Consider an arbitrary instance of Partition (Garey and Johnson 1979): Partition:
Given an integer number 2K, a set of n positive integers Z = {z1, z2, . . . , zn−1, zn}
and∑zj∈Zzj = 2K, is there a partition of Z into two disjoint subsets Z1 and Z2 such that Z = Z1∪ Z2 and
∑
zj∈Z1zj = ∑
zj∈Z2zj = K?
Figure A.2: Problem Instance Corresponding to the Example.
j=n+4 K Dj=3 + Regional Vault (i=3) Retail Vault (i=1) j=n+1 K Dj= + j=n+2 K Dj= + … j=1 1 z Dj= + j=2 2 z Dj= + j=3 3 z Dj= + j=4 4 z Dj= + j=5 5 z Dj= + j=n-4 4 − += n j z D j=n-3 3 − += n j z D j=n-2 2 − += n j z D j=n-1 1 − += n j z D j=n n j z D+= Retail Vault (i=2) j=n+3 K Dj=2 +
• Before proceeding with the proof, given an instance of Partition, we provide the construction of the instance of Problem P1 using the following example. Refer to Figure A.2.
• There are one regional vault, two specified retail vaults, and n + 4 branches in the network. That is Mrv = 1, r = 2, Mbr = n + 4, and MB = 0. In the
Table A.4: Transportation Cost gij between Vault i and Branch j.
i\ j j = 1 j = 2 . . . j = n j = n + 1 j = n + 2 j = n + 3 j = n + 4 i = 1 z1(2X + 1) z2(2X + 1) . . . zn(2X + 1) KX KX 2K5X 3K5X i = 2 z12X z22X . . . zn2X K4X K4X 2KX 3K9X i = 3 z16X z26X . . . zn6X K4X K4X 2K9X 3KX
by the regional vault i = 3; branch j = n + 2 is supplied by retail vault i = 1; branch j = n + 3 is supplied by retail vault i = 2.
• Z = {z1, z2, . . . , zn−1, zn}. Each variable in the set Z = {z1, z2, . . . , zn−1, zn}
corresponds to a branch. The demand values of Dj−and D+j , j = 1, 2, . . . , n+4, are given as follows.
D−j = 0, ∀ j.
D+j = zj, j = 1, 2, . . . , n ; Dj+= K, j = n+1, n+2; D
+
n+3 = 2K; D+n+4= 3K.
• For two specified retail vaults (i = 1, 2) and the existing regional vault (i = 3), their existing capacities are Ce
1 = K, C2e = 2K, and C3e = 6K, respectively. • Values of the transportation cost parameter gij are set in Table A.4 with X ≥ 1
by selecting daij and dtij accordingly for given value of cu, ct, ca, fa, fu, ba, bt,
ljU, ljIU, and lIMj , where X is an integer number.
• The capacity limit for the retail vault is cℓ = 3K and h = 1. The unit cost of
incremental capacity at the vaults is co = 1.
• For the current assignment, costs incurred to vault 1, 2, 3 are KX, 2KX, and 19KX, respectively. So, the total cost is 22KX currently.
For the instance of Problem P1 constructed above, we consider the following question:
Decision Problem: Does there exist a assignment σ with two specified retail vaults being upgraded such that the total cost Φσ
1 ≤ 11KX + 4K?
The decision problem is clearly in class NP. Also, it is easy to verify that the construction of the decision problem can be done in polynomial time. We now show that there exists a assignment σ with two specified retail vaults being upgraded such that Φσ1 ≤ 11KX +4K if and only if there exists a solution to the Partition problem. If part: Suppose there exists a partition of Z into two disjoint subsets Z1 and Z2 such that Z = Z1∪ Z2 and
∑
zj∈Z1zj = ∑
zj∈Z2zj = K. With |B| = 2, assignment σ
specifies the assignment to one regional vault and two upgraded regional vaults. Note that both retail vault 1 and 2 are upgraded, consider the following assign- ment σ: x1j = 1, j = zi ∈ Z1, n + 1, n + 2; x2j = 1, j = zi ∈ Z2, n + 3; x3j = 1, j = n + 4.
In σ, all three vaults have new capacities of 3K: C1p = C2p = C3p = 3K. Therefore, both retail vault 1 and 2 are upgraded to regional vaults with incremental capacity costs 2K and K. There is no incremental capacity cost for the original regional vault 3. So, the total incremental capacity cost is 3K in σ. Since in σ, branches j = zj ∈
Z1, n + 1, n + 2 are assigned to new regional vault 1, branches j = zj ∈ Z2, n + 3 are assigned to new regional vault 2, and branches j = n + 4 is assigned to the original regional vault 3, the total transportation cost is 11KX + K (4KX + K for vault 1, 4KX for vault 2, and 3KX for vault 3). Thus, the assignment σ gives the total cost Φσ1 = (11KX + K) + 3K = 11KX + 4K.
Only if part: Suppose there exists a assignment σ with two specified retail vaults being upgraded such that the total cost Φσ
1 ≤ 11KX + 4K. We first show that if the total cost Φσ
1 ≤ 11KX + 4K both retail vaults must be upgraded.
Proof: The following assignment σ is optimal: x1j = 1, j = zi ∈ Z, n + 1, n + 2;
x2j = 1, j = n + 3; x3j = 1, j = n + 4.
In σ, three vaults have new capacities: C1p = 4K; C2p = 2K; C3p = 3K. Thus, retail vault i = 1 is upgraded to new regional vault with incremental capacity costs 3K. There is no incremental capacity cost for vault i = 2 and i = 3. Since in assignment σ2, branches j = zi ∈ Z, n + 1, n + 2 are assigned to new regional vault
i = 1, branches j = n + 3 is assigned to new regional vault i = 2, and branches j = n + 4 is assigned to the original regional vault i = 3, the total transportation cost is 11KX + 2K (6KX + 2K for vault i = 1, 2KX for vault i = 2, and 3KX for vault 3). Thus, with X ≥ 1, the total cost is Φσ
1 = 11KX + 2K + 3K > 11KX + 4K. Claim 2: The retail vault i = 2 is only upgraded, then Φσ
1 > 11KX + 4K.
Proof: The following assignment σ is optimal: x1j = 1, j = n + 1; x2j = 1, j = zi ∈ Z, n + 3; x3j = 1, j = n + 2, n + 4.
In σ, three vaults have new capacities: C1p = K; C2p = 4K; C3p = 4K. Thus, retail vault i = 2 is upgraded to new regional vault with incremental capacity costs 2K. There is no incremental capacity cost for vaults i = 1 and i = 3. So, the total incremental capacity cost is 2K in σ. Since in σ, branches j = n + 1 is assigned to new regional vault i = 1, branches j = zi ∈ Z, n + 3 are assigned to new regional
vault i = 2, and branches j = n + 2, n + 4 are assigned to the original regional vault i = 3, the total transportation cost is 14KX (KX for vault i = 1, 6KX for vault i = 2, and 7KX for vault i = 3). Thus, with X ≥ 1, the total cost is Φσ
1 = 14KX + 2K > 11KX + 4K.
Claim 3: Both retail vaults i = 1 and i = 2 must be upgraded.
Proof: Note that the total cost is 22KX without any upgrades of vaults. Since Φσ1 ≤ 11KX + 4K, the result follows from Claims 1 and 2.
Claim 4: All branches j = zi ∈ Z cannot be assigned to vault i = 1.
Proof: Suppose all branches j = zi ∈ Z are assigned to vault i = 1. To find the
minimum cost solution all other branches must be assigned to the vaults having minimum gij. Thus we the following assignment σ: x1j = 1, j = zi ∈ Z, n + 1, n + 2;
x2j = 1, j = n + 3; x3j = 1, j = n + 4; with Φσ1 = 11KX + 5K > 11KX + 4K. This contradicts with the fact that Φσ1 ≤ 11KX + 4K.
Claim 5: All branches j = zi ∈ Z cannot be assigned to vault i = 2.
Proof: Suppose all branches j = zi ∈ Z are assigned to vault i = 2. To find the
minimum cost solution all other branches must be assigned to the vaults having minimum gij. Thus we the following assignment σ: x1j = 1, j = n + 1; x2j = 1, j = zi ∈ Z, n + 3; x3j = 1, j = n + 2, n + 4; with Φσ1 = 14KX + 2K > 11KX + 4K. This contradicts with the fact that Φσ1 ≤ 11KX + 4K.
As a consequence of Claims 4 and 5, we assume that Z1 (respectively, Z2) is a subset of branches zj ∈ Z are assigned to vault i = 1 (respectively, i = 2). Since a
branch can only be assigned to one vault, sets Z1 and Z2 are disjoint sets. In the minimum cost solution, a branch zj ∈ Z must be assigned to either vault i = 1 or
i = 2, we have Z = Z1∪ Z2. Claim 6: ∑zj∈Z1zj =
∑
zj∈Z2zj = K and there exists a solution to Partition
Problem.
Proof: In order to obtain minimum cost solution we have the following assignment σ: x1j = 1, j = zj ∈ Z1, n + 1, n + 2; x2j = 1, j = zj ∈ Z2, j = n + 3; x3j = 1, j = n + 4. Since both vaults must be upgraded and cℓ = 3K, we must have ∑
zj∈Z1zj = ∑
zj∈Z2zj = K. Thus, Φ σ
1 = 11KX + 4K and there exists a solution to Partition Problem.