El conflicto en el materialismo histórico

We show in this section that there is under some mild conditions an asymptotic equivalence between the V^EHICLE R^OUTING PROBLEM WITH S^TOCHASTIC D^EMANDS and the T^RAVELING S^ALESMAN P^ROBLEM, as well as between the V^EHICLE R^OUT

-INGPROBLEM WITH STOCHASTIC DEMANDS AND CUSTOMERS and the PROBABILISTIC

T^RAVELINGS^ALESMANP^ROBLEM.

One might think that it is important for the VRPSD and the VRPSDC to opti-mize the (expected) length of the route as well as to optiopti-mize the route in a way, such that those customers which cause a restocking action with high probability, are located close to the depot. We show that the distribution of the amount of goods in the vehicle converges to the uniform distribution under certain condi-tions. In this case the probability that restocking is required at a certain customer depends mainly on the distribution of its demand and only slightly on the par-ticular shape of the tour. In other words, if the amount of goods in the vehicle is close to the uniform distribution, it is sufficient to optimize the (expected) length of the tour.

Let us assume for the moment that the vehicle is loaded in a probabilistic way, such that the amount of goods is initially distributed according to the uniform distribution. Then the VRPSD and the TSP, and the VRPSDC and the PTSP are equivalent, if the basic restocking strategy is used. These equivalences can be verified easily using the Markov chain model introduced in the previous section.

Together with the convergence of the amount of goods in the vehicle towards the uniform distribution, this observation is the basic idea behind our analyses.

We now start with a more general mathematical result. After that we show under which conditions this result can be used to derive concrete results for the VRPSD and for the VRPSDC.

Theorem 1. Let (An)_n∈N be a series of doubly stochastic m× m matrices, whose entries are all bounded from below by a constant L > 0. Then lim

n→∞

Qn i=1

A_i = _m¹J , where J is a matrix of ones.

Proof. This theorem is trivially true for m = 1, so we can assume m > 1 for the following proof. Since the product of doubly stochastic matrices is a dou-bly stochastic matrix and for any doudou-bly stochastic matrix A = (ai j) we have

m−1P

j=0

a_{i j}= 1 ∀i ∈ {0, 1, . . . , m − 1} and ai j ≥ 0 ∀i, j ∈ {0, 1, . . . , m − 1}, it is suffi-cient to show that the entries in each row converge to the same value. We prove this by showing that the difference between the smallest and the largest entry in each row converges to 0.

We show by induction that the difference between the smallest and the largest entry in each row of

Qn i=1

A_i is bounded from above by (1 − 2L)ⁿ. It is clear that this expression is always non-negative, because L is trivially bounded from above by 1/2.

For n= 1 this is clearly the case. The smallest entry is bounded from below by L and since we have doubly stochastic matrices, the largest entry is bounded from above by 1− L. The difference of the the largest entry and the smallest entry can be at most(1−L)−L = 1−2L = (1−2L)¹. Now suppose the statement holds for n∈ N. Here we use A⁽ⁿ⁾=Qⁿ

i=1

A_i as an abbreviation for the product of the first n matrices. Let i ∈ {0, 1, . . . , m − 1} be a fixed row index and let li(n) and u_i(n) be the smallest, respectively the largest entry in the i-th row of A⁽ⁿ⁾. The entries in the i-th row of A⁽ⁿ⁺¹⁾ are convex combinations of the entries in the i-th row of A⁽ⁿ⁾, where the corresponding coefficients of each of these convex combinations is a column of A_n+1. Since the entries of A_n+1 are bounded from below by L we have

u_i(n + 1) ≤ (1 − L)ui(n) + Lli(n) = ui(n) − L(ui(n) − li(n)) and

l_i(n + 1) ≥ (1 − L)li(n) + Lui(n) = li(n) + L(ui(n) − li(n)).

The difference of these values is

u_i(n + 1) − li(n + 1) ≤ ui(n) − L(ui(n) − li(n)) − li(n) − L(ui(n) − li(n))

= ui(n) − li(n) − 2L(ui(n) − li(n))

= (1 − 2L)(ui(n) − li(n))

≤ (1 − 2L)ⁿ⁺¹.

In the last step we used the induction hypothesis and since lim

n→∞(1−2L)ⁿ= 0 we finished the proof.

In the previous section we have seen that we can model the distribution of the amount of goods in the vehicle with a Markov chain. It is possible to identify the matrices A_iof theorem 1 with transition matrices of that Markov chain. Using the basic restocking strategy these transition matrices are doubly stochastic, but unfortunately it is a very strong assumption that their entries are all bounded from below by a constant L> 0. In this case each customer has a strictly positive probability for each possible demand and this assumption is obviously too strong for practical applications. In the remaining part of this section we show that we can use theorem 1 also under more moderate conditions.

The main idea here is to partition the matrices into blocks of consecutive matrices, such that the product of all the matrices within a block fulfills the con-ditions of theorem 1. Due to the associative law the product of all the matrices can be obtained by first calculating the products of the matrices within each block and then multiplying the resulting products. In this way we are able to state theorem 1 using much weaker assumptions on the matrices used. These weaker assumptions are comprised in the following definition and as we will see later, it is not even required that these assumptions hold for all of the matrices.

Definition 5 (greatest common divisor property, gcd property). A cyclic m× m matrix A= (ai j) with only non-negative entries and with strictly positive entries a_i

10, a_i20, . . . , a_il0, i₁< i2< . . . < il in the first column fulfills the greatest common divisor property (short: gcd property), ifgcd(il− i1, i_l− i2, . . . , i_l− i_l−1, m) = 1.

Instead of using the differences relative to i_l we could have also used differ-ences relative to any other of the elements i_k, k ∈ {1, 2, . . . , l}. The definition is also invariant with respect to the chosen column. We prove both invariances in appendix A.

At first glance this definition looks quite technical, but it is usually fulfilled by a lot of the transition matrices for practical instances of the VRPSD and the

VRPSDC. In particular, the greatest common divisor property is fulfilled for a transition matrix, if the corresponding customer has two different demands of amount k and k+ 1 with strictly positive probabilities. For the VRPSDC we have always a strictly positive entry at the first position in the first column (unless the probability for visiting this customer is exactly 1) and therefore the condition is even slightly weaker in this case. We discuss this issue more in detail in section 2.4. Here we continue with deriving a convergence result as in theorem 1 under more mild assumptions comprised by the gcd property.

We start with proving the following important lemma using the definition of the gcd property and some modular arithmetic.

Lemma 2. Let A be a m× m matrix with only non-negative entries and with ki

strictly positive entries in row i, i ∈ {0, 1, . . . , m − 1}, and let B be a cyclic m × m matrix with only non-negative entries fulfilling the greatest common divisor prop-erty. Then the product C = AB contains only non-negative entries and has at least min{ki+ 1, m} strictly positive entries in row i, for each i ∈ {0, 1, . . . , m − 1}.

Proof. The elements of the set I = {i1, i₂, . . . , i_n} ⊂ Z/mZ, n ∈ N, with gcd(i1, i₂, . . . , i_n, m) = 1 can be used to generate any element of Z/mZ by a linear combination of i₁, i₂, . . . , i_n. This is a well known algebraic result, see e.g.

Baldoni et al.[2008].

For our proof we need the following useful property. Given a set S⊂ Z/mZ with|S| < |Z/mZ|, and I as above, then for the set S⁰= {s+i | s ∈ S, i ∈ I ∪{0}}

the inequality |S| < |S⁰| holds. Otherwise we cannot generate all elements of Z/mZ by linear combinations of i1, i₂, . . . , i_n.

Let i∈ {0, 1, . . . , m − 1} be a fixed row index with ki < m, let S = {j | ai j >

0} ⊂ Z/mZ and let S⁰= {j | ci j > 0} ⊂ Z/mZ. Furthermore let bj10, b_j20, . . . , b_j

l0, j₁ < j2 < . . . < jl be the strictly positive entries in the first column of B. Due to the properties of our matrices, we have

S⁰

which concludes the proof for all rows with less than m strictly positive entries.

Now let i ∈ {0, 1, . . . , m − 1} be a fixed row index with ki = m. Since each column of B has at least 1 strictly positive entry, the number of strictly positive

entries in the i-th row of C = AB is m which concludes the proof for rows with exactly m strictly positive entries.

Using this lemma we can obtain a sufficient condition for the case that the product of a number of transition matrices contains only strictly positive entries.

Corollary 1. Let A₁, A₂, . . . , A_k be cyclic m× m matrices with only non-negative entries and with at least 1 strictly positive entry in each row, and let m− 1 of these matrices fulfill the greatest common divisor property. Then B=Q^k

i=1

A_i contains only strictly positive entries.

Proof. Since the matrices are cyclic and have at least 1 strictly positive entry in each column the number of strictly positive entries in the series of products is never decreasing in any of the rows. Due to lemma 2 a multiplication with one of the matrices, which fulfills the gcd property, increases the number of strictly positive entries in each row by at least 1 (if it has not been maximum already).

Putting these observations together the corollary follows easily.

To generalize the convergence result of theorem 1 we need the following lemma.

Lemma 3. Let A= (ai j) be a m×m matrix containing only strictly positive entries, with a smallest entry l and a largest entry u and let B be a m× m column stochastic matrix. Then u⁰−l⁰≤ u−l, where u⁰and l⁰are the largest, respectively the smallest, strictly positive entries in AB.

Proof. The elements of AB are convex combinations of elements of A and there-fore the elements of AB are bounded from below by l and from above by u.

Now we are able to generalize the convergence result of theorem 1.

Theorem 2. Let (An)_n∈N be a series of doubly stochastic, cyclic m× m matrices, whose entries are all bounded from below by a constant L > 0. Let the great-est common divisor property be fulfilled by infinitely many of these matrices and let k₁ < k2 < . . . be the indices of the matrices that fulfill the gcd property. If max{ki+1− ki | i ∈ N} ≤ k for a constant k ≥ k1 then lim

n→∞

n

Q

i=1

A_i = _m¹J , where J is a matrix of ones.

Proof. To proof this theorem we have to combine the previous results. Here we show like in theorem 1 that the differences between the largest and the

smallest element in each row of the successive products

It is possible to partition the matrices into blocks of at most k(m − 1) con-secutive matrices, such that the matrices in each block fulfill the conditions of corollary 1. Let 1 = j1 < j2 < . . . be the indices at which the first, second, . . . block begins. Then we can bound the smallest entry of the products

j_l+1−1

Due to corollary 1 these matrices fulfill the conditions of theorem 1 and thus we have lim the difference between the largest and the smallest element in each row of the product that the difference between the largest and the smallest element of

j

Let us quickly connect these results with the original stochastic vehicle rout-ing problems. What we have seen on a more abstract level is the followrout-ing. If we impose some mild conditions on the transition matrices of our Markov chain model, the amount of goods in the vehicle after a customer has been processed is approaching the uniform distribution as we continue on the tour and more and more customers are visited. For the basic restocking strategy which we are using here, the transition matrices are characterized only by the customers’ de-mands (and in the case of the VRPSDC additionally by the probability that the customers require to be visited). Therefore, the mild conditions on the transition matrices are transferred to mild conditions on the customers’ demands. At the very beginning of this chapter we have already observed that it is sufficient to optimize the (expected) length of the route if the amount of goods in the vehicle is distributed according to the uniform distribution. Together with our conver-gence result, this shows that it is sufficient to focus on optimizing the (expected) length of the route in many situations. In the next section we will examine the

convergence speed for a typical demand distribution and in section 2.4 we will discuss the impacts of the results for the VRPSD and the VRPSDC more in detail.

2.3 Convergence Speed for Binomial