Deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time

Abstract

In today's time-based competition, the unit cost of a high-tech product declines significantly over its short life cycle while its demand increases. In this paper, we extend the classical economic order quantity model to allow for not only time-varying demand but also fluctuating unit cost. In addition, we also allow for shortages and partial backlogging. We then prove that the optimal replenishment schedule not only exists but also is unique. In addition, we also show that the total cost is a convex function of the number of replenishments, which simplifies the search for the optimal number of replenishments to find a local minimum. Moreover, we further simplify the search process by providing an intuitively good starting search point.

Appendices

Appendix A. Proof of Lemma 1

Let

We then have

If c _v′(t)⩾C′(t)>0, then we know from (A2) that ∂TC _i /∂t _i ⩾0. Therefore, for any given i, TC _i is increasing with t _i . This implies that TC _i (s _i−1, t _i , s _i )⩾TC _i (s _i−1, s _i−1, s _i ) for any fixed i. Consequently, we obtain

This completes the proof of (a).

Next, if (t _i) ⩽ − (t,t _i ) < 0, with t⩽t _i , then we know from (A2) that ∂TC _i /∂t _i ⩽0. Thus, for any given i, TC _i is decreasing with t _i . This implies TC _i (s _i−1, t _i , s _i )⩾TC _i (s _i−1, s _i , s _i ), for any i. Therefore, we know

which proves part (b).

Appendix B. Proof of Lemma 2

From (11), we know that the optimal value of s _i (ie, s _i ^*) is the interior point between t _i and t _i+1 because if s _i =t _i or t _i+1, then Equation (11) does not hold. TC(n, {s _i }, {t _i }) is a continuous (and differentiable) function minimized over the compact set [0, H]²ⁿ. Hence, there exists an absolute minimum. The optimal value of t _i (ie, t _i ^*) cannot be on the boundary since TC(n, {s _i }, {t _i }) increases when any one of the t _i 's is shifted to the end points 0 or H. Consequently, there exists at least an inner optimal solution satisfied (11) and (12). Since s ₀=0 and t ₁ ^* is unique, if we can prove that both s _i ^* generated by (12) and t _i+1 ^* by (11) are uniquely determined, then we prove Lemma 2.

For any given s _i and t _i , from (11), we set

If C′(t)>c _v′(t)>−S _t (s _i , t) for all t, then we have

By the mean value theorem, we get

for some t _i ⩽k _i ⩽s _i . By taking the first derivative of F(x) with respect to x, we obtain

Consequently, there exists a unique t _i+1 (>s _i ) such that F(t _i+1)=0, which implies that solution to Equation (11) uniquely exists. Similarly, from (12), we set

Likewise, if C′(t)>c _v′(t)>−S _t (s _i , t) for all t⩾s _i , then we have

Taking the first derivative of G(x) with respect to x, we obtain

As a result, we know that there exists a unique s _i (>t _i ) such that G(s _i )=0. Thus, the solution to Equation (12) uniquely exists. Therefore, we complete the proof of Lemma 2.

Appendix C. Proof of Theorem 1

For any fixed n, differentiating TC(n, {s _i }, {t _i }) with respect to t ₁ and simplifying terms, we obtain

If we relax s _n to be any number, then we know from (11) and (12) that

which implies that TC(n, {s _i }, {t _i }) without the boundary condition of s _n =H is an increasing function of t ₁. From (11) and (12), we know that if t ₁=0 (or H), then s _n (t ₁)=0 (or >H). As a result, for any given n, there exists a unique t ₁ such that TC(n, {s _i }, {t _i }) in (10) is minimized with s _n =H. This proves (a).

To prove (b), we know from (11) that the optimal value of s _i (ie, s _i ^*) is the interior point between t _i and t _i+1 because if s _i =t _i or t _i+1, then Equation (11) does not hold. TC(n, {s _i },{t _i }) is a continuous (and differentiable) function minimized over the compact set [0, H]²ⁿ. Hence, there exists an absolute minimum. The optimal value of t _i (ie, t _i ^*} cannot be on the boundary since TC(n, {s _i },{t _i }) increases when any one of the t _i s is shifted to the end points 0 or H. Consequently, (11) and (12) are the necessary and sufficient conditions for the absolute minimum.

Appendix D. Proof of Theorem 2

Let us set

where R(n)=nc _f and T(n,0,H) = ∑ _{i = 1} ⁿ (P _i + c _h I _i + c _b B _i + c _l L _i ) − nc _f . Firstly, we know that R(n) is an increasing convex function of n. Next, by Bellman's principle of optimality,²² we know that the minimum value of T(n, 0, H) is

Let t=H and hence T ^*(n, 0, H) < T ^*(n−1, 0, H). The strict inequality follows because the minimum in (D2) occurs at an interior point. Thus, T ^*(n, 0, H) is strictly decreasing in n. Recursive application of (D2) yields the following relations:

In order to prove that T ^*(n, 0, H) is strictly convex in n, we choose H ₁ and H ₂ such that

and

Employing the principle of optimality on (D5) again, we have

and

respectively. Since H is an optimal interior point in T ^*(n+1, 0, H ₁) and T ^*(n+2, 0, H ₂), we know that

and

Utilizing the fact that

and

where w is the replenishment time between (a, b), and both a and b are the time at which the inventory level drops to zero in the cycle (a, b). We then obtain

where t _n ^*(n, 0, H) and t _n+1 ^*(n+1, 0, H ₁) are the corresponding last replenishment time when n orders are placed in [0, H], and n+1 orders are placed in [0, H ₁], respectively.

Similarly,

Subtracting (D12) from (D13), we have

since k(t)=[(c _h/θ)+c _v(t)]e^θ(H−t) is a decreasing function for all t<H. This implies that T ^*(n, 0, H)−T ^*(n+1, 0, H) is a strictly increasing function of H. Thus,

Again, by (D2) and the principle of optimality, we have

Let t=H in (D16), we have

By (D15) and (D17), we have

which implies T ^*(n, 0, H) is convex in n, and hence, TC(n)=R(n)+T ^*(n, 0, H) is also convex in n.