1985). Analysis and Control of Production Systems. Englewood Cliffs, NJ: Prentice-Hall.

Hu, X., Wu, E., & Jin, Y. (2008). A Station-oriented Enumerative Algorithm for Two-sided Assembly Line Balancing. European Journal of Operational Research, 186 (1). 435-440.

Kim, Y.K., Kim, K., & Kim, Y.J. (2000). Two-sided Assembly Line Balancing: A Genetic Algorithm Approach. Production Planning & Control, 11 (1). 44-53.

Ozbakir, L., & Tapkan, P. (2010). Balancing fuzzy multi-objective two-sided assembly lines via Bees Algorithm. Journal of Intelligent & Fuzzy Systems, 21. 317-329.

Özcan, U., & Toklu, B. (2009). A Tabu Search Algorithm for Two-sided Assembly Line Balancing. The International Journal of Advanced Manufacturing Technology, 43 (7). 822-829.

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Before outlining the gradient search procedure, the requirements and assumptions that are needed in order to use this procedure must first be established.  This paper focuses on optimization problems in which the objective is to maximize a concave function f (x) of several variables, x = (x₁, x₂, …, x_n) for n = 1, 2, …, with no constraints on the feasible region (Hillier and Lieberman, 2010, p. 557).  Thus, before the procedure can be applied the problem must be in the previously stated form.  Therefore the objective function, f (x), must be checked for concavity by showing that the hessian matrix, H(x), is negative semi-definite.  Note that if a function is negative definite, then it is also negative semi-definite so it follows that the function would be concave (Ravindran, Reklaitis, and Ragsdell, 2006, pp. 80-81).  Furthermore, assume that the objective function f (x) is differentiable which means that the gradient, denoted by ∇ f (x) = for n = 1, 2, …, can be calculated at each point of x.  The significance of the gradient is that the change in x that maximizes the rate at which f (x) increases is the change that is proportional to ∇ f (x) (Hillier and Lieberman, 2010, p. 558).  Basically this means that from our current trial, or location on the hill, the gradient when evaluated at a particular point creates a directed line segment, or path, that can be followed until that point of evaluation is reached.  Since the problem is unconstrained, which implies that there are no obstacles on the hill, it makes sense to move in the direction of the gradient as much as possible because it will yield an efficient procedure to get to the optimal solution, which is the top of the hill.  Thus by noting the requirements and assumptions of the problem, the gradient search procedure can now be summarized.

To initialize the gradient search procedure an acceptable error tolerance, ε, and initial trial solution x₀ must be chosen.  After this is done, go to the stopping rule which states to evaluate  and check if  ≤ ε for i = 0, 1, 2, … (Hillier and Lieberman, 2010, p. 559; Winston, Venkataramanan, and Goldberg, 2003, pp. 703-705).  This means that the gradient of the current trial solution should be evaluated and if the magnitude of the gradient is less than or equal to the desired error tolerance, then stop all further iterations.  However, if  ε, then another iteration is performed which takes us to step one of the gradient search procedure.

The first step of the procedure is to express f (x_i + t ∇ f (x_i)) where i = 0, 1, 2, … as a function of t by setting x_i+1 = x_i + t ∇ f (x_i) and then substitute these expressions into f (x).  This step takes a function of several variables and reduces it to a function of a single variable, which makes it much easier to optimize (Hillier and Lieberman, 2010, p. 559; Winston, Venkataramanan, and Goldberg, 2003, pp. 703-705).  We have now expressed f (x_i + t ∇ f (x_i)) as a function of t which completes step one of the procedure.

Step two begins by using a search procedure for a one-variable unconstrained optimization problem to find t = t^* that maximizes f (x_i + t ∇ f (x_i)) over t ≥ 0 (Hillier and Lieberman, 2010, p. 559; Winston, Venkataramanan, and Goldberg, 2003, pp. 703-705).  Some commonly used search procedures for one-variable unconstrained optimization problems are the Bisection Method, Newton’s Method and the Golden Section Search.  Alternatively one could use calculus to find the value of t that maximizes f (x_i + t ∇ f (x_i)), t^*, by taking the derivative of the function and setting it equal to zero (Hillier and Lieberman, 2010, p. 559).  By calculating t^* we have completed step two and now can move to the third and final step in the gradient search procedure.

The final step is to reset x_i+1 = x_i + t ∇ f (x_i) and then go to the stopping rule (Hillier and Lieberman, 2010, p. 559; Winston, Venkataramanan, and Goldberg, 2003, pp. 703-705).  This resetting takes us from our current trial solution and moves us in the direction of the gradient at each iteration (Hillier and Lieberman, 2010, pp. 558-559).  Thus, an iteration has been completed and now it is necessary to determine whether the current trial solution is optimal with respect to the chosen ε.  Hence the outline of the gradient search procedure has been formed so now this procedure will be illustrated through the use of an example.

With the basic methodology of the gradient search procedure developed we will now use an example to work through the gradient search procedure step-by-step.  Suppose we were given the problem  to maximize  (Hillier and Lieberman, Introduction to Operations Research Information Center, 2010).  Certainly this is a nonlinear function consisting of multiple variables, that is x₁ and x₂, and it is unconstrained.  In order to use the gradient search procedure to solve this unconstrained optimization problem we must first make sure that this function is concave, since this is one of our requirements.  Concavity can be tested for by calculating the hessian matrix, H(x), and show that it is negative semi-definite.  Before the hessian matrix can be calculated we must first calculate .   In this two variable optimization problem H(x) = .  One way to test for negative semi-definiteness is to test – H(x) =  for positive semi-definiteness.  First we must make sure that all the entries on the main diagonal, {4, 2}, are greater than or equal to zero, which they are.  Next we check to make sure the matrix is symmetric, which it is.  If a matrix, A, is not symmetric use the transformation , where A^T is the transpose of matrix A.  Note that this transformation will produce a symmetric matrix but will not have any effect on the test for definiteness.  Lastly, the leading principal determinants must be greater than or equal to zero.  So the det [4] = 4 ≥ 0 and det  = (4*2) – (-2 * -2) = 8 – 4 = 4 ≥ 0, therefore the leading principal determinants are greater than or equal to zero.  Thus by satisfying these conditions we conclude that –H(x) is not only positive semi-definite but actually positive definite, which implies that H(x) is negative definite and the object function is concave.  Since the requirements of this problem type have been satisfied the gradient search procedure can be used to determine an optimal solution.

To initialize the gradient search procedure we must choose a starting point, say x₀ = (x₁, x₂) = (1, 1), and error tolerance, ε (Hillier and Lieberman, Introduction to Operations Research Information Center, 2010).  Typically the error tolerance is under 0.1 because the smaller the error tolerance the closer our solution is to optimality.  However for this example we will choose ε = 0.5, which means that our solution will not be as close to optimality compared to using ε = 0.01 (Hillier and Lieberman, Introduction to Operations Research Information Center, 2010).  With our initial trial solution, x₀ = (1, 1), and acceptable error tolerance, ε = 0.5, we must go to the stopping rule to see if x₀ is the solution to our problem.  The stopping rule states to evaluate  and check if  ≤ ε.  Thus, ∇ f (x₀) = ∇ f (1, 1) = (-2, 0) and the magnitude of the gradient,  = 2.  Notice that 2  0.5 so we will begin our first iteration.

The first step of the gradient search procedure is to express f (x_i + t ∇ f (x_i)) where i = 0, 1, 2, … as a function of t by setting x_i+1 = x_i + t ∇ f (x_i) (Hillier and Lieberman, 2010, p. 559; Winston, Venkataramanan, and Goldberg, 2003, pp. 703-705).  Since x₀ = (1, 1) and ∇ f (x₀) = (-2, 0) it follows that x₀₊₁= x₁  = x₀ + t ∇ f (x₀) = (1, 1) + t (-2, 0) = (1, 1) + (-2t, 0) = (1-2t, 1).  Now we take this expression and substitute it back into f (x).  Hence, f (x_i + t ∇ f (x_i)) = f (1-2t, 1) = 2(1-2t)(1) – 2(1-2t)² – 1² = -8t² + 4t -1.  We have now expressed f (x_i + t ∇ f (x_i)) as a function of t which completes step one of the procedure.

Step two begins by using a search procedure for a one-variable unconstrained optimization problem to find t = t^* that maximizes f (x_i + t ∇ f (x_i)) over t ≥ 0 (Hillier and Lieberman, 2010, p. 559; Winston, Venkataramanan, and Goldberg, 2003, pp. 703-705).  Instead of using a search procedure to solve this problem, calculus was used to find the value of t that maximizes f (x_i + t ∇ f (x_i)).  The previous step resulted in f (x_i + t ∇ f (x_i)) = -8t² + 4t -1, so (-8t² + 4t -1) = -16t + 4.  When we set this equal to zero we get -16t + 4 = 0 which implies that t^* = 0.25.  By calculating t^* we have finished step two and now can move to the third and final step in the gradient search procedure.

The final step is to reset x_i+1 = x_i + t ∇ f (x_i) and then go to the stopping rule (Hillier and Lieberman, 2010, p. 559; Winston, Venkataramanan, and Goldberg, 2003, pp. 703-705).  Therefore, we have calculated that x₁ = (1, 1) + 0.25 (-2, 0) = (1, 1) + (-0.5, 0) = (0.5, 1), which concludes our first iteration.  Now we need to determine if our first iteration trial solution, x₁, is the solution to our problem by using the stopping rule.  So ∇ f(x₁) = ∇ f (0.5, 1) = (0, -1) and  = 1.  Since 1 0.5 we must perform a second iteration.

In step one of the second iteration we get x₂  = x₁ + t ∇ f (x₁) = (0.5, 1) + t (0, -1) = (0.5, 1-t).  Taking this and substituting it into the objective function f(x) it follows that  f (0.5, 1-t) = -t² + t – 0.5.  With the completion of step one, we now move to step two of the gradient search procedure and find t^* by taking the derivative of f (x₁ + t ∇ f (x₁)), setting it equal to zero and solving for t.  When this is done the result is  (-t² + t – 0.5) = -2t + 1 = 0 and t* = 0.5.  Finally, we reset x₂ = (0.5, 1) + 0.5 (0, -1) = (0.5, 0.5) and go to the stopping rule.  Thus ∇ f(x₂) = ∇ f (0.5, 0.5) = (-1, 0) and  = 1.  Hence, 1 0.5 so we execute a third iteration.

This process will continue until a solution is found that satisfies the acceptable error tolerance we have defined.  The iterations have been summarized in the following table to show the calculations that were made till a satisfactory solution was found (Hillier and Lieberman, Introduction to Operations Research Information Center, 2010).

Notice for x₃ = (0.25, 0.5) that ∇ f(x₃) = (0, -0.5) and = 0.5.  Since 0.5 0.5 all further iterations are stopped and our approximate optimal solution becomes x^* = (0.25, 0.5).

In the following figure the path of the trial solutions from the previous table have been plotted with solid arrows and the next three trial iterations are represented by dashed line segments (Hillier and Lieberman, Introduction to Operations Research Information Center, 2010).

As you can see by the dashed arrows it appears that the optimal point of this function converges to (x₁^*, x₂^*) = (0, 0).  We can verify this by using calculus to solve the problem.

Given the problem of maximizing  it follows that when we set   ∇ f (x) = 0 we get = -4x₁ + 2x₂ = 0 and  = 2x₁ – 2x₂ = 0.  Now we have two equations and two unknowns so x₁ and x₂ can be solved for uniquely.  When this is done we get an optimal solution of (x₁^*, x₂^*)  =  (0, 0).  As you can see from this example, the larger your error tolerance, ε, the distance between your solution and the optimal solution is increased.  If we were to choose a very small ε, for example ε = 0.01, then our optimal solution to the previous example would be (0.004, 0.008), which is much closer to (0,0) compared to (0.25, 0.5) (Hillier and Lieberman, Introduction to Operations Research Information Center, 2010).

Imagine standing at the bottom of a hill you wish to climb to the top of.  Even though you cannot see your goal you can see the ground that lies directly in front of you so you begin to walk in the direction of upmost slope as long as you are still climbing.  You continue these iterations, following a zigzag path up the hill, until you reach a point when the slope is zero in all directions, which means ∇ f (x) = 0.  Since we established that the hill is concave, you are standing on the top of the hill.  In this paper we acknowledged the assumptions, summarized, and developed the gradient serach procedure with the use of an example.  Although this procedure has its faults it is very applicable to many problem types and it ultimately got you to the top of the hill.

   References

Hillier, F.S., & Lieberman, G.J. (2010). Introduction to Operations Research. 9th ed. New York, NY: McGraw-Hill Higher Education.

Hillier, F.S,, & Lieberman, G.J. (2010). Introduction to Operations Research Information Center. Introduction to Operations Research, 9/e. Retrieved from http://highered.mcgraw-hill.com/sites/0073376299/information_center_view0/.

Ravindran, A. G., Reklaitis, V., & Ragsdell, K.M. (2006). Engineering Optimization: Methods and Applications. 2nd ed. Hoboken, NJ: John Wiley & Sons.

Winston, W.L., Venkataramanan M.A., & Goldberg, J.B. (2003). Introduction to Mathematical Programming. 4th ed. Vol. 1. Pacific Grove, CA: Thomson/Brooks/Cole.

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Before developing and stating any conclusions from the article, it is essential to identify all assumptions that have been made.  In order to conceptualize the TSDP consider a distribution system consisting of a warehouse that acts as a transshipment node, which is a node that can both receive and send goods from other nodes, and many customer locations (Winston & Venkataramanan, 2003, 400).  Each location has a delivery demand and a pick-up demand while the warehouse is the origin and destination of each demand, respectively.  The warehouse is assumed to have the ability to maintain inventory levels of products at customer locations and the ability to make decisions regarding the size of delivery demands and the timing of replenishments for each customer.  Furthermore, it is assumed that the size of the pick-up demands follow a given proportional ratio of the size of the delivery demands.  In regards to the actual routing of the delivery system it is assumed that the route starts and ends at the node of origin, also known as the depot.  Each node  in N, where N = {1, 2, …, n} represents the node set of customer locations with node 0 referring to the node of origin is visited exactly once.  Split delivery, which is a method of ordering a larger quantity to secure a lower price and dividing the delivery into smaller quantities spread out over time in order to control inventory, is not permitted.  The distance from node i to node j is equal to the distance from node j to node i, Cij = Cji for all i and j. Lastly, all delivery quantities are loaded at the depot and all quantities picked up must be unloaded only at the depot (Ganesh & Narendran, 2008, p. 1222).  With the assumptions stated it is now possible to formulate a mixed integer linear program (MINLP) to solve TSDP.

In order to express the TSDP as a MINLP the notation of the problem must be defined.  Let S be the set of customer locations, T be the set of nodes with depot 0; T = [S  {0}], N be the total number of nodes; N = |S|, {n: 1, 2, …, N} represents customer locations with n = 0 denoting the depot, Cij be the cost to travel from node i to node j, d_i be the delivery request for node i; i = 1, 2, …, N, p_i be the pick-up request for node i; i = 1, 2, …, N and Q be the capacity of the vehicle.  The decision variables, which are variables whose values are under our control and influence the performance of the system, are Di which is the load remaining to be delivered by the vehicle when departing from node i, Pi which is the cumulative load picked up by the vehicle when departing from node i and Xij which is equal to one if the vehicle travels directly from i to j and zero otherwise.  It should be noted that the distance matrix Cij satisfies the triangular inequality which means that ||z|| = ||x + y|| < ||x|| + ||y|| (refer to figure 1).

Furthermore, at every node in a path the sum of the loads picked up and the quantities remaining to be delivered must be less than or equal to the vehicle capacity.  With the variables and constraints defined the TSDP can be formulated as follows:

The objective function is to minimize the total cost traveled.  Constraints (2) and (3) restrict the visiting of a node to exactly one.  Constraint (4) states that the remaining delivery amount after departing from node i summed with the load picked up at node i must be less than or equal to the vehicle capacity.  Constraint (5) ensures that the total delivery for a tour is loaded on the vehicle.  Constraint (6) states that there is no pick-up at node 0, which is the depot.  Constraints (7) thru (10) decrease the amount remaining to be delivered; increase the amount picked up and eliminates the possibility of sub-tours.  Constraints (11) and (12) are non-negativity conditions and constraint (13) creates the binary condition of whether node i to node j is traveled or not.  Although the TSDP can be formulated as a mixed integer linear program it has been proven that the TSP is NP- complete which means that the TSDP is also NP-complete (Ganesh & Narendran, 2008, 1223).  Being a member of the non-deterministic polynomial (NP) class implies that computational effort can increase rapidly for problems of even moderate sizes which makes seeking an optimal solution impractical.  Hence, the two-phase heuristic procedure TASTE was developed in order to calculate near optimal solutions within a reasonable computational time.

The first phase of TASTE constructs an initial solution using a constructive heuristic (CH) procedure based on the cheapest agglomeration of nodes.  Distance and load-feasibility, which checks to make sure that the vehicle should balance the route with respect to delivery and pick-up capacities, are considered and ensured with the computation of the net load.  The algorithm proceeds as follows: Step one is to calculate the net load N(j) for each node j; N(j) =  (p_j – d_j).  Note that N(j) is unrestricted.  Step two forms a tour from the depot to all nodes carrying negative net loads.  Then nodes with positive net load are added to the tour such that the distance traveled is minimal.  Once the tour is complete an initial tour, σ₀, is created and phase one of TASTE is complete.

To illustrate phase one of TASTE consider the following simple example in figure 2.

The results of the first step, which is to calculate the net load for all nodes, are shown in table 1.

Once the net loads have been determined the next step is to create a tour from the depot (node 0) to the nodes with negative net loads.  By traveling to the locations with negative net load, which means that the delivery demand is greater than the pick-up demand, this ensures that the capacity of the vehicle will not be exceeded.  If the vehicle started its tour by traveling to locations with a positive net load, implying that there is a larger pick-up demand than delivery demand, then it is possible that the vehicle capacity could be exceeded.  In the above example two locations have the same negative net load, therefore, to break the tie visit the node that is nearest to the depot first.  Figure 3 depicts the tour with negative net loads.   Lastly to obtain the initial tour, σ₀, add positive net load nodes to the tour such that the distance traveled is minimized.  The initial tour for the above example is shown in figure 4.

Phase two of TASTE is the improvement algorithm that uses meta-heuristics, which is a general solution method that provides both a general structure and strategy guidelines for developing a specific heuristic method to fit a particular kind of problem, because they are the most promising and effective solution methods for the TSP and vehicle routing problems (VRP) (Hillier & Lieberman, 2010, p. 607).  The meta-heuristic implemented in the procedure of TASTE is simulated annealing (SA).  SA is a local search meta-heuristic in the sense that it conducts local searching while guiding the process intelligently, offering the possibility of accepting, in a controlled manner, solutions that do not descend along the path of search (Ganesh & Narendran, 2008, pp. 1224-1225).  The first step of SA is to generate a solution X₀ for the problem, which was created at the end of phase one in TASTE.  Arrive at an initial temperature T₀, number of iterations at each step I, choose the cooling schedule temperature reduction δ, where 0 ≤ δ ≤ 1, based on experience or preliminary studies and set T_cur = T₀, X_cur = X₀,

X_best = X_cur and Z_best = Z(X_cur).  Step two is for count = 1 to I, randomly generate a new solution X_count.  If          Z(X_count) is better than Z(X_cur) set X_cur = X_count, otherwise calculate ΔC = best objective function value to date – current objective function value = Z(X_count) – Z(X_cur) and set X_cur = X_count with probability .  On the other hand, if Z_best is worse than Z_cur set X_best = X_cur, Z_best = Z_cur and update the count.  The third and final step of SA is to set T_cur = T_cur – δ.  If the stopping criteria of temperature level (for example, temperature < 10) or another stopping criteria is met then stop with Z_best and X_best as the SA solution, otherwise return to step two with T_cur (Winston & Venkataramanan, 2003, pp.806-807).  Although SA is a good procedure for solving routing problems it imposes a trade-off between computational time and the quality of the solution. Therefore TASTE combines SA with evolutionary computation (EC) to produce enhanced simulated annealing (ESA) which makes each tour determine its appropriate temperature instead of forcing a uniform cooling schedule by using Or-opt exchange, a local search procedure that generates neighborhoods.

The Or-opt algorithm begins by considering an initial route with n vertices and set t = 1 = location of first node in tour and s = 2 = chain of consecutive vertices from position t to t + 2 that must be removed and inserted between all remaining pairs of consecutive vertices on the route. Note that s is an arbitrary value.  If one or more insertions decreases the cost of the route then choose new route, otherwise set t = t + 1, if t ≤ n + 1 repeat the previous step.  Lastly, set t = 1 and s = s-1, if s > 0 go to step two, in which a segment is removed and inserted between all remaining consecutive vertices, otherwise stop (Ganesh & Narendran, 2008, p. 1226).  The Or-opt with two nodes, s = 2, will be shown in figures 5-7.

With the SA and Or-opt algorithms defined it is now possible to describe the ESA procedure with a clear understanding of each steps purpose.  Step one of ESA is to create an initial solution, which is obtained at the end of phase one of TASTE.  Step two is to set the initial temperature T_max, cooling rate α (0 ≤ α ≤ 1) and the iteration number I = 0.  Step three uses Or-opt for neighborhood generation.  Step four calculates the objective functions of the newly created routes and the initial route.  Step five ranks the newly created routes in ascending order.  Step six calculates the maximum temperature, T_max, for each newly created route.  Step seven compares the values of the objective functions for the newly created routes with the objective function value of the initial route using the third step of the SA algorithm previously discussed.  Finally, step eight states to stop if the solution converges or the maximum number of iterations has been reached, otherwise go to step three (Ganesh & Narendran, 2008, p. 1226).  After formulating the TASTE procedure, it was coded in C to analyze the effectiveness and efficiency of the algorithm.

Benchmark problems for the TSDP do not exist so TASTE was tested on the standard data sets (TSPLIB) of symmetric TSPs, which are available on the University of Heidelberg website, derived TSDP data-sets from TSP data-sets, and randomly generated instances.  Along with possessing a reasonable computational time TASTE resulted in an average deviation of 2.32% from the optimal solutions when compared with data-sets published in literature for the classical TSP, 0.43% average deviation with respect to the randomly generated data-sets for the TSDP and 11.98% when compared to derived data-sets from the existing TSP data-sets and used published solution as lower bounds (Ganesh & Narendran, 2008, pp. 1228, 1230).

In this paper the concepts of reverse logistics were discussed then formulated as a linear program.  Due to the time complexity of the linear program, TASTE, a two-phase heuristic for solving routing problems with simultaneous delivery and pick-ups, was developed with in-depth descriptions and simple examples of the algorithms being used.  After comparing TASTE with several systems it is reasonable to state that TASTE is a fast and reliable procedure to obtain near optimal solutions for the TSDP.  The field of reverse logistics is limited with respect to published literature but the work from the article TASTE: a two-phase heuristic to solve a routing problem with simultaneous delivery and pick-up and others leave room for developing new ideas and heuristics in the field of reverse logistics, in particular focusing on methodologies that account for time windows.

References

Ganesh, K., & Narendran, T.T. (2008). “TASTE: A Two-phase Heuristic to Solve a Routing Problem with Simultaneous Delivery and Pick-up.” The International Journal of Advanced Manufacturing Technology, 37. 1221-1231.

Hillier, F.S., & Lieberman, G.J. (2010). Introduction to Operations Research. 9^th ed. New York, NY: McGraw-Hill Higher Education.

Winston, W.L, & Venkataramanan, M. (2003). Introduction to Mathematical Programming: Operations Research. 4^th ed. Duxbury.

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These advances in semi-automation and automation technologies are poised to drastically change military operations.  Unmanned Aerial Systems (UAS) and Unmanned Ground Systems (UGS) are an increasingly attractive option for the US military, especially in an era of reduced manpower and fiscal constraints.  Automation has the potential to reduce accidents, function in more difficult conditions than a human driver can endure, and reduce fuel consumption and maintenance requirements.  Automated vehicles can mitigate or eliminate crew or driver rest requirements, allowing for continuous operations or reducing the vehicle fleet required to move a given amount of cargo.  In addition to those benefits, tasks that can be completed by a “robot” can remove Soldiers from danger, reducing the risk of injury or death.  While some tasks will not translate between the commercial and defense sectors, much of the technological innovation can be transposed.  This reduces the costs to the Department of Defense for research and development, and allows for future commercial development to be incorporated into defense platforms.

However, discussions involving automation often result in debates about specific terms, and ways to best define a system’s “smartness.”  Various solutions have been proposed, and agencies each have their own definitions.  This can make communication within acquisitions and between industry participants difficult.  The Department of Defense acquisition system is designed to acquire systems with required capabilities, or software to fill a particular need.  Semi-automation and automation are a unique and different problem set, because they are in effect a hardware and software solution to a problem.  Requirements must not only address a system’s physical capabilities, but also the actions of the system under certain conditions.

The scope of this model focuses on conducting military convoy operations under a variety of conditions, but only consisting of three vehicles for a sixty minute mission.  This model does not address combinations of automation categories, such as “Leader-Follower” combined with “Driver Assist,” “teaming” situations where individual automated vehicles become “smarter” when networked with other vehicles, nor learning curves where a system may require less input over time as the system “learns.”  The purpose of this initiative is to establish a quantitative mathematical model to visually depict: human input frequency versus system configurations, robotic decision making ability versus system configuration, and the relationship between human input frequency and robotic decision making ability.

This framework models and discusses “Human Input Frequency” (H) and “Robotic Decision Making Ability” (R).  Human input frequency is defined as instructions received, as a fraction of the total operating time.  This can be never be zero, as any system requires a human being to turn it on for the first time, no matter how “smart” it is, and provide the information critical to accomplishing the mission.  Input is the transmission of instructions, while interaction is simply the transmission of data such as location or video feeds.  A system can simultaneously receive input, interact with the operator, and execute its assigned task.  “R” is defined as the ability of the system to perform a given set of tasks.  This list can change based on the system, but the list was primarily focused on its potential application in vehicles.

•  Human Input Frequency: the frequency with which an operator must provide instructions to the robot. This is a ratio of the total time of input divided by the total time of task/mission accomplishment.  This will produce a value between 0 and 1.  This value cannot be zero.  This is not the same as frequency of feedback. (Human Input Frequency is not equal to Interaction Frequency, which may be initiated by either the human or the system.)  A system may receive instructions and operate at the same time.
• Interaction Frequency: the frequency of communication of any type between the operator and the system. Can be initiated by either the operator or the system.
• Complexity of Instructions/Tasks: a function of the skill and experience level (human and/or automated) required to accomplish a task under current conditions. This is not dependent on the machine’s intelligence.  Do not assume that a system is “dumb” because it requires complex instructions.
• Robotic Decision Making Ability: the number of variables that the system can perceive and is allowed AND able to control. This was scored based on fifteen factors, including everything from acceleration, reacting to road signs, and terrain the system could traverse.

Automation can be partitioned into sub-categories.  This project focused on the logistical platforms, which conduct military convoy operations, consisting of three or more tactical wheeled vehicles moving supplies and/or personnel from one location to another as a group with a company commander. Calculations were based on a convoy of three tactical wheeled vehicles executing a sixty minute mission.  Below are the various levels of automation identified for conducting convoy operations.

•  Status Quo: vehicles are operated entirely by human operators. The scenario would have three vehicles and three operators with continuous input/control.  The system does not make decisions (R=0).
• Remote Control: a system which allows an operator to drive a vehicle or platform from a location other than inside the vehicle. This system configuration can be wired or wireless, but the operator does not require sensory data from the system.  The scenario would have three vehicles and three operators (assuming a 1:1 ratio) with continuous input (H) and control (R).  The system does not make decisions (R=0).
• Tele-operation: a system which allows an operator to drive a vehicle or platform from a location other than inside the vehicle. This system configuration can be wired or wireless, but the operator does require sensory data from the system.  The scenario would have three vehicles and three operators (assuming a 1:1 ratio) with continuous input (H) and control (R).  The system does not make decisions but does interact (R=0).
• Driver Warning: a system within the vehicle which provides visual, auditory or other forms of notification to platform drivers of potential dangers, obstacles, and accidents. The scenario would have three vehicles and three operators with continuous input (H) and control (R).  The system does not make decisions but does interact (R=0).
• Driver Assist: a system within the vehicle which provides assistance to drivers by assuming temporary control of one or more vehicle functions such as braking. The scenario would have three vehicles and three operators with near continuous input (H) and control (R).  The system does make decisions under certain conditions.  (R ≠ 0).
• Leader-Follower: a system involving two or more vehicles which allows a “follower” vehicle to mimic the behavior of the “leader” vehicle in the order of movement.
  • Tethered (LF1): vehicles within the convoy are physically connected. The lead vehicle has an operator but follower vehicles do not.  The system does make decisions under certain conditions (R ≠ 0).
  • Un-Tethered (LF2): vehicles within the convoy are connected via non-physical means (radio, etc.).  The lead vehicle has an operator but follower vehicles do not.  The system does make decisions under certain conditions (R ≠ 0).
  • Un-tethered/Unmanned/Pre-driven (LF3): vehicles within the convoy are connected via non-physical means. No operators are present in any of the vehicles.  The system does make decisions (R ≠ 0).
  • Un-tethered/Unmanned/Uploaded: vehicles within the convoy are not tethered, and map data is uploaded from a database. No operators are present in any of the vehicles.  The lead vehicle operates in a way-point mode while the remaining vehicles are “followers” and mimic the behavior of the preceding vehicle.  The system does make decisions (R ≠ 0).
• Way Point Mode: a mode of operation where the system follows a path generated by a series of waypoints. The differentiation between leader-follower and waypoint is that in waypoint each vehicle is independent and can move within formation, while the “follower” vehicles in leader-follower are dependent and cannot function if the lead vehicle is disabled.
  • Pre-recorded “Breadcrumb” (WA1): the system must be initially “taught” by a pre-recorded route driven by an operator. No operators are present in the vehicles after the initial teaching.  The system does make decisions under certain conditions (R ≠ 0).
  • Uploaded “Breadcrumbs” (WA2): digital map data is provided to the system and waypoints plotted by an operator. No operators are present in the vehicles during mission execution.  The system does make decisions under certain conditions (R ≠ 0).
• Full Automation:
  • Uploaded “Breadcrumbs” with Route Suggestion (FA1): digital map data is provided to the system, with specified origin and destination. The system will provide suggested routes.  No operators are present in the vehicles during mission execution.  The system does make decisions under certain conditions (R ≠ 0).
  • Self Determining (FA2): the system determines its own route given that the grid coordinates of the origin and destination are provided, performs all safety-critical driving functions, monitor roadway conditions, and interpret sensory information to identify obstacles, relevant signage, and dynamically re-route to traverse alternate routes through uncharted, all weather environments for the entire mission. No operators are present in the vehicles during the mission.  The system makes almost all the decisions (R ≠ 0).

The most current modeling of an automation scale uses an exponential curve to depict the increasing intelligence of systems. Although this scale is generally accepted across the robotics community, there has never been any mathematical modeling to support an exponential curve for a vehicle’s “smartness.” Thus, the only valid information to support the above claim is a logical understanding of the positive correlation which exists between a vehicle’s “smartness” and the system configuration modes.

The framework described in this article is “a” way to consider the various implementations of semi-automation and automation and should allow both acquisitions personnel and sustainers to discuss both the minimal requirements and the maximum boundaries of automation that is “safe.”

Analysis began by calculating the percentage of human input frequency required for each respective system configuration of automation with respect to the given scenario, which states that three tactical wheeled vehicles are conducting a sixty minute mission.

The results of the analysis are highlighted in the “Human Input Frequency vs. System Configuration” chart. A mathematical equation could be calculated using regression analysis, which will accurately model the relationship between H and the system configurations. Beyond the pure mathematics of the model, it is possible to identify “breaks” where the system configurations require a significant change in H, i.e. SQ, DW, DA, RC, and TO form the first “group” with a human input (H) requirement of approximately 100%, LF1, LF2, LF3, and LF4 form the second “group” with a significantly lower H range when compared to the first “group,” and WA1, WA2, FA1, and FA2 form the final group which asymptotically converges to an approximate H value of 0%.

With a clear understanding of the relationship between human input frequency and system configuration, the next step was to mathematically model a system configuration and its respective decision making ability. In order to differentiate each level of automation, decision making probabilities were calculated for each system configuration based on their respective ability to execute tasks, e.g. acceleration, deceleration, interval/formation integrity, etc., independent of human input. The results from the analysis are displayed in the “Robotic (“Smartness”) vs. System Configuration” chart.

Based on the results, it is clear that R cannot be modeled accurately using an exponential curve. The curvature and number of inflection points indicates that an exponential distribution would not accurately model the system. Regression analysis determined that a polynomial equation accurately models the distribution. The one deficiency of these results is that it does not incorporate H into the model. It is necessary to integrate the human input frequency into the model because it is not sufficient to model the probability of a particular system configuration’s ability to execute the established tasks on a first attempt without understanding the amount of time which the system could potentially experience each task.

Hence, the results from the “Human Input Frequency vs. System Configuration” chart were used as a weighting coefficient and multiplicatively combined with the results from the “Robotic (“Smartness”) vs. System Configuration” chart to produce the final mathematic model in the “Robotic (“Smartness”) vs. System Configuration Weighted by H” chart.

This model now incorporates the probability of each system configuration being able to execute the identified tasks independently of additional human input. Thus, the system must now have the authority to execute a given task, and the ability. Regression analysis was used to create a mathematical equation that is extremely accurate for modeling R and the system configuration when weighted by H. Below is an “Overlay of H and R with Respect to System Configuration (Weighted)” chart, which can be used to summarize the above results.

Therefore, the results of the analyses conclude that the null hypothesis of an exponential function to model R should be rejected in favor of the alternate hypothesis, which states that R cannot be modeled using an exponential function. In particular, a polynomial function creates the “best fit” by minimizing the sum of the squared residuals between the actual and projected values of R Weighted by H for each system configuration.

After successfully modeling H and R against the various system configurations, the final analysis to pursue was to create a mathematical model which accurately models R in terms of H. The model displayed below in the “Mathematically Modeling H and R” chart can be used to estimate a military convoy operation “Smartness” based on the amount of human input required.

Regression analysis was used to create the “curve of best fit,” which was a sixth order polynomial distribution. However, a third order polynomial distribution is shown above because it is more visually logical than the sixth order polynomial equation. Even though the sixth order polynomial distribution minimized the sum of the squared residuals, the distribution included a significant portion of negative values, which are not possible. Therefore, the third order polynomial distribution makes more logical sense, even though it does not “fit” the data as well as other alternatives. The above model now provides organizations throughout the robotics community a way to quantify H and R, and determine where a specific scenario may fall on the scale of automation.

The potential implications are staggering for national security and defense.  Automation could allow for equal or greater combat power with drastically reduced personnel requirements, greater efficiencies in fuel consumption and maintenance, and increased utilization of platforms and assets.  This model can serve as the basis for further discussion and analysis as the Department of Defense wrestles with the incorporation of automation into military operations.  Increased use of automation will create a variety of challenges ranging from the technological and operational to the legal and moral.  Further research can be conducted to better quantify the impacts on manning and training, maintenance, fuel efficiency, and deployability.

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Creating an integrated system is not limited to the private sector. One area in the government where an integrated methodology would significantly improve the quality, timeliness, and cost of the output would be establishing a linkage between the Joint Capabilities Integration and Development System (JCIDS) process, the Joint Capability Technology Demonstration (JCTD) process, and the Defense Acquisition Process (DAP). The purpose of this article is to identify sub-optimality, from a high level perspective, residing in the current JCIDS and JCTD processes, and highlight disconnect between the two relative to their linkage to the DAP.

The JCIDS process exists to support the Joint Requirements Oversight Council (JROC) in identifying, assessing, validating, and prioritizing joint military capability requirements, and provide requirements related advice (CJCSI 3170.01J, 2013, p. 2). The most critical aspect of the JCIDS process is to allow the JROC to manage and prioritize capability requirements within and across the Joint staff. The review and validation of capability requirement documents is the most visible aspect of the JCIDS process, including staffing, review, and validation, enabling trade-offs and prioritizations within and between portfolios (CJCSI 3170.01J, 2013, p. 2). The iterative manner in which JCIDS operates will next be defined.

Validated initial Capability Based Assessment (CBA) documents drive the early JCIDS process, which in turn drives capability requirement documents related to materiel and non-materiel solution approaches. Since the JCTD process develops and demonstrates potential materiel solutions the remainder of this article will relate to materiel approaches/solutions. Knowledge gained early in the DAP is leveraged to refine capability requirements and make effective cost, performance, schedule, and quantity trade-offs in the succeeding capability requirement documents, which then drive the development, procurement, and fielding of materiel solutions that satisfy the validated capability requirements and close/mitigate the associated capability gaps (4.2.2. Pre-Materiel Development Decision, 2013).

Prior to entering the DAP for a new start or updated capability requirement, the JCIDS process completes the CBA. In turn, the CBA underpins an Initial Capabilities Document (ICD) or a partial draft Capabilities Development Document (CDD) and a draft Analyses of Alternatives (AoA) Study Plan to serve as pre-DAP entry documents (4.2.2. Pre-Materiel Development Decision, 2013). The pre-DAP entry documents provide the Milestone Decision Authority (MDA) with an in-depth understanding of the operational capability gaps, identify an appropriate range of materiel solution approaches, identify near-term opportunities for rapid response, analyze trade space, and plan for future technical efforts. This allows an informed decision to be made with respect to the project’s viability and milestone entry point (4.2.2. Pre-Materiel Development Decision, 2013).

Once the appropriate pre-DAP documents, which may be accompanied with any prior analytic, experimental, prototyping, and/or technology demonstration efforts, are developed a Materiel Development Decision (MDD) review may be requested. A successful MDD review ends when the MDA approves entry into the DAP. This decision is documented in a signed Acquisition Decision Memorandum (ADM), which specifies the approved Milestone entry point and approved AoA Guidance (10-017, 2010).

The typical entry point for a project is the Materiel Solution Analysis (MSA) Phase, also known as Milestone A. The objective of the MSA phase is to select and adequately describe a preferred materiel solution approach to satisfy the phase-specific entrance criteria for the next program milestone designated by the MDA. A Milestone A AoA will focus on concept refinement (4.2.3 Materiel Solution Analysis Phase, 2013). Once sufficient analysis has been executed, an MDA review to move past Milestone A will occur. Upon successful completion of the Milestone A MDA review, the next milestone is usually a decision to invest in technology maturation and preliminary design in the Technology Development (TD) Phase (4.2.4. Technology Development Phase, 2013). A Milestone B MDA review is then informed by an approved CDD, an approved AoA, and an approved Cost-Benefit Analysis (C-BA) which evaluates the systems’ performances, effectiveness, cost, and affordability of achieving the CDD requirements. Typically, based on the conclusions of the AoA, a Milestone B MDA review will result in progressing to the Engineering and Manufacturing Development (EMD) Phase. During the EMD Phase, the JCIDS process updates the AoA, if necessary, and a Capabilities Production Document (CPD) is produced for the Milestone C MDA review (4.2.5. Engineering and Manufacturing Development Phase, 2013).  A successful Milestone C MDA review typically progresses the materiel program to the Production and Deployment Phase of the DAP (4.2.6. Production and Deployment Phase, 2013).

The JCTD is an Office of the Secretary of Defense (OSD) sponsored process that can synchronize the DAP and JCIDS. JCTDs are intended to develop and demonstrate a technological approach/solution that has potential to rapidly transition to the field to resolve an immediate battlefield deficiency.

The JCTD technological approach is in the Science and Technology Development and Demonstration Phase. Before the JCTD initiation there has been no MDD or ADM from a MDA to enter the DAP. If and when the JCTD results, i.e. the Operational Utility Assessment (OUA) and Business Case Analysis (BCA), are presented and prove to the MDA that the JCTD technology is at an acceptable level of maturity to close the operational battlefield deficiency, is cost-effective, and affordable, the MDA can transition the technology to the field for immediate use and further evaluation; however, this transition does not establish a Program of Record (POR).

If one of the JCTD goals is to make the technological approach that is being explored and experimented a Program of Record (POR), then a synergistic JCTD can rapidly develop and transition a technology to MS B or C faster than the normal JCIDS process. The JCTD is expected to be more expeditious because the normal JCIDS regulations are circumvented by using OSD and other stakeholder funding allocations to hire all omnipotent contractors to conduct the same elements that would normally occur in the JCIDS CBA, Post Independent Assessment (PIA), ICD, AoA, MS A, AoA update, Cost-Benefit Analysis (C-BA) and CDD. JCTD technologies may have the potential to transition to a DAP POR, but require the same JCIDS underpinnings and Milestone B entrance criteria as other materiel solutions that enter the DAP to become a POR.

OSD requires JCTDs to be sponsored by a Combatant Command (COCOM). Typically, proponents are neither the JCTD sponsor, nor the JCTD Coordinator for OSD. The JCTD Coordinator is also at the COCOM. Typically, the proponent’s role is to support the COCOM in the execution of the JCTD in accordance with the JCTD Implementation Directive (ID). The voting members of the JCTD Integrated Management Team (IMT) are the OSD Oversight Executive (OE), COCOM (Operational Manager), Technical Manager (also the fiscal manager), and the Transition Manager (normally a Product Manager (PM)). Note that the proponent is not a voting member of the IMT.

If the JCTD IMT uses the funds allocated by OSD and agencies typically represented by the IMT to execute a JCTD cohesively, efficiently, and thoroughly, then the technology build, subsequent demonstrations performed, and assessments made during the OUA along with the BCA can equate to the JCIDS CBA, ICD, AoA, and C-BA, which can underpin a CDD or CPD, and support an MDA’s decision to accept the technology as a POR.

The successful integration of JCIDS and the JCTD into the DAP will require the IMT to be restructured to include the proponent as an equal voting member. By including the proponent in the decision making process, the necessary JCIDS components can be included in the JCTD ID, Management Plans, Methodologies, Assessment Plans, Data Collection Plans, and other events to support both the JCIDS and JCTD processes.

 References

10-017, D. (2010, September 13). Development Planning to Inform Materiel Development Decision (MDD) Reviews and Support Analyses of Alternatives (AoA). Washington DC, USA.

4.2.2. Pre-Materiel Development Decision. (2013, May 23). Retrieved December 2013, from Defense Acquisition Guidebook: https://acc.dau.mil/CommunityBrowser.aspx?id=638309

4.2.3 Materiel Solution Analysis Phase. (2013, May 23). Retrieved December 2013, from Defense Acquisition Guidebook: https://acc.dau.mil/CommunityBrowser.aspx?id=638310&lang=en-US

4.2.4. Technology Development Phase. (2013, September 03). Retrieved December 2013, from Defense Acquisition Guidebook: https://acc.dau.mil/CommunityBrowser.aspx?id=638311&lang=en-US

4.2.5. Engineering and Manufacturing Development Phase. (2013, September 03). Retrieved December 2013, from Defense Acquisition Guidebook: https://acc.dau.mil/CommunityBrowser.aspx?id=638312&lang=en-US

4.2.6. Production and Deployment Phase. (2013, April 22). Retrieved December 2013, from Defense Acquisition Guidebook: https://acc.dau.mil/CommunityBrowser.aspx?id=638313&lang=en-US CJCSI 3170.01J. (2013, August 01). Joint Capabilities Integration and Development System (JCIDS). USA.

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In order to determine whether the proposed system will be more desirable, it is necessary to establish a baseline for the sudden-death overtime system using Markov chain analysis.  The article Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League (2004) analyzes the impact of the coin toss on the outcome of the game and the efficiency at eliminating ties.   A Markov chain model to analyze the sudden-death overtime system was developed using the following notation: [a b T], where a is the number of points scored by team A in overtime, b is the number of points scored by team B in overtime, and T is the team that has possession of the football (Jones, 2004, p. 331).  In this system, a team can win by scoring in any manner.  The most typical methods of scoring are to either score a touchdown or kick a field goal.  Let the probability of a given team scoring a touchdown be represented by α, such that α ≥ 0, and the probability of kicking a field goal be represented by β, such that β ≥ 0, on a possession.  Notice that α + β < 1 since it does not take into consideration a team scoring a safety.  Thus γ = α + β will represent the probability that a team scores provided they have possession of the football (Jones, 2004, p. 331).

In Table 1, the possible states of the system have been represented in which state one is the point in overtime where team A has possession of the football, state two is the point in overtime where team B has possession of the football, state three denotes team A winning the game, and state four denotes team B winning the game (Jones, 2004, p. 331).  Notice that states three and four are absorbing since upon entering either of these states the game will be over.  A directed graph of the Markov process appears in Figure 1 (Jones, 2004, pp. 331-332).

 Now the transition matrix P_sd can be formed such that P_sd = .  Once again notice that states three and four are absorbing which means that the transition matrix can be rearranged as P_sd =  (Jones, 2004, p. 332).  If c represents the number of absorbing states and m represents the number of non-absorbing states then I is a c x c identity matrix, 0 is a c x m zero matrix, R is a m x c matrix that gives the probabilities of going from non-absorbing to absorbing states, and Q is a m x m matrix that gives the probabilities of going from non-absorbing to non-absorbing states.  With transition matrix P_sd rearranged in the above form, (I – Q)^-1 can be calculated which yields the expected number of times the Markov process will be in a given state before being absorbed.  Furthermore, (I – Q)^-1 R = provides the probability of absorption by any given absorbing state.  Thus, if team A receives the football in the overtime period then team A wins with probability of which means that team B wins with probability of (Jones, 2004, pp. 332-333).  In theory, the transient states will eventually enter one of the absorbing states but in reality this may not be the case since a game can end in a tie due to no team scoring within the fifteen minute period.  Since there is a time constraint this implies that there will be a finite number of possessions in the overtime period, which will be represented by n.  If team A receives the football to begin overtime then they will have n/2 possessions and team B will have n/2 possessions.  Thus the probabilities for the possible outcomes of the game can be determined as the product of [1 0 0 0]  (Jones, 2004, p. 333).  Now that a general solution set has been formed for the sudden-death overtime system a general solution set must be calculated for the first-to-six system.

The first-to-six system states that the first team to score six points in overtime wins the game.  In this system the sample space of possible events is larger compared to the sudden-death system.  For example, if team A wins the coin toss and elects to receive the football then they can win the game by scoring a touchdown, but if they kick a field goal team B will get possession of the football.  If team B scores a touchdown they win the game, otherwise the game will continue, within the fifteen minute period, until either team A or B has at least six points.  If the overtime period expires, then the team with the most points will win or the game will result in a tie.

Table 2. The states for a finite Markov chain representing the first-to-six overtime rule.

In Table 2 the possible states of the system have been represented with respect to the previously defined notation [a b T] that was used during the sudden-death overtime procedure (Jones, 2004, p. 333).  Certainly this policy will decrease the chances that the receiving team wins the game on the first possession, but the question that this paper focuses on is how will this new system change the probability of winning the game for the team that kicks the football to start the overtime period?  To determine an answer to this question the transition matrix for the first-to-six system is constructed as  . Once again states nine and ten are absorbing since upon entering these states the game is finished which means that the transition matrix can be rearranged such that P_ft6 = .  Assuming that team A receives the football, it follows that (I – Q)^-1 R can be calculated and the probability that team A wins is  and the probability that team B wins is   (Jones, 2004, p. 334).  This is the general solution set for the first-to-six system that will be used to answer whether the first-to-six system will affect the likelihood of team A winning the game.  In order to get numerical solutions to compare, data from the 2002 NFL football season will be used.

In the 2002 NFL regular season there were 6049 total possession in which 1270 possessions resulted in a touchdown, 737 possessions resulted in a field goal and the average number of possession per period were approximately 6.  So it follows that α =  and β =  , which means that γ = α + β = 0.210 + 0.122 = 0.332.  Using the sudden-death overtime system if team A receives the football then theoretically team A will win with probability  = 0.599 , team B wins with probability of  = 0.401, but these probabilities assume that a tie cannot occur.  So to take the result of a tie into account the initial state of the system [1 0 0 0] must be multiplied with our n^th transition matrix, , where  n = 6.  Therefore  [1 0 0 0] = [0.089 0 0.546 0.365], which means that for the sudden-death system approximately 8.9% of games end in a tie, team A wins 54.6% of the games, and team B wins 36.5% of the games.  The probabilities of game outcomes for the first-to-six policy can be calculated in a similar manner.  Assuming that team A receives the football it follows that team A wins approximately 49.1% of the games; team B wins is 39.3% of the games and 11.6% of the games end in a tie (Jones, 2004, p. 335).  Based on the above data, it is clear that the first-to-six policy does in theory decrease the probability that receiving team will win the game in overtime.

To provide additional statistical support for the thesis that the first-to-six policy does decrease the probability of the receiving team winning in overtime, I decided to carry out my own research.  In order to create a larger statistical model, I accumulated data that ranged from the 2002 NFL season till the 2010 NFL season.  After gathering this statistical data I used the general solution sets for the sudden-death system and the first-to-six system that were derived above and graphed the results in Table 3.

            Table 3.  The statistical NFL overtime analysis from the 2002 season till 2010 season.

Assuming that team A receives the football to start the overtime period, the first row of the chart represents that theoretical probability that team A wins for the given season using the current sudden-death system and the second row represents the theoretical probability that team B wins for the given season using the sudden-death system.  Rows three and four represent the theoretical probabilities that team A or team B wins, respectively, for the given season using the proposed first-to-six system.  From Table 3 it has been shown that the probability of team A winning the game was not only approximately 3.53% lower per year for the first-to-six system but in fact the first-to-six system produced more desirable outcomes for every season when compared with the sudden-death system.

With the increase in demand for an exciting overtime system, NFL management has several types of systems that they could implement which will provide an overall satisfactory product for the players, owners, and most importantly fans.  The question of whether or not the proposed first-to-six system would be more balanced in terms of victor in comparison with the sudden-death system was answered.  After reviewing the article, Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League, the general solution sets were found for both systems.  Next actual data from a football season was entered in the previous formulations, the numerical results were compared, and it was concluded that the first-to-six system does decrease the probability that the team who receives the football to start the overtime period will win with respect to the sudden-death system.  Further research was executed and involved expanding the data set by using data from multiple seasons.  The larger research did support the hypothesized thesis that the first-to-six system would make the outcomes of overtime games fairer.

References

Jones M. (2004) Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League. The College Mathematics Journal, 35 (5). 330-336.

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            Forecasting Petrol Demand with Linear Trend and Exponential Smoothing Models

     With the growing concern of global warming, it is more important now than ever before to be conscious of fossil fuel emissions.  In an effort to have smaller carbon footprints, many people are purchasing environmentally friendly products such as smart cars and recycled materials.  Even with this effort, a large amount of greenhouse gases emitted into the atmosphere daily still remains troublesome.  The transportation industry contributes one-sixth of all global greenhouse gas emissions.  The article Forecasting petrol demand and assessing the impact of selective strategies to reduce fuel consumption (Li, Rose and Hensher, 2010) will be analyzed as it forecasts Australia’s automobile petrol demand up to the year 2020 based on the best performing forecasting model selected from eight models.  Although research in this area is abundant mainly focused on theoretical differences, this article provides empirical comparison of fuel demand forecasts by using different forecasting models.  In the following, two of the eight forecasting models, linear trend and single exponential smoothing, will be developed with the use of a simple examples, results from the article will be discussed, and the article will be critiqued.

     The linear trend model is one of the simpler models used to predict demand, yet it can be very accurate in approximating short-term demand.  Before discussing the results from the article, a straightforward example will illustrate the work behind the final results.  Suppose table one provided the following data (Taylor, Introduction to Management Science, 2003):

Table 1

The main objective of the linear trend model is to create a line of best fit such that the total over-estimation equals the total under-estimation.  Therefore the formula for this model is y = a + bx; b = ; a = , where a is the intercept of the line at period zero, b is the slope of the line, x is the time period, y is the forecast for demand for period x, n is the total number of periods,  is the average of the time period and  is the average of the forecasted demand.  From table one, we can plug the data into these formulas and calculate the line of best fit, which could be used to predict future demand.  When the data is plugged into the formulas, the line of best fit equates to y = 35.2 + 1.72x.  The linear trend line would then predict that the demand for period thirteen would be y = 35.2 + 1.72(13) = 57.56, and so on (Taylor, Introduction to Management Science, 2003).  With the linear trend model developed it is necessary to establish the concepts behind the single exponential smoothing model.

The single exponential smoothing is a type of moving-average forecasting technique which weighs past data in an exponential manner so that the most recent data carriers more weight in the moving average.  Suppose one wanted to use the exponential smoothing model to forecast future demand for the data provided in table two (Taylor, Introduction to Management Science, 2003).

Table 2

The formula for this model is F_{t + 1} = aD_t + (1 – a)F_t , where F_{t + 1} is the forecast for the next period, D_t is the actual demand in the present period, F_t is the previously determined forecast for the present period and a is a weighting factor known as the smoothing constant.  In this model one must choose a smoothing constant such that 0 <= a <= 1 (Hyndman, Koehler, Ord and Snyder, 2008, p. 30).  It should be noted that this choice is subjective and generally based on trial-and-error.  However when a is large, more weight goes to recent data while a small a respects the value of historical data.  From the results in table two, the next step was to compare a = 0.3 and a = 0.5 to determine which value of a would produce a more precise forecast.  Since there is no previous data to insert into the formula F_{t + 1} = aD_t + (1 – a)F_t when t = 0, it is assumed that the model accurately predicted the demand for the first month with one-hundred percent certainty so F₁ = D₁.  The formula can now be used to calculate further forecasts, with respect to a, such as F₃ = aD₂ + (1 – a)F₂ = 0.3(40) + (1 – 0.3)(37) = 37.9 and so forth.  From the results in table two, when a = 0.3 the forecasted demand for January of year two is 51.79 and the forecasted demand is 53.61 when a = 0.5.  Note: at this point there is no longer previously known demand so no further forecasts can be calculated meaning that the forecast for any month after January of year two for a = 0.3 is 51.79 and 53.61 for a = 0.5 (Taylor 2003).  Now that these simple examples have been used to illustrate the linear trend and single exponential smoothing models, the results of the article can be discussed in detail.

In order to understand the results from the article, how the data was collected and implemented must be understood.  Quarterly time series data was used including total road petrol consumption (TPC), real gross domestic product (GDP) and real petrol price (RPP) for Australia over the period 1977 quarter one to 2006 quarter four.  Next, the total data was divided into two parts.  The first part dates from 1977 quarter one to 2005 quarter one which were used for model and estimation purposes, while the remaining data from 2005 quarter two to 2006 quarter four was used to examine the forecasting effectiveness of different forecasting models (Li, Rose and Hensher, 2010, p. 411).  By applying the same methodology as seen in the above examples, the forecasts for linear trend and single exponential smoothing resulted as follows: (Li, Rose and Hensher, 2010, p. 414).

To identify which method provided the best forecast, with minimal forecasting error, the mean forecasting error was defined by using the mean absolute deviation (MAD), such that MAD =  (Li, Rose and Hensher, 2010, p. 412).  This means that the average of the absolute value of the error is calculated and the method with the smallest MAD will be considered the most desirable.  From the above charts, the MAD for the linear trend model is MAD =  = 172.77, similarly the MAD for the single exponential smoothing model is MAD =  = 194.  Between these two choices, it is clear that the linear trend method is more accurate.  However this article explored a total of eight forecasting models determining the quadratic trend model as the most desirable with MAD = 170.14.  Therefore, the quadratic trend model was used to predict future petrol demands for the years of 2007 to 2020 (Li, Rose and Hensher, 2010, pp. 414-415).  The article concluded that petrol demand in Australia will increase annually due to the continued growth of household incomes and population.  After comparing many possible policies to help control the disbursement of global gases into the atmosphere, it was found that a carbon tax, a tax imposed on energy sources releasing carbon dioxide, of AU $0.50/kg can reduce automobile kilometers by 5.9%, resulting in reduced demand for petrol and a reduction in carbon dioxide emissions of 1.5% (Li, Rose and Hensher, 2010, p. 407).  With the process and results of the article acknowledged, it is necessary to discuss the advantages and disadvantages of the article.

Having such a large sample of data was very beneficial to the results in providing accurate forecasts.  Furthermore, this article did not just analyze forecasts from one model but eight.  By using the linear trend, quadratic trend, exponential trend, single exponential smoothing, Holt’s linear method, Holt-Winters’ method, ARIMA and the partial adjusted model, the researchers were able to determine the most precise method and base any further calculations using that method.  With a large pool of historical data and several methods explored, the results and conclusions in the article are considered reliable.  However, there is one limitation in the article that needs to be discussed.  Econometric modeling and forecasting methods can be divided into four categories, but the article only focused on models that estimate relationships between explanatory and dependent variables over time and models that depict relationships between the past and current values, and forecast the future based entirely on historical outcomes (Li, Rose and Hensher, 2010, pp. 411-412).  This was done because time series data was collected meaning there were observations of a single event over multiple periods of time.  Thus the research was limited to two types of econometric models.  In critiquing the linear trend model, a line of best fit does not change in trend and is limited to accurately forecasting with respect to short time frames.  Conversely, the exponential smoothing model can better account for the change in trend.  This can be achieved by choosing a small smoothing constant when there is stable data without trend and a higher smoothing constant for data with trends.

Global warming is a phenomenon that has many people considering alternative ways of living.  Exploring the Australian automobile petrol demand  due to the transportation industry as one of the main contributors of greenhouse gases,  the article Forecasting petrol demand and assessing the impact of selective strategies to reduce fuel consumption (Li, Rose and Hensher, 2010) used eight different forecasting models to accurately predict future petrol demands. Predicting future demand helps implement policies to limit the amount of carbon dioxide in the atmosphere.   In total, this paper explored the linear trend and single exponential smoothing forecast models.  These models were first demonstrated with simple examples, then results from the article were discussed and finally the advantages and disadvantages of the article were acknowledged.  It appears that there exists a positive correlation between population growth and the amount of petrol use.  Forecasting models can be used as a guide to help society make better decisions and protect the atmosphere from harmful pollutants like greenhouse gases.

References

Hyndman, R.J., Koehler, A.B., Ord, J.K, & Snyder, R.D. (2008). Forecasting with Exponential Smoothing. Springer-Verlag Berlin Heidelberg

Li, Z., Rose, J.M., & Hensher, D. (2010). Forecasting petrol demand and assessing the impact of selective strategies to reduce fuel consumption. Transportation Planning and Technology 33 (5). 407-421.

Taylor, B.W. (2003). Introduction to Management Science. 8^th edition. Retrieved from homepage.mac.com/thubsch/MAT540/ch05-8th.ppt

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