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We analyze an infinite horizon discrete time inventory model with deterministic but non-stationary demand for a single product at a single stage. There is a finite cycle of vectors of characteristics of the environment (demand, fixed ordering cost, variable procurement cost, holding cost) which is repeated after a finite number of periods. Future cost is discounted. In general, minimization of the sum of discounted total cost over the cycle does not give the minimum of the sum of discounted total cost over the infinite horizon. We construct an algorithm for computing of an optimal strategy over the infinite horizon. It is based on a forward in time dynamic programming recursion.

Standard finite horizon inventory models with deterministic but non-stationary demand (see, for example, [

If demands and other characteristics of the environment that differ between periods exhibit some finite cycle, we can obtain a numeric solution of an infinite horizon inventory model. In this case, after a finite number of periods, the same finite cycle of characteristics of the environment is repeated (Stationary characteristics of the environment are a special case of this, with cycle length equal to one). In the present paper, we deal with such a case. We allow fixed ordering cost, variable procurement cost, and holding cost that differ between periods. We develop an algorithm for computing of an optimal procurement strategy in this model that minimizes the sum of discounted total costs over the infinite horizon of the model. The optimal procurement strategy determines the optimal procurement cycle, at the end of which the inventory is zero. That is, except for a finite number of periods at the beginning of the model, the optimal procurement strategy is an infinite repetition of the procurement strategy over the optimal procurement cycle.

Throughout the paper,

Consider the inventory system with the length of the planning horizon equal to five periods used in [

Without discounting of future cost, the unique optimal procurement strategy is

We consider an infinite horizon discrete time inventory model. Periods are numbered by positive integers. Each period

where

That is, the environmental vectors exhibit the finite cycle of length

We assume that

and

It follows from (2) and (3) that the sum of procurement and holding cost in each period is not lower than procurement cost in the immediately following period. Inequality (4) implies that there does not exist a period

This follows from the fact that

All arguments used in this paper remain valid and the algorithm described in Section 4 can be used when (4) does not hold but Conditions 1 and 2 given below the definition of

We denote by

The purchasing firm discounts future cost by discount factor

subject to

We will use the term “optimal procurement strategy” for an optimal solution to the problem (5)-(8) and the term “feasible procurement strategy” for a procurement strategy that satisfies constraints (6)-(8). In the construction of the algorithm in the next section, we will use the following lemma. It is an analogue of a well known result from the analysis of finite horizon lot sizing models without discounting of future cost that was used in [

Lemma 1 Let

Proof. Suppose that the claim of the lemma does not hold for some optimal procurement strategy

Thus,

by reducing

Thus,

Therefore, the sum of discounted procurement and holding cost is decreased.

Lemma 1 has an obvious corollary.

Corollary 1 Let

i.e.,

We begin this section with formulation of criteria that we will use in the description of the algorithm for solving the problem (5)-(8).

The sufficient condition for not placing an order in period

If (9) holds then it is cheaper to satisfy demand in period

Taking into account (3) and the fact that

If an order was placed in period

Let

Throughout the paper, we assume that, whenever the firm is indifferent between placing an order in two periods, it places it in the later one. Then the sufficient condition for placing an order in period

Denote by

All arguments used in this paper remain valid and the algorithm described in Section 4 can be used when (4) does not hold but the following conditions are satisfied. We illustrate their use in the example at the end of this section.

Condition 1 There exists

Condition 2 For each

We let

It follows from the assumption that there exists

For each

Lemma 2 Let

Then, for each optimal procurement strategy for the first

Proof. Take (arbitrary)

The optimality of

Since (using (3))

and

Of course,

Consider period

Inequality (16) is equivalent to

and (17) is equivalent to

From inequalities (18) and (19) we can compute the critical value of

for each

Suppose that set

For each

Proposition 1 Assume that there exist

satisfies

Proof. Using Lemma 1, the optimal procurement strategy for the first

Suppose that there exists feasible procurement strategy

Taking into account (4), we can assume without loss of generality that

(where

This contradicts the fact that

The algorithm is based on solving a succession of problems with a finite number of periods. Proposition 1 implies that we can stop when we find

exists. The following lemma shows that such

Lemma 3 There is

Proof. For each

Taking into account (4), (2), Lemma 1, and Corollary 1, there is a finite set to which element of

The stopping rule in the algorithm can be simplified if there exists

In the algorithm, we use the equality sign for the assignment of a new value to the variable whenever such expression is correct from the mathematical point of view. Otherwise, we use the symbol

Algorithm 1 Step 1: Set

Step 2: Set

Step 3: If

Step 4: For each

and let

Step 5: Let

If

If

Step 6: Let

set

Step 7: Let

If

Step 8: If there exists

go to step 9. Otherwise, go to step 2.

Step 9: Set

The algorithm does not give the optimal value of the objective function (5). Using values computed by the algorithm and setting

We could use the stopping rule specified in the algorithm and solve finite horizon problems by the WagnerWhitin algorithm, modified for the case of discounting of future cost. Nevertheless, our algorithm has several advantages in comparison with their algorithm. Firstly, it saves calculations by identifying periods in which an order should be placed. Secondly, it saves calculations by identifying periods in which an order will not be placed. Thirdly, when a period is removed from the set of candidates for placing an order in some iteration, it is no longer considered in the following iterations. Moreover, it is enough to compare only successive elements of the set of candidate periods. From the point of view of elimination of candidate periods, our algorithm is similar to Wagner-Whitin algorithm [

We have constructed an algorithm for computing an optimal procurement strategy in an infinite horizon inventory model with non-stationary deterministic demand, a finite cycle of environmental vectors, and discounting of future cost. It is based on solving a succession of finite horizon inventory optimization problems. The formulation of the stopping rule is made possible by the fact that the cycle of environmental vectors is finite.

It is worth noting that our algorithm can also be used to solve a finite horizon problem. This also holds when future cost is not discounted (i.e.,

The research reported in this paper was supported by the grant VEGA 1/0181/12 from the Slovak Ministry of Education, Science, Research, and Sport. VEGA did not play any role in the study design or in the writing of the article or in the decision to submit it for publication.