Multi-period risk minimization purchasing models for fashion products with interest rate, budget, and profit target considerations

Abstract

Traditionally, in the fashion industry, purchasing decisions for retailers are made based on various factors such as budget, profit target, and interest rate. Since the market demand is highly volatile, risk is inherently present and it is critically important to incorporate risk consideration into the decision making framework. Motivated by the observed industrial practice, we explore via a mean-variance approach the multi-period risk minimization inventory models for fashion product purchasing. We first construct a basic multi-period risk optimization model for the fashion retailer and illustrate how its optimal solution can be determined by solving a simpler problem. Then, we analytically find that the optimal ordering quantity is increasing in the expected profit target, decreasing in the number of periods of the season, and increasing in the market interest rate. After that, we propose and solve several extended models which consider realistic and timely industrial measures such as minimum ordering quantity, carbon emission tax, and carbon quota. We analytically derive the necessary and sufficient condition(s) for the existence of the optimal solution for each model and show how the purchasing budget, the profit target, and the market interest rate affect the optimal solution. Finally, we investigate the supply chain coordination challenge and analytically illustrate how an upstream manufacturer can offer implementable supply contracts to optimize the supply chain.

Appendix: All Proofs

Proof of Lemma 3.1

From (3.6a), (3.6b), (3.6c), we have Problem (BP-1) given as follows,

$$\begin{aligned} &\min_{q}\quad V \Biggl( \sum_{k = 1}^{N} \frac{\pi_{_{R,k}}(q)}{(1 + i)^{k}} \Biggr) \end{aligned}$$

(A.1)

$$\begin{aligned} &s.t.\quad\ \, E \Biggl( \sum_{k = 1}^{N} \frac{\pi_{_{R,k}}(q)}{(1 + i)^{k}} \Biggr) \ge T_{\sum} \end{aligned}$$

(A.2)

$$\begin{aligned} &\hphantom{\min_{q}}\quad \sum_{k = 1}^{N} \frac{cq}{(1 + i)^{k}} \le W_{\sum}. \end{aligned}$$

(A.3)

From (3.4) and (3.5), we have $A(i,N)= \frac{(1 + i)^{N} - 1}{i(1 + i)^{N}}$, and $B(i,N)= \frac{(1 + i)^{2N} - 1}{(1 + i)^{2N}[(1 + i)^{2} - 1]}$. Put them into (A.1), (A.2) and (A.3) yields (A.4),

$$\begin{aligned} &\min_{q}\quad B(i,N)\hat{\pi}_{R}(q) \\ &s.t.\quad\ \, A(i,N)\bar{\pi}_{R}(q) \ge T_{\sum} \\ &\hphantom{\min_{q}}\quad A(i,N)cq \le W_{\sum}. \end{aligned}$$

(A.4)

It is straight forward to show that (A.4) and Problem (BP-2) are the same.

Problem (BP-2)

$$\begin{aligned} &\min_{q}\quad \hat{\pi}_{R}(q) \end{aligned}$$

(A.5a)

$$\begin{aligned} &s.t.\quad\ \, \bar{\pi}_{R}(q) \ge T_{\sum} /A(i,N) \end{aligned}$$

(A.5b)

$$\begin{aligned} &\hphantom{\min_{q}}\quad cq \le W_{\sum} /A(i,N). \end{aligned}$$

(A.5c)

Thus, Problems (BP-1) and (BP-2) are equivalent, and the optimal solution for the original basic Problem (BP-1) is the same as the one for Problem (BP-2). □

Proof of Proposition 3.2

By definition in (3.9), we have $q_{R,BP^{*}}(T_{\sum} ) = \arg_{q \in [0,q_{R,E^{*}}]} [\bar{\pi}_{R}(q) = T_{\sum} /A(i,N)]$. Notice that: (i) $d\hat{\pi}_{R}(q)/dq > 0$ and hence the objective function $\hat{\pi}_{R}(q)$ [P.S.: (A.5a)] is an increasing function of q, (ii$) \bar{\pi}_{R}(q$) is a concave function of q and it is increasing $\forall q \in [0,q_{R,E^{*}}]$. Thus, if $q_{R,BP^{*}}(T_{\sum} ) \le W_{\sum} /[cA(i,N)]$[constraint (A.5c)] holds, the optimal solution for Problem (BP-2) is given by $q_{R,BP^{*}}(T_{\sum} )$. If $q_{R,BP^{*}}(T_{\sum} ) > W_{\sum} /[cA(i,N)]$, $q_{R,BP^{*}}(T_{\sum} )$ violates the purchasing budget constraint (A.5c) and is infeasible. However, since $q_{R,BP^{*}}(T_{\sum} )$ is the minimum ordering quantity which can achieve the expected profit target (A.5b), having a lower quantity will violate the expected profit target constraint and is hence infeasible. As a consequence, we can see that (a) Problem (BP-2), and equivalently Problem (BP-1), has an optimal solution if and only if $q_{R,BP^{*}}(T_{\sum} ) \le \frac{W_{\sum}}{cA(i,N)}$. (b) If $q_{R,BP^{*}}(T_{\sum} ) \le \frac{W_{\sum}}{cA(i,N)}$, the optimal ordering quantity for every period is equal to $q_{R,BP^{*}}(T_{\sum} )$. □

Proof of Corollary 3.3

From Proposition 3.2, we know that the necessary and sufficient condition for the existence of solution for Problem (BP-1) is given by$q_{R,BP^{*}}(T_{\sum} ) \le \frac{W_{\sum}}{ cA(i,N)}$. From it, we can see that $\frac{W_{\sum}}{cA(i,N)}$ will get larger if the purchasing budget W _∑ is larger, and $q_{R,BP^{*}}(T_{\sum} )$ will get smaller if the expected profit target T _∑ is smaller. These two situations both increase the likelihood that $q_{R,BP^{*}}(T_{\sum} ) \le \frac{W_{\sum}}{cA(i,N)}$ will be satisfied. We thus have Corollary 3.3. □

Proof of Corollary 3.4

If the optimal ordering quantity of Problem (BP-1) exists (when the condition in Proposition 3.2a holds), the optimal solution is given by $q_{R,BP^{*}}(T_{\sum} )$. By definition, we have $q_{R,BP^{*}}(T_{\sum} ) = \arg_{q \in [0,q_{R,E^{*}}]}[\bar{\pi}_{R}(q) = T_{\sum} /A(i,N)]$. From the analytical expression, it is obvious that $q_{R,BP^{*}}(T_{\sum} )$ is: (a) increasing in the expected profit target T _∑,(b) decreasing in the number of periods of the season N, and (c) increasing in the market interest rate i. □

Proof of Proposition 4.1

Problem (MP) extends Problem (BP-1) by including a constraint on MOQ.

From Proposition 3.2, we notice that Problem (BP-1) and Problem (BP-2) are equivalent. As such, we can rewrite Problem (MP) as Problem (MP-2) in the following,

Problem (MP-2)

$$\begin{aligned} &\min_{q}\quad \hat{\pi}_{R}(q) \end{aligned}$$

(A.6a)

$$\begin{aligned} &s.t.\quad\ \, \bar{\pi}_{R}(q) \ge T_{\sum} /A(i,N) \end{aligned}$$

(A.6b)

$$\begin{aligned} &\hphantom{\min_{q}}\quad cq \le W_{\sum} /A(i,N) \end{aligned}$$

(A.6c)

$$\begin{aligned} &\hphantom{\min_{q}}\quad q \ge q_{MOQ}. \end{aligned}$$

(A.6d)

Define:

$$ Q_{MOQ}(T_{\sum} ) = \max \bigl(q_{R,BP^*}(T_{\sum} ),q_{MOQ}\bigr). $$

(A.7)

$$ q_{R,MP2^*} = \textrm{the optimal ordering quantity for Problem (MP-2)}. $$

(A.8)

Notice that following the same argument as in the proof of Proposition 3.2, Problem (MP-2) has an optimal solution if and only if $Q_{MOQ}(T_{\sum} ) \le \frac{W_{\sum}}{cA(i,N)}$, which relates to constraint (A.6c). If this necessary and sufficient condition holds, we have two cases, namely Case (i) $q_{MOQ} \le q_{R,BP^{*}}(T_{\sum} ) \le \frac{W_{\sum}}{cA(i,N)}$, and Case (ii) $q_{R,BP^{*}}(T_{\sum} ) < q_{MOQ} \le \frac{W_{\sum}}{ cA(i,N)}$. This completes the proof for part (a).

For parts (b) and (c), as we discussed in the proof of Proposition 3.2, since the objective function (A.6a) aims at minimizing risk and $\hat{\pi}_{R}(q)$ is an increasing function of q, the optimal q is the smallest possible q which satisfies all the constraints. As such, we have:

• For Case (i): If $q_{MOQ} \le q_{R,BP^{*}}(T_{\sum} ) \le \frac{W_{\sum}}{ cA(i,N)}$, then $q_{R,MP2^{*}} = q_{R,BP^{*}}(T_{\sum} )$.

• For Case (ii): If $q_{R,BP^{*}}(T_{\sum} ) < q_{MOQ} \le \frac{W_{\sum}}{ cA(i,N)}$, then $q_{R,MP2^{*}} = q_{MOQ}$.

Since Problem (MP-2) and Problem (MP) have the same optimal solution, we have: $q_{R,MP2^{*}}= q_{R,MP^{*}}$, which implies Proposition 4.1(b) and Proposition 4.1(c). □

Proof of Corollary 4.2

(a) Following the same logic and argument as in the proof of Corollary 3.4. (b) A direct result from Proposition 4.1(c). □

Proof of Lemma 5.1

By definition, we have $\bar{\pi}_{R}(q)= (p - c)q - (p - v)\int_{0}^{q} F(x)dx$,

$$\hat{\pi}_{R}(q)= (p - v)^{2}\xi (q),\qquad \bar{ \pi}_{R,TAX}(q)= (p - c - t)q - (p - v)\int_{0}^{q} F(x)dx,\quad \mbox{and } $$

$\hat{\pi}_{R,TAX}(q) = (p - v)^{2}\xi (q)$. Directly comparing them implies that for any fixed quantity q and carbon emission tax t>0, we have: (a$)\bar{\pi}_{R,TAX}(q) < \bar{\pi}_{R}(q$) and (b$)\hat{\pi}_{R,TAX}(q)=\hat{\pi}_{R}(q$). □

Proof of Proposition 5.2

First of all, similar to the proof for Proposition 3.2, observe that Problem (CP) and Problem (CP-2) below have the same optimal solution.

Problem (CP-2)

$$\begin{aligned} &\min_{q}\quad \hat{\pi}_{R,TAX}(q) \end{aligned}$$

(A.9)

$$\begin{aligned} &s.t.\quad\ \, \bar{\pi}_{R,TAX}(q) \ge T_{\sum} /A(i,N) \end{aligned}$$

(A.10)

$$\begin{aligned} &\hphantom{\min_{q}}\quad (c + t)q \le W_{\sum} /A(i,N) \end{aligned}$$

(A.11)

$$\begin{aligned} &\hphantom{\min_{q}}\quad q \le q_{CQ}. \end{aligned}$$

(A.12)

Define:

$$ q_{R,CP2^*} = \textrm{the optimal ordering quantity for Problem (CP-2)}. $$

(A.13)

It is easy to observe that Problem (CP-2)’s optimal solution $q_{R,CP2^{*}}$ exists and is equal to $q_{R,BP,TAX^{*}}(T_{\sum} )$ if and only if $q_{R,BP,TAX^{*}}(T_{\sum} ) \le \frac{W_{\sum}}{(c + t)A(i,N)}$ and $q_{R,BP,TAX^{*}}(T_{\sum} ) \le q_{CQ}$. Since, $q_{R,CP2^{*}}= q_{R,CP^{*}}$, Proposition 5.2 results. □

Proof of Corollary 5.3

Following the same logic and argument as in the proof of Corollary 3.4. For the impact of carbon emission tax t, it is a direct result from Proposition 5.2 and the definition of $q_{R,BP,TAX^{*}}(T_{\sum} )$. □

Proof of Proposition 6.1

Following the proofs and the results of Proposition 4.1 and Proposition 5.2, Proposition 6.1 is derived. □

Proof of Lemma 7.1

• (a) First of all, from (7.4), we know that $q_{SC,TAX^{*}} = \arg \max_{q}E[\sum_{k = 1}^{N} \frac{\pi_{SC,k,TAX}(q)}{(1 + i)^{k}} ]= \arg \{ \max_{q}A(i,N)\bar{\pi}_{SC,TAX}(q)\}=\arg \{ \max_{q}\bar{\pi}_{SC,TAX}(q)\}$. Simple calculus confirms that $\bar{\pi}_{SC,TAX}(q)$ is a concave function and hence $q_{SC,TAX^{*}}=\arg_{q}\{ d\bar{\pi}_{SC,TAX}(q)/dq = 0\}= F^{ - 1}[(p - m - t)/(p - v)]$.

• (b) Since the presence of carbon quota is imposed by the governing body and the whole supply chain will be affected, the best possible quantity cannot exceed q _CQ and hence we have $q_{SC^{*}}=\min \{ q_{SC,TAX^{*}},q_{CQ}\}$.

 □

Proof of Proposition 7.2

By coordination, we aim at making the optimal ordering quantity for Problem (MCP) equal to $q_{SC^{*}}$. Notice that by definition, we have $q_{SC^{*}}=\min \{ q_{SC,TAX^{*}},q_{CQ}\}$. Thus, the feasibility condition on $q_{SC^{*}} \le q_{CQ}$ is implied (and always satisfied). As a result, under the markdown money contract with MOQ =0 (i.e. q _MOQ =0), the only feasibility check is $q_{SC^{*}} \le \frac{W_{\sum}}{(c + t)A(i,N)}$. When $q_{SC^{*}} \le \frac{W_{\sum}}{(c + t)A(i,N)}$ holds, we have,

$$ q_{R,MM - CP^*}^{MM} = \arg_{q}\biggl\{ (p - c - t)q - (p - v - \alpha )\int_{0}^{q} F(x)dx = T_{\sum} /A(i,N)\biggr\}. $$

(A.14)

With (A.14), we have the following,

$$\begin{aligned} &q_{R,MM - CP^*}^{MM} = q_{SC^*} \\ &\quad {} \Leftrightarrow (p - c - t)q_{SC^*} - (p - v - \alpha )\int _{0}^{q_{SC^*}} F(x)dx = T_{\sum} /A(i,N) \\ &\quad {} \Leftrightarrow \alpha = \alpha^{*} \equiv (p - v) - \biggl( \frac{A(i,N)(p - c - t)q_{SC^*} - T_{\sum}}{A(i,N)\int_{0}^{q_{SC^*}} F(x)dx} \biggr). \end{aligned}$$

 □

Proof of Proposition 7.3

Similar to the proof of Proposition 7.2, observe that by definition, we have $q_{SC^{*}}=\min \{ q_{SC,TAX^{*}},q_{CQ}\}$. Thus, the feasibility condition on $q_{SC^{*}} \le q_{CQ}$ is always satisfied. Under the wholesale pricing contract with MOQ =q _MOQ , the only feasibility check is $q_{SC^{*}} \le \frac{W_{\sum}}{(c + t)A(i,N)}$ as q _MOQ is a control variable of the manufacturer. When $q_{SC^{*}} \le \frac{W_{\sum}}{(c + t)A(i,N)}$ holds, according to Proposition 6.1, setting $q_{MOQ}= q_{SC^{*}}$ can coordinate the supply chain by making $q_{R,MCP^{*}} = q_{SC^{*}}$. □