A Lagrangian model to analyze time dependent resource allocation inefficiency due to intrinsic responses

Transcription

1 1 A Lagrangian model to analyze time dependent resource allocation inefficiency due to intrinsic responses By Juha-Pekka Nikkarila, and Juhani S. Hämäläinen Finnish Defence Forces Technical Research Centre (PVTT), Riihimäki, Finland Abstract The request for an effective resource usage has motivated the development of methods and techniques to find an optimal solution of the allocation problem. However, many approaches neglect the time dependency, which is naturally present in dynamic situations. This motivates us to construct a model for resource allocation in terms of functional analysis analogous to Lagrangian mechanics. The proposed model optimizes resource allocation for a given time interval. The model deals with a given set of resources which are allocated to perform a set of prioritized tasks. In particular, we consider the resource consumption in the task exhange, similarly to the energy dissipation in physics due to friction. The purpose of this study and our research question is to examine the optimal solution of the resource allocation problem in the presence of inefficiency. This will be analyzed through numerical examples, where we compare the optimal allocations with and without friction. Friction brought three main differences to the optimal solution: smoother resource exchange, the tasks will be completed slower and less outcome was achieved. We also resolve the amount of the resources consumed due to the changing priorities, and studied how stable the optimal allocation is. The model may be further applied to search for cost-effective resource allocations. 1. Introduction Resource allocation is related to the resource management for example in industry, defence or in economics and it may also be applied in computing resources (Bird and Smith (2011), Chichilnisky and Kalman (1980), Maruyama (1981), Farzin (2006)). Ideally all resources would be available for effective use. However, this is seldom possible, at least when considering the human resources. This encourages us to consider effectively available resources for the given tasks. The simplest model would be to consider some percentage of the resources as effective ones. However, the simplest model neglects the inefficiencies variation which may occur due to several reasons. The reasons could be the needs for learning, the needs for different equipment and for example the physical distance between the tasks. Also the resources have intrinsic inefficiencies due to (partial) skills, motivation, experience etc. In order to consider the described problem, we shall construct a time dependent resource allocation model. The constructed Lagrangian model is based on variational calculus (Cherkaev and Cherkaev (2003), Pars (1962), Hanz (2003)). The model includes the inefficiencies in the task exchange, which are modeled similarly as friction is modeled in physics.

2 2. Lagrangian model of inefficiency in resource allocation We have considered resource allocation in a static situation (Hämäläinen and Åkesson (2012)). Now we introduce a time dependent resource allocation model, which will be written in a functional form and solved numerically with time discretization. Following the similar procedure as in (Hämäläinen and Åkesson (2012)), we define the comparison value CV for one task and allow the included variables to have a time dependencies, for example the priority P (t), the resource suitability T (t) and the cost of a resource C(t). The CV is defined as a definite integral over the given time interval [0, t ], and it represents the outcome of the resources performance CV = t 0 P (t)t (t)x(t)/c(t) dt. (2.1) This generalizes to M tasks and N resources directly by summing the components. The inefficiency of the allocation will be introduced by adding a friction-like term into the Lagrangian. The role of the friction shall be to unfavour the resource exchanges between the tasks, i.e. to smoothen the allocation. The Lagrangian (CV ) to be maximized finally reads as: CV = t 0 M N i=1 j=1 ( P i (t)t ij (t) X ) ij(t) C j (t) bijẋ2 ij(t) dt, (2.2) where the time-dependent terms P i (t), T ij (t) and C j (t) are the priority of the task i, the resource j suitability for the task i and the cost of the resource j, respectively. The usage of the resource is also time dependent X ij (t) (the resource j allocation to conduct the task i). The second part in the functional (2.2) is the friction term where b ij is the friction coefficient for exchanging the resources between the tasks i and j. The corresponding rate of change is given by the term Ẋij(t). The most effective resource allocation functions can be obtained by maximizing Eq. (2.2) (or minimizing its opposite number which we later denote by F ) with respect to the resource function X(t), subject to the constraints. The frictionless case is provided by coefficients b ij equal to zero. The model have three constraints on X(t). Resources cannot be allocated more than 100 percent at any instant of time. Resource will not be allocated to unsuitable tasks,. i.e. if T ij = 0 then X ij = 0. Tasks will be completed if enough resources are available Discretization of the problem In practical calculations, we minimize the opposite number of CV (F ) K M N ( ) F = b ij Ẋij(t 2 k ) P i (t k )T ij (t k )X ij (t k ) t, (2.3) k=1i=1 j=1 where t k = k K t and t = 1 K t and K is the number of the time steps. The discretized constraints are given in a form X ij (t k ) 1 j, t k (2.4) i which means that each resource may be at maximum 100% usage at every time step. The readiness of the i th task after k 2 time steps is given by

3 k 2 R(t) = T ij (t k )X ij (t k ) t (2.5) k=1 j and each task is required to be completed between the total time interval. The third constraint, i.e. no resource allocation for unsuitable tasks, takes the form X ij (t k ) = 0 if T ij (t k ) = 0. (2.6) 3. Numerical example Let us consider an illustrating example of three tasks and three available resources assigned to complete the tasks. For simplicity, we consider a time interval of [0, 1] and assume the following time dependence for the priorities: P 1 (t) = 0.5, P 2 (t) = 1 t, P 3 (t) = 1.5 t 2. (3.1) Note that the definite integral of all priorities over time [0, 1] equals to 1 2 even though their time dependencies differ. The suitability of the resource j for the task i is defined by a matrix element T ij and tells how many times the j th resource can complete the i th task if it is completely allocated to it during the considered time interval. For simplicity, we let the resources have equal costs normalized to 1 i.e. C(t) = 1 j. Finally, the suitability matrix of the example is T = (3.2) where the tasks and the resources are sorted by rows and columns, respectively. The values of the suitability matrix signifies that for example the first resource is 6.25 = times more effective to conduct the third task than the second resource and 5.0 = faster than third resource. Further assumption following the constraints and Eq. (2.5) is that the tasks shall be completed during the following time intervals: the first task [0; 0.8], the second task [0; 1] and the third task [0.6; 0.8]. The friction coefficients are set to constants b = b ij i, j with the numerical values 0 (frictionless case), , 0.04 and Results of the example The time was discretized into 10 time steps and the optimized results were calculated using MATLAB. The results show the influence of friction to the allocation by comparing it to the frictionless case. We considered the resources exchanges, the readiness of the tasks and the completing times of the tasks and calculated the outcome (CV ) of the models. The task completion curves in the optimal allocations are presented in Fig. 1 for variable friction coefficients. For the frictionless allocations the results show that at the beginning all resources are assigned to the second task even though for example the third resource is not efficient on working with the task. This is mainly due to the highest priority of the second task at the beginning and consequently, it is completed already during the third time step. From the 4 th to the 7 th time step all resources are allocated to the first task. At the 8 th and at the 9 th time steps the most of the resources are used to the third task, which is therefore completed, and only the second resource remains

4 Figure 1. Tasks completions for variable friction coefficients. The tasks readiness changes slower when the friction increases. Figure 2. Comparing the optimal resource allocations with and without friction. working on the first task. When the third task is finished all resources are allocated to the first task. The optimal solution (with friction) behaves differently from the frictionless case in three ways. The total outcome of the optimized solution is smaller (CV is without friction and with friction (b = 0.01)), which is due to the costs related to reallocating resources. The resources change slower between the tasks and consequetly, the readiness of the tasks increase slower. Fig. 2 consists of a comparison between the

5 Figure 3. The optimal solution is stiffer when the friction coefficient is larger. optimal resource allocations. The exchange between the tasks is more undesirable in the case of friction enforcing also less suitable resources to work on with the tasks. The effect of the friction coefficient is shown to act similarly as in physics. The exchange between the tasks consumes resources and is therefore more undesirable solution. Variation of the friction coefficients effects on the task completions (Fig. 1) as well as on

6 the optimal allocations. The first resource s allocation variation respect to the friction coefficient is presented in Fig. 3. It is seen how the resource liquidity in tasks exchange decreases when the friction coefficient increases. The optimized solution with friction approaches to the frictionless solution when the friction coefficient approaches to zero. The obtained result is mainly a doublecheck of the computations. We observed that the optimization can not be conducted if the friction coefficient is increased too much, because of the higher cost of the reallocation. However, this would not occur if the numerical values of the model allowed the resources to accomplish the tasks without reallocation. 4. Conclusions and summary We have considered a time dependent resource allocation problem and in particular the inefficiency of the allocation due to friction in the resource exchange. We have constructed a Lagrangian model and demonstrated its usability by numerical examples, which showed how friction leads to a smoother resource exchange. The friction behaves similarly as in physics since it causes slower reallocations of the resources and less outcome. As friction consumes resources, there appear delays in completing the task (readiness curves) compared with the frictionless case, observed in our example as well. In the optimization, the calculated comparison values were decreased when the friction coeffiecient increased. When the coefficient was increased enough, the model turned out to be unsolvable. REFERENCES Hämäläinen J. and Åkesson B A linear algebra based model to support comparison of resource allocation strategies. Proceedings of 5th IMA International Conference on Influence and Conflict. Sandhurst Sarah L. Bird and Burton J. Smith 2011 PACORA: Performance Aware Convex Optimization for Resource Allocation. Conference: HotPar 11, 3rd USENIX Workshop on Hot Topics in Parallelism. G. Chichilnisky and P.J. Kalman 1980 Application of Functional Analysis to Models of Efficient Allocation of Economic Resources., Journal of Optimization Theory and Applications vol. 30, no. 1, January 1980 T. Maruyama (1981) A variational problem relating to the theory of optimal economic growth, Proc. Japan Acad. Ser. Math. Sci. Vol. 57 no. 7 (1981), Y. H. Farzin and K. K. Ahao (2006) When is it Optimal to Exhaust a Resource in a Finite Time?, Working Papers Fondazione Eni Enrico Mattei A. Cherkaev and E. Cherkaev 2003 Calculus of Variations and Applications. Lecture Notes, Draft L.A. Pars 1962 An Introduction to the Calculus of Variations., Dover Publications, Inc., Mineola, New York J. Hanz 2003 The original Euler s calculus-of-variations method: Key to Lagrangian mechanics for beginners.