An Exact Procedure for Solving P 2

Based on previous results, in this section, we propose a procedure to solve the problem based on partitioning the demand space. The idea is to list all possible forms of the optimal solution, and determine the corresponding valid regions in the demand space. Although this procedure solves P2optimally, it works for problems that are small in size (e.g., problems

with 3-4 resources). In the next section, we will present two heuristic procedures that can be used to solve realistic size problems. The exact method based on partitioning the demand space is summarized as follows where each step of the procedure is explained for a system with three resources.

1. List all the possible sharing groups. For example, the three-resource model has fol- lowing combinations:{(1),(2),(3)},{(1,2),(3)},{(1),(2,3)},{(1,3),(2)},{(1,2,3)}. The numbers in()are the indexes of resources in the same sharing group.

2. For the group which has more than one resource, list all the possible combinations of reallocation. For example, the group {(1,2,3)}has three resources. Let →and ← denote the direction of reallocation. The cycle-free combinations include{(1→2→ 3)},{(1→2←3)},{(1←2→3)},{(1←2←3)},{(1→3→2)},{(1→3←2)},

{(1←3→2)},{(1←3←2)}. According to Proposition 8, inequality(4.13)must be satisfied by every pair of the resources in the combinations.

3. Each group may work in capacity constraint binding or nonbinding status. Each resourcei may have a positive βi orβi=0. Thus, corresponding combinations are

generated. For example, group{(1→2→3)}has the following combinations:

(a) Capacity constraint binding,β1=0,β2=0,β3=0.

(b) Capacity constraint binding,β1=0,β2=0,β3>0.

(c) Capacity constraint binding,β1=0,β2>0,β3=0.

(d) Capacity constraint binding,β1=0,β2>0,β3>0.

(e) Capacity constraint binding,β1>0,β2=0,β3=0.

(f) Capacity constraint binding,β1>0,β2=0,β3>0.

(g) Capacity constraint binding,β1>0,β2>0,β3=0.

(h) Capacity constraint binding,β1>0,β2>0,β3>0.

(i) Capacity constraint nonbinding,β1=0,β2=0,β3=0.

(j) Capacity constraint nonbinding,β1=0,β2=0,β3>0.

(k) Capacity constraint nonbinding,β1=0,β2>0,β3=0.

(l) Capacity constraint nonbinding,β1=0,β2>0,β3>0.

(m) Capacity constraint nonbinding,β1>0,β2=0,β3=0.

(n) Capacity constraint nonbinding,β1>0,β2=0,β3>0.

(o) Capacity constraint nonbinding,β1>0,β2>0,β3=0.

(p) Capacity constraint nonbinding,β1>0,β2>0,β3>0.

4. Apply the result of Proposition 11 to find the expressions of the optimal shadow prices and selling prices, then further obtain the optimal reallocation quantities by Proposition 12. The resources must satisfy the corresponding inequalities given in Propositions 10 and 11. The amounts of reallocations must be nonnegative.

5. The resources in different sharing groups must satisfy the inequalities given in Propo- sition 13. (If one or more of the inequalities are not satisfied, the corresponding grouping can not be optimal.)

6. Summarizing the inequalities obtained in steps 2, 4 and 5 for each combination (which represents each possible combination of steps 1, 2 and 3) gives rise to a re- gion of the demand space in which the combination is optimal. For example, that {(1),(3→2)} with β1= 0, β2 =0, β3 =0, the capacity constraint of group (1)

is binding and the capacity constraint of group(3→2) is nonbinding is one of the combinations.

The algorithm above solves P2 by obtaining all the valid regions of the optimal solu-

tion. However, as the dimension of the problem increases, the number of regions increases exponentially. Therefore, this procedure is good for problems with small dimension. For moderate or large dimensional problems, it is necessary to seek other methods which solves the problem efficiently.

In the remainder of this section, we focus on a special case which allows us decom- pose P2 into smaller independent subproblems. As a result, the above procedure, based

on partitioning the demand space, may be applied to each subproblem. The structure of P2 is complicated because of the reallocation imposed by flexibility. If we add more con-

straints to the model in a way that the reallocation will be less flexible, the problem can be decomposed into small subproblems as explained below.

Let ki j+kjl ≥kil ∀i,j,l. This inequality indicates that the minimum unit reallocation

cost between resourcei andl can be obtained by reallocating one unit resource fromi to l directly. We will call this inequality as the “triangle assumption”. This is a reasonable assumption in many cases. For example, consider that the reallocation cost is the trans- portation cost between two locations. Suppose that there are three locations, A, B and C. Generally, the transportation cost between A and C can be assumed to be less than or equal to the sum of transportation costs from city A to B and from B to C. Under the “triangle assumption”, we have the following result:

i j l ij z il z jl z δ

Figure 4.3: Pure consumer or supplier

Proposition 14. Ifki j+kjl≥kil ∀i,j,l,P2has an optimal solution such that each resource

is either a pure supplier or a pure consumer in an optimal solution.

Proof: Suppose P2 has an optimal solution in which resource j is a supplier of l and a

consumer ofias shown is Figure 4.3. Consideriis a supplier oflwithzil =0. LetAdenote

the optimal objective value ofP2excluding the reallocation cost among resourcesi, jandl.

Then, the optimal objective value isA+z∗

i jki j+z∗_jlkjl+z_il∗kil. Ifki j+kjl>kil, if we send a

flowδ≤min(zi j,zjl)with the direction as shown in Figure 4.3, the objective value becomes

A+ (z∗_{i j}−δ)ki j+ (z∗jl−δ)kjl+ (z∗il+δ)kil

= A+z∗_{i j}ki j+z∗jlkjl+z∗ilkil+δ(kil−ki j−kjl) < A+z∗_{i j}ki j+z∗jlkjl+z∗ilkil.

which is a contradiction with the assumption of the optimality. Therefore,ki j+kjl>kil can

not be true. So, only whenki j+kjl =kil, resource j can be a supplier and a consumer at

the same time. Without loss of generality, suppose thatz∗_{i j} ≤z∗_jl. If we send a flowδ=z∗_{i j} with the direction as shown in Figure 4.3, each resource becomes either a pure supplier or a pure consumer. That is, resourcesiand jare suppliers, and resourcel is a consumer.¥

Proposition 15. Ifki j+kjl ≥kil ∀i,j,l i6= j6=l, and if for j∈S,(γi−_α2_ixi)+−(γj−_α2_jxj)+<

kji, resource jcannot be a supplier of resourceiin an optimal solution.

Remark: Proposition 15 provides a necessary condition to check if some resource could be a supplier or consumer of another resource in the optimal solution.

Proof: Suppose that in an optimal solution, resource jis a supplier of resourcei. Resource jmust be a pure supplier and resourceimust be a pure consumer. (γj−2xj

αj )

shadow price of j (i.e., λ∗_j) when it is neither a supplier nor a consumer. When resource jis a pure supplier, then the shadow price of j must be greater than (γj−2xj

αj )

+_{. Similarly,}

as a pure consumer, the shadow price λ∗_i of i must be less than (γi−2xi

αi ) +_{. That is, λ}∗ i < (γi−2xi αi ) +_{, λ}∗ j>( γj−2xj αj ) +_{. Then} λ∗_i −λ∗_j <(γi−2xi αi ) +₋₍γj−2xj αj ) +_<k ji,

where the second inequality follows from the assumption of the proposition. This result is in contradiction based on Proposition 7. Therefore, resource j cannot be a supplier of resourceiin an optimal solution. ¥

When the dimension of the problem, n, increases, the number of partitions explodes. The procedure to solveP2by partitioning the demand space becomes computationally in-

feasible. The assumptionki j+kjl>kil gives the solution ofP2a nicer structure, and enable

us to decompose the problem into several subproblems. The algorithm to decompose the problem into subproblems is given as follows:

Decomposition under the “triangle assumption” 1. DefineS={1,2, ...,n}.

2. Leti=0.

3. IfS= /0, seti∗=iand stop the algorithm. Otherwise, leti:=i+1 and defineGi=/0.

Choose a resource j∈S, updateStoS\ {j}, GitoGi∪ {j} and mark resource j as

not visited.

4. If all the resources inGiare visited, go to step 3. Otherwise, choose a resource j∈Gi

which is not visited. LetSj be the set of resources that can be suppliers or consumers

of resource j. UpdateSasS\Sjand updateGiasGi∪Sj.

5. Mark resource jas visited and go to step 4.

When the algorithm stops, i∗ _{is the smallest number of optimal sharing groups inP} 2. The

setsG1,G2, ...,Gi∗ are mutually exclusive.P₂can be solved separately for each of these sets asi∗distinct subproblems.