A Dynamic Programming Approach to the Multiple Criteria Knapsack Problem.

Transcription

1 A Dynamic Programming Approach to the Multiple Criteria Knapsack Problem. Maria Boye Charlotte Wilson May 5,

2 Contents Introduction The multiple criteria knapsack problem. The mathematical formulation A dynamic programming approach Solving a time-dependent multiple criteria single-machine scheduling problem 8 3. Theory The theory of the dynamic programming approach Example Concluding remarks 3

3 Introduction This paper introduces the knapsack problem and proposes a solution method using dynamic programming to solve the knapsack problem. The knapsack problem can be described as a hiker who can only carry one backpack. A backpack has a limited capacity, so the hiker has to consider what he needs on his trip. Further, some goods are of higher value than others, e.g., it is more important to bring rope on a mountain expedition than a bathing suit. Therefore each good has a certain objective value. The problem the hiker faces is to pack the backpack with an objective value as large as possible under the constraint that he can only carry the one backpack and that the backpack has a capacity constraint. Additionally this paper will consider the case of multiple criteria, meaning that the hiker s objective function is of several different criteria. The mathematical formulation of the integer multiple criteria knapsack problem and a solution method to the problem is provided in section. Further, an example is constructed to illustrate the theory. Other kinds of problems, like the time-dependent multiple criteria singlemachine scheduling problem can be solved by using the solution method described in section for the multiple criteria knapsack problem. Additionally an application is considered in the end of the paper. The time-dependent multiple criteria single-machine scheduling problem is interesting since it can be used to determine which jobs to select from a number of jobs. An application of the problem is for NASA to determine which space research projects to carry out on the given space flight and which projects to put on hold for a later flight. The multiple criteria knapsack problem In this section the mathematical formulation needed to define a solution method to the multiple criteria knapsack problem is presented. Further the solution method is presented and illustrated with an example.. The mathematical formulation The integer multiple criteria knapsack problem (MCKP) can be formulated as

4 vmax s.t. f(x)=cx ax b x j integer, j =,..., n wherecisam n-matrix with nonnegative entries c ij, i =,...,m, j =,...,n and m is the criteria number. The i th row of C is denoted by c i and the j th column is denoted by c j. Hence f i (x) =c i x, i =,...,m are the m objective functions. The m objective functions are often conflicting since maximizing one criteria can decrease another criteria. The constraint ax b is the capacity constraint (budget constraint) for the knapsack problem. An important assumption is that the weight coefficients a j, j =,...,n and the capacity constraint b are positive integers. Note that the operation v max is vector maximization. To avoid trivial solutions let <a j <b, j =,..., n and n j= a j >b. Furthermore let X denote the set of feasible solutions i.e., X = {x N n () : ax b}. It is often convenient to consider the MCKP with the right-hand-side k =,,,...,bof the capacity constraint. This problem is denoted k-mckp. In general the knapsack problem can be solved by a process of sequential decision making where states and transitions are determined by the feasibility constraint of the original problem (). The multiple criteria are treated as a vector cost function defined on the states and decisions of the process. The link to the dynamic programming course is when only one criteria, i.e. m =, is used for the knapsack problem. The formulation is then the same as the one used in the textbook [StoLuc99] t= β t U (c t ) max {(c t,k t+)} t= s.t. c t + k t+ f(k t ) c t, k t+ k given t =,,..., n () To this problem the functional equation is of the form v(k) = s.t. max[u(c)+βv(y)] c,y c + y f (k) c, y (3) 3

5 A way of solving () is to generate the efficient solutions. Definition A feasible solution x X is an efficient solution of () if there is no other feasible solution x X such that f i (x) f i (x ) for all i and f i (x) >f i (x ) for at least one i. Let X e denote the set of efficient solutions to () and Y e is the image of X e in the objective space, i.e., Y e = f(x e ). Y e is called the set of nondominated solutions of (). To illustrate the theory an example with criteria is constructed v max f(x) =[x + x +7x 3 +3x 4, 4x + x +3x 3 + x 4 ] s.t. x + x +x 3 + x 4 3 x j, integer, j =,...,4 (4) The set of feasible solutions i.e., the set of combinations which respects the capacity constraints of the problem above are: X = The set of efficient solutions and the image of the efficient solutions (nondominated solutions) are X e = 3 (5) Y e = {[ 6 ], [ 7 9 ], [ 9 7 ], [ 4 ]} These solutions are the same as the solutions which will be found by dynamic programming approach below. 4

6 . A dynamic programming approach Here a dynamic programming (DP) approach to solve the MCKP will be presented. For further dynamic programming solution methods see the article Dynamic Programming Approaches to the Multiple Criteria Knapsack Problem [KlaWie]. Start with defining states Q as where Q := {q(),q(),...,q(b)} (6) q(k) :={x N n : ax = k, k =,..., b} are all the nonnegative integer solutions to the k-mckp. The set of final state is given by Q F := {q(),q(),...,q(b)} (7) e.g., all states. If the decision is to increase x j by in the solution x q(k) then the right-hand-side k will be increased by a j. This corresponds to a transition from the state q(k) to q(k + a j ). For the example (4) the possible transitions between states are represented by the arcs in the network given in figure Figure Each arc is identified by the objective value (c j, c j ) of each transition and the corresponding variable x j. Therefore each arc is denoted by the vector [j, (c j,c j )]. 5

7 The set of all nondominated solutions of the k-mckp is denoted by G(q(k)). The original MCKP can then be solved by applying the following recursive equations: G(q()) = {} (8) G(q(k)) = v max{g(q(k a j )) + c j : j S, k a j }, k =,..., b(9) where S := {,..., n} denotes the index set of the variables x j, j =,...,n. Note that (8) is equivalent with the functional equation (3). Since all states are final, the image of all efficient solutions (nondominated solutions) is obtained as the vector-maximum of the union of the sets G(q(k)), k =,...,b, i.e., Y e = v max k=,,...,b G(q(k)). Note that in this formulation multiple nondominated solutions may occur in each step of the recursion since the variables can be selected in a different order. In order to avoid multiple solutions a reduction has to be applied in each stage additionally to the reduction due to dominated solution. To illustrate the recursion the example (4) is used. G(q()) = {[ ]} G(q()) = v max{g(q()) + c,g(q()) + c,g(q()) + c 4 } {{[ ]} [ ] {[ ]} [ ] {[ ]} [ 3 = v max +, +, + 4 {[ ] [ ] [ ]} 3 = v max,, 4 {[ ] [ ]} 3 =, 4 G(q()) = v max{g(q()) + c,g(q()) + c,g(q()) + c 3,G(q()) + c 4 } {{[ ] [ ]} [ ] {[ ] [ ]} [ ] 3 3 = v max, +,, +, {[ ]} [ ] {[ ] [ ]} [ ]} ,, {[ ] [ ] [ ] [ ] [ ] [ ] [ ]} = v max,,,,,, ]} 6

8 {[ ] [ ] [ ]} =,, G(q(3)) = v max{g(q()) + c,g(q()) + c,g(q()) + c 3,G(q()) + c 4 } {{[ ] [ ] [ ]} [ ] {[ ] [ ] [ ]} [ = v max,, +,,, {[ ] [ ]} [ ] {[ ] [ ] [ ]} [ ]} , +,,, {[ ] [ ] [ ] [ ] [ ] [ ] = v max,,,,,, [ ] [ ] [ ] [ ] [ ]} 9 7 8,,,, {[ ] [ ] [ ] [ ]} =,,, ], The image of the efficient solutions is Y e = v max {G(q()),G(q()),G(q()),G(q(3))} {[ ] [ ] [ ] [ ] [ ] [ = v max,,,,, [ ] [ ] [ ] [ ]} 6 7 9,,, {[ ] [ ] [ ] [ ]} =,,, ] The set of efficient solutions X e canbefound by going backward through the recursion. The efficient solutions are 3 X e = which are the same solutions as found earlier in (5). In section the mathematical formulation for the MCKP has been presented and a solution method has been explained and illustrated. 7

9 3 Solving a time-dependent multiple criteria single-machine scheduling problem In this section the dynamic programming approach will be used to solve a time-dependent multiple criteria single-machine scheduling problem (TDM- CSP) [WieKla]. Further an example is made to illustrate the problem. A single-machine scheduling problem with time-dependent multiple criteria is presented. Here job processing times will depend on when the jobs have been started. The minimization of the completion time may be one of the criteria of interest. In general all the criteria to be optimized are monotone functions of time. The scheduling problem is limited by a machine capacity constraint. The formulation of the problem is based upon the classical knapsack problem extended by multiple criteria, as explained earlier. Note that in this paper all jobs are not enforced to be scheduled. 3. Theory The time-dependent multiple criteria single-machine scheduling problem can be described as a selection of choosing jobs over time in which the total associated profit is to be optimized subject to capacity constraints. There can only be chosen one job at a time and time is a continuous variable. The profit depends on which job is considered at time t. Further the amount of capacity used on each job is a positive constant. Let x j be the variable representing job j. Let S be as before the index set of the variables for the jobs x j, j =,..., n, i.e., S = {,...,n}. A schedule of jobs is defined as x := {x j } p j= which is a sequence of elements x j S, j =,..., p satisfying x {{x j } p j= : p N, x j S, j =,...,p}. Note that p n. In a schedule a job can be repeated or only executed once. Further assumptions are that all jobs are released at time and only one job can be processed at a time. There are no due dates for the jobs. Each job j S has a weight ax j which represents the job s usage of the resource. The total weight of a schedule is a(x) := where a xj is the weight coefficient of the job x j in the schedule, i.e., the total weight of a schedule is the sum of the weights of all jobs in a schedule. The p j= a xj 8

10 machine on which the jobs are executed has a capacity constraint a(x) b which means that the total weight of a schedule can not exceed the capacity of the machine. In order to avoid trivial solutions let <a xj b, j =,...,n. A feasible schedule of jobs is a sequence of jobs in S such that the total weight of a schedule does not exceed the capacity constraint. Therefore let the set of feasible schedules for the TDMCSP be given by X = {x : a(x) b}. Since all the constants in the constraint are positive integers the set of feasible schedules X will be finite. Namely p b for all {x j } p j=, since the a x j are positive integers. A job is evaluated by the m profits the job yields. Let c xj (t), j S, be a unit vector profit for job j at time t. The elements c ix j (t), i =,...,m, j =,...,n, are defined to be real-valued functions of time t and are not assumed to be continuous. Let c xj (t), j =,..,n, be positive functions measuring the processing time of the job j if the job has been started at time t. c xj (t) thus represents the duration of a job j, j =,...,n. Let the other components c ixj (t), i =,..., m, j =,...,n, represent the profits for the decision maker for each job. The profit of a schedule is the sum of the profits for all jobs in the schedule f i (x) := p j= c ixj (t j (x)), i=,...,m where t j (x) represents the time at which the j th job of the schedule x is started. t j (x) is calculated as t (x) = t s+ (x) = t s (x) +c xs (t s (x)), s =,...,p which means that the first job is started at time and the next job will started at the end of the first job and so on. The completion time of a schedule is represented by f (x). The TDMCSP can be formulated as v max s.t. f (x) =[f(x), f(x),...,f m (x)] T a(x) b 9

11 where the operator v max denotes the maximization of [ f (x),f (x),...,fm(x)] T, i.e., v max [f (x),f (x),...,fm(x)] T T := v max[ f (x),f (x),...,f m (x)] Now the problem can be described as in section given that the objective functions are linear. According to section the efficient solutions can be found using the dynamic programming approach. Let X e denote the set of efficient schedules and let Y e be the image of X e. 3. The theory of the dynamic programming approach To make the most of the theory in section it is convenient to consider the capacity constraint of the TDMCSP with the right-hand-side k =,,,...,b. This problem is denoted k-tcdmcsp. With this notation the dynamic approach explained earlier can be used. A feasible schedule x := {x j } p j= of the k-tcdmcsp is a sequence of jobs x j S, j =,...,p, p k, so j= a x j = k for k =,...,b. The set of states Q are the same as before, (6), and so are the set of final states Q F (7). To have optimality in the TDMCSP the following assumption is necessary: Assumption. For all t,t, if t t, then (a) t + c x j (t ) t + c x j (t ) for all j =,...,n p and (b) c ix j (t) c ix j (t) for all i =,..,m, and j =,..., n. Part (a) in the assumption states that if a job j is started at time t (an early start) instead of time t (a late start) then the next job can be started earlier than with job j starting at time t. Note that if c xj (t), j=,...,n are monotone increasing functions of time then assumption (a) is satisfied. Part (b) in the assumption states that all other components of the objective functions c ixj (t), i=,...,m, j =,...,n, are monotone decreasing functions of time. Part (b) can be interpreted as the profit of job j is decreasing in time.

12 The assumption is a typical requirement when working with the job selection problem. The assumption states that the earlier the job is started, the earlier it is completed and the bigger profit it will make. The following principle of optimality for the TDMCSP exists. Theorem Principle of optimality for the TDMCSP. Under the assumption, an efficient solution sequence of jobs x p = {x j } p j= of the k-tdmcsp completed at time t p+ (x p ) has the property that each solution subsequence x s = {xj} s, j= s p completed at time ts+ (x s ), where t s+ (x s ) t p+ (x p ), is an efficient solution sequence of jobs for the ( s j= a x j ) TDMCSP. For a proof is referred to the article Wiecek et al. [WieKla]. Note that if the assumption is not satisfied then the theorem will be invalid. Theorem establishes that to solve the TDMCSP the DP approach explained earlier in section. can be used. 3.3 Example An example extending the example in the article by Wiecek et al. [WieKla] is made. It is an example of the time-dependent multiple criteria singlemachine scheduling problem with 3 criteria (m = 3) and4jobs(n =4). The fixed budget is given by b =3and the cost coefficients ax j, x j =,, 3, 4 of each job are given by a =, a =, a3 =, a4 =. The first criteria defines the processing time of each job if the job is added to the schedule x at time t. The time can be interpreted as the duration of the corresponding job. The two other criteria which are maximized could represent the profit and the usage of the resources yielded by the jobs if they are selected at time t. The objective vectors c x j (t), xj =,, 3, 4, related to each job are defined as c (t) = 4 t 3 t c (t) = 4 8 t 4 t c 3(t) = t + 4 c 4(t) = t Note that all the objective functions are linear. The resulting TDMCSP is v max f (x) =[ f,f,f 3,f 4 ] s.t. a(x) 3 t + 45 t 3

13 The possible transitions between states are the same as the transitions in the example which illustrated the DP-approach, therefore see figure. By applying the DP-approach described in the section the following set of nondominated criteria vectors G(q(k)), k =,,, 3, are found: G(q()) = {} G(q()) = G(q()) = G(q(3)) = The image of X e is 8 Y e = The set of efficient solutions are X e = {{,, 4}, {4,, 4}, {,, }, {4,, }, {4,, }, {, }, {4, }, {3, }, {3, 4}, {3, }, {4}, {3}} where {,, 4} corresponds to the schedule where job is processed twice in the beginning and the schedule ends with job 4. In section 3 the theory for using the solution method of the multiple criteria knapsack problem on the TDMCSP is described. At the end the solution method was illustrated by an example.

14 4 Concluding remarks In this paper an introduction to the knapsack problem is presented. Further the theory of a dynamic programming approach to the multiple criteria knapsack problem has been introduced and illustrated by an example. The use of the theory gives the same solution as when the example is calculated by hand. The multiple criteria knapsack problem can be used to solve other kinds of problems, e.g., the time-dependent multiple criteria single-machine scheduling problem. This is explained in section 3. Further an application was created and the problem was solved using the dynamic programming approach. The consequences for NASA of choosing the wrong jobs can be economic fatale since a single space flight is very expensive. Therefore NASA would like to find the optimal way for choosing jobs and here the approach illustrated in section 3 can be used to find an efficient solution. 3

15 References [KlaWie] Klamroth, Kathrin and Wiecek, Margaret M.. Dynamic Programming Approaches to the Multiple Criteria Knapsack Problem. [StoLuc99] Stokey, Nancy L. and Lucas, Jr. Robert E Recursive Methods in Economic Dynamics. Harvard University Press. [WieKla] Klamroth, Kathrin and Wiecek, Margaret M.. A timedependent multiple criteria single-machine scheduling problem. European Journal of Operational Research. 35 ()