Robust production planning with budgeted cumulative demand uncertainty

We are given $T$ periods, a demand $d_{t}\geq 0$ in each period $t$, $t\in[T]$ ($[T]$ denotes the set $\{1,\ldots,T\}$), production, inventory and backordering costs and a selling price, denoted by $c^{P}$, $c^{I}$, $c^{B}$ and $b^{P}$, respectively, which do not depend on period $t$. Let $x_{t}\geq 0$ be a production amount in period $t$, $t\in[T]$. We assume that the production amounts $x_{1},\ldots,x_{T}$, called the production plan, can be under some linear constraints. Namely, let $\mathbb{X}\subseteq\mathbb{R}^{T}_{+}$ be a set of production plans, specified by linear constraints, for instance:

$$\mathbb{X}=\{\boldsymbol{x}=(x_{1},\ldots,x_{T}):\;x_{t}\geq 0,l_{t}\leq x_{t}\leq u_{t},L_{t}\leq\sum_{i\in[t]}x_{i}\leq U_{t},t\in[T]\}\subseteq\mathbb{R}^{T}_{+},$$

where $l_{t}$, $u_{t}$ and $L_{t}$, $U_{t}$ are prescribed capacity and cumulative capacity limits, respectively. Accordingly, we wish to find a feasible production plan $\boldsymbol{x}\in\mathbb{X}$, subject to the conditions of satisfying each demand and the capacity limits, which minimizes the total production, storage and backordering costs minus the benefit from selling the product.

Set $D_{t}=\sum_{i\in[t]}d_{i}$ and $X_{t}=\sum_{i\in[t]}x_{i}$, i.e. $D_{t}$ and $X_{t}$ stand for the cumulative demand up to period $t$ and the cumulative production up to period $t$, respectively. We do not examine additional production processes, for example with setup times and costs, which lead to NP-hard problems even for special cases (see, e.g., [17, 14]). The problem under consideration is a version of the capacitated single-item lot sizing problem with backordering (see, e.g., [13, 37]). It can be represented by the following linear program:

	$\displaystyle\min$	$\displaystyle\sum_{t\in[T]}(c^{I}I_{t}+c^{B}B_{t}+c^{P}x_{t}-b^{P}s_{t})$			(1)
	s.t.	$\displaystyle B_{t}-I_{t}=D_{t}-X_{t}$	$\displaystyle t\in[T],$		(2)
		$\displaystyle\sum_{i\in[t]}s_{i}=D_{t}-B_{t}$	$\displaystyle t\in[T],$		(3)
		$\displaystyle X_{t}=\sum_{i\in[t]}x_{i}$	$\displaystyle t\in[T],$		(4)
		$\displaystyle B_{t},I_{t},s_{t}\geq 0$	$\displaystyle t\in[T],$		(5)
		$\displaystyle\boldsymbol{x}\in\mathbb{X}\subseteq\mathbb{R}^{T}_{+},$			(6)

where $I_{t}$, $B_{t}$ and $s_{t}$ are inventory level, backordering level and sales of the product at the end of period $t\in[T]$, respectively. We assume that the initial inventory and backorder levels are equal to 0. There is an optimal solution to (1)-(6) which satisfies $B_{t}I_{t}=0$ for each $t\in[T]$, so inventory storage from period $t$ to period $t+1$ and backordering from period $t+1$ to period $t$ are not performed simultaneously. Indeed, if $B_{t}>0$ and $I_{t}>0$, then we can modify the solution so that $B_{t}=0$ or $I_{t}=0$ without violating the constraints (2)-(3) and increasing the objective value. Using this observation, we can rewrite (1)-(6) in the following equivalent compact form, which is more convenient to analyze:

$$\min_{\boldsymbol{x}\in\mathbb{X}}\left(\sum_{t\in[T]}\max\{c^{I}(X_{t}-D_{t}),c^{B}(D_{t}-X_{t})\}+c^{P}X_{T}-b^{P}\min\{X_{T},D_{T}\}\right).$$

(7)

Define

$$f_{I}(X_{t},D_{t})=\left\{\begin{array}[]{lll}c^{I}(X_{t}-D_{t})&\text{if }t\in[T-1],\\ c^{I}(X_{t}-D_{t})+c^{P}X_{T}-b^{P}D_{t}&\text{if }t=T,\end{array}\right.$$

$$f_{B}(X_{t},D_{t})=\left\{\begin{array}[]{lll}c^{B}(D_{t}-X_{t})&\text{if }t\in[T-1],\\ c^{B}(D_{t}-X_{t})+c^{P}X_{T}-b^{P}X_{t}&\text{if }t=T.\end{array}\right.$$

Hence, after considering two cases, namely $X_{t}\leq D_{t}$ and $X_{t}>D_{t}$, for each $t\in[T]$, problem (7) can be represented as follows

$$\min_{\boldsymbol{x}\in\mathbb{X}}\sum_{t\in[T]}\max\{f_{I}(X_{t},D_{t}),f_{B}(X_{t},D_{t})\}.$$

(8)

From now on, we will refer to (8) instead of (1)-(6). Observe that $f_{I}(X_{t},D_{t})$ is nonincreasing function of $D_{t}$, $f_{B}(X_{t},D_{t})$ is nondecreasing function of $D_{t}$. Furthermore, the function $\max\{f_{I}(X_{t},D_{t}),f_{B}(X_{t},D_{t})\}$ is convex in $X_{t}$ and $D_{t}$, since both functions $f_{I}$ and $f_{B}$ are linear in $X_{t}$ and $D_{t}$.

We now admit that demands in problem (8) are subject to uncertainty. In practice, a knowledge about uncertainty in a demand is often expressed as $\pm\Delta$, where $\Delta$ is a possible deviation from a nominal demand value. This means that the actual demand will take some value within the interval determined by $\pm\Delta$, but it is not possible to predict at present which one. In consequence, a simple and natural interval uncertainty representation is induced.

A demand has a twofold interpretation, namely an actual demand $d_{t}$ in period $t$ or a cumulative demand $D_{t}$ up to period $t$, $t\in[T]$. The former interpretation is often considered in the literature. However, in this case the deviations $\Delta$ cumulate in subsequent periods, due to the interval addition, $\widetilde{D}_{t}=\widetilde{\sum}_{t\in[T]}\tilde{d}_{t}$, $d_{t}\in\tilde{d}_{t}=[\hat{d}_{t}-\Delta,\hat{d}_{t}+\Delta]$, where $\hat{d}_{t}$ is the nominal value of the demand in period $t$. This may be unrealistic in practice, because the deviation for cumulative demand in period $t$ becomes $t\Delta$ (see Figure 1a). Therefore, in this paper we use a model of uncertainty in the cumulative demands rather than in actual demands, expressed by symmetric intervals $\widetilde{D}_{t}=[\widehat{D}_{t}-\Delta,\widehat{D}_{t}+\Delta]$, $t\in[T]$, prescribed, where $\widehat{D}_{t}$ is the nominal value of the cumulative demand up to period $t\in[T]$ (see Figure 1b). Such a model is able to take the uncertain demands in periods into account ($\tilde{d}_{t}$ lies in $\widetilde{D}_{t}$) as well as dependencies between periods. We allow different deviations for each period. Accordingly, the value of cumulative demand $\widetilde{D}_{t}$ is only known to belong to the interval $[\widehat{D}_{t}-\Delta_{t},\widehat{D}_{t}+\Delta_{t}]$, $t\in[T]$, where $0\leq\Delta_{t}\leq\widehat{D}_{t}$ is the maximum deviation of the cumulative demand from its nominal value $\widehat{D}_{t}\geq 0$. Each feasible vector $\boldsymbol{D}=(D_{1},\ldots,D_{T})$ of cumulative demands, called scenario, must satisfy the following two reasonable constraints: $D_{t}\geq 0$ for every $t\in[T]$ and $D_{t}\leq D_{t+1}$ for every $t\in[T-1]$. Since the vector of nominal cumulative demands should be feasible, we also assume that $\widehat{D}_{t}\leq\widehat{D}_{t+1}$, $t\in[T-1]$. Notice that scenario $(D_{1},\ldots,D_{T})$ induces a vector of actual demands in periods $t$, i.e. $d_{1}=D_{1}$, $d_{t}=D_{t}-D_{t-1}$, $t=2,\ldots,T$.

In this paper we study the following two special cases of the interval, symmetric uncertainty representations, that are common in robust optimization, in which scenario sets are defined in the following way (see, e.g., [9, 10, 35]):

	$\displaystyle\mathcal{U}^{d}$	$\displaystyle=\{\boldsymbol{D}=(D_{t})_{t\in[T]}\,:\,D_{t}\leq D_{t+1},D_{t}\in[\widehat{D}_{t}-\Delta_{t},\widehat{D}_{t}+\Delta_{t}],|\{t\,:\,D_{t}\neq\widehat{D}_{t}\}|\leq\Gamma^{d}\},$		(9)
	$\displaystyle\mathcal{U}^{c}$	$\displaystyle=\{\boldsymbol{D}=(D_{t})_{t\in[T]}\,:\,D_{t}\leq D_{t+1},D_{t}\in[\widehat{D}_{t}-\Delta_{t},\widehat{D}_{t}+\Delta_{t}],\parallel\boldsymbol{D}-\widehat{\boldsymbol{D}}\parallel_{1}\leq\Gamma^{c}\}.$		(10)

The first representation (9) is called the discrete budgeted uncertainty, where $\Gamma^{d}\in\{0\}\cup[T]$, and the second one (10) is called the continuous budgeted uncertainty, where $\Gamma^{c}\in\mathbb{R}_{+}$. The parameters $\Gamma^{d}$ and $\Gamma^{c}$, called budgets or protection levels, control the amount of uncertainty in $\mathcal{U}^{d}$ and $\mathcal{U}^{c}$, respectively. If $\Gamma^{d}=\Gamma^{c}=0$, then all cumulative demands take their nominal values (there is only one scenario). On the other hand, for sufficiently large $\Gamma^{d}$ and $\Gamma^{c}$ the uncertainty sets $\mathcal{U}^{d}$ and $\mathcal{U}^{c}$ are the Cartesian products of $[\widehat{D}_{t}-\Delta_{t},\widehat{D}_{t}+\Delta_{t}]$, $t\in[T]$, which yields the interval uncertainty representation discussed in [28].

In order to compute a robust production plan, we adopt the minmax approach (see, e.g., [28]), in which we seek a plan minimizing the maximal total cost over all cumulative demand scenarios. This leads to the following minmax problem:

$$\textsc{MinMax}:\;OPT=\min_{\boldsymbol{x}\in\mathbb{X}}\max_{\boldsymbol{D}\in\mathcal{U}}\sum_{t\in[T]}\max\{f_{I}(X_{t},D_{t}),f_{B}(X_{t},D_{t})\},$$

(11)

where $\mathcal{U}\in\{\mathcal{U}^{d},\mathcal{U}^{c}\}$, i.e. to the one of computing an optimal production plan $\boldsymbol{x}^{*}$ of (11) along with its worst-case cost $OPT$. Indeed, $\boldsymbol{x}^{*}$ is a robust choice, because we are sure that it optimizes against all scenarios in $\mathcal{U}$ in which the amount of uncertainty allocated by an adversary to cumulative demands is upper bounded by a budget provided. Furthermore, the budget enables to control the level of robustness of a production plan computed. More specifically, an optimal production plan to the MinMax problem under $\mathcal{U}^{d}$ optimizes against all scenarios, in which at most $\Gamma^{d}$ cumulative demands take values different from their nominal ones at the same time. Moreover, by changing the value of $\Gamma^{d}$, from $0$ to $T$, one can flexibly control the level of robustness of the plan computed. An optimal production plan to the MinMax problem under $\mathcal{U}^{c}$ optimizes against all scenarios in $\mathcal{U}^{c}$ in which the total deviation of the cumulative demands from their nominal values, at the same time, is at most a bound on the total variability $\Gamma^{c}$. In this case one can also flexibly control the level of robustness of the plan computed by changing the value of $\Gamma^{c}$ from $0$ to a big number, say $\sum_{t\in[T]}\Delta_{t}$.

The MinMax problem contains the inner adversarial problem, i.e.

$$\textsc{Adv}:\;\max_{\boldsymbol{D}\in\mathcal{U}}\sum_{t\in[T]}\max\{f_{I}(X_{t},D_{t}),f_{B}(X_{t},D_{t})\}.$$

(12)

The Adv problem consists in finding a cumulative demand scenario $\boldsymbol{D}\in\mathcal{U}$ that maximizes the cost of a given production plan $\boldsymbol{x}\in\mathbb{X}$ over scenario set $\mathcal{U}$. In other words, an adversary maliciously wants to increase the cost of  $\boldsymbol{x}$.

Throughout this paper, we study the Adv and MinMax problems under two standing assumptions about cumulative demand uncertainty intervals $\widetilde{D}_{t}$, $t\in[T]$. Under the first one, the intervals are non-overlapping, i.e. $\widehat{D}_{t}+\Delta_{t}\leq\widehat{D}_{t+1}-\Delta_{t+1}$, $t\in[T-1]$. This assumption is realistic, in particular at the tactical level of planning, for instance in the master production scheduling (MPS) (see, e.g. [6]), where the lengths of periods are big enough (for example, they are equal to one month). This restriction leads to more efficient methods of solving the problems under consideration. Notice that it allows us to drop the constraints $D_{t}\leq D_{t+1}$, $t\in[T-1]$, in the definition of scenario sets $\mathcal{U}^{d}$ and $\mathcal{U}^{c}$. Under the second assumption, called the general case, we impose no restrictions on the interval bounds, i.e. they can now overlap.

In this section we consider the non-overlapping case, i.e we assume that $\widehat{D}_{t}+\Delta_{t}\leq\widehat{D}_{t+1}-\Delta_{t+1}$ for each $t\in[T-1]$. We first focus on the Adv problem, i.e the inner one of MinMax.

From Lemma 1 and the integrality of $\boldsymbol{\delta}^{*}$, it follows that an optimal solution $\boldsymbol{D}^{*}$ to (12) is such that $D_{t}^{*}\in\{\widehat{D}_{t}-\Delta_{t},\widehat{D}_{t},\widehat{D}_{t}+\Delta_{t}\}$ for every $t\in[T]$. Accordingly, we can provide an algorithm for finding an optimal solution to (13)-(15). An easy computation shows that the objective function (13) can be rewritten as follows:

$$\sum_{t\in[T]}\max\{f_{I}(X_{t},\widehat{D}_{t}),f_{B}(X_{t},\widehat{D}_{t})\}+\sum_{t\in[T]}c_{t}\delta_{t},$$

(17)

where

$$c_{t}=\max\{f_{I}(X_{t},\widehat{D}_{t}-\Delta_{t}),f_{B}(X_{t},\widehat{D}_{t}+\Delta_{t})\}-\max\{f_{I}(X_{t},\widehat{D}_{t}),f_{B}(X_{t},\widehat{D}_{t})\},\;t\in[T],$$

are fixed coefficients. The first sum in (17) is constant. Therefore, in order to solve Adv, we need to solve the following problem:

	$\displaystyle\max$	$\displaystyle\sum_{t\in[T]}c_{t}\delta_{t}$		(18)
	s.t.	$\displaystyle\sum_{t\in[T]}\delta_{t}\leq\Gamma^{d},$		(19)
		$\displaystyle 0\leq\delta_{t}\leq 1,\;\;t\in[T].$		(20)

Problem (18)-(20) can be solved in $O(T)$ time. Indeed, we first find, in $O(T)$ time (see, e.g., [27]), the $\Gamma^{d}$th largest coefficient, denoted by $c_{\sigma(\Gamma^{d})}$, such that $c_{\sigma(1)}\geq\cdots\geq c_{\sigma(\Gamma^{d})}\geq\cdots\geq c_{\sigma(T)}$, where $\sigma$ is a permutation of $[T]$. Then having $c_{\sigma(\Gamma^{d})}$ we can choose $\Gamma^{d}$ coefficients $c_{\sigma(i)}$, $i\in[\Gamma^{d}]$, and set $\delta^{*}_{\sigma(i)}=1$. Having the optimal solution $\boldsymbol{\delta}^{*}$, we can construct the corresponding scenario $\boldsymbol{D}^{*}$ as in the proof of Lemma 1 (see (16)). Hence, we get the following theorem.

We now show how to solve the MinMax problem for the non-overlapping case in polynomial time. We will first reformulate Adv as a linear programming problem with respect to $X_{t}$. Then, the linearity will be preserved by adding linear constraints for $X_{t}$. Writing the dual to (18)-(20), we obtain:

	$\displaystyle\min\;$	$\displaystyle\Gamma^{d}\alpha+\sum_{t\in[T]}\gamma_{t}$			(21)
	s.t.	$\displaystyle\alpha+\gamma_{t}\geq c_{t},$	$\displaystyle t\in[T],$		(22)
		$\displaystyle\gamma_{t}\geq 0,$	$\displaystyle t\in[T],$		(23)
		$\displaystyle\alpha\geq 0,$			(24)

where $\alpha$ and $\gamma_{t}$ are dual variables. Using (21)-(24), equality (17), and the definition of $c_{t}$, we can rewrite (13)-(15) as:

	$\displaystyle\min\;$	$\displaystyle\sum_{t\in[T]}\pi_{t}+\Gamma^{d}\alpha+\sum_{t\in[T]}\gamma_{t}$			(25)
	s.t.	$\displaystyle\pi_{t}\geq f_{I}(X_{t},\widehat{D}_{t}),$	$\displaystyle t\in[T],$		(26)
		$\displaystyle\pi_{t}\geq f_{B}(X_{t},\widehat{D}_{t}),$	$\displaystyle t\in[T],$		(27)
		$\displaystyle\alpha+\gamma_{t}\geq f_{I}(X_{t},\widehat{D}_{t}-\Delta_{t})-\pi_{t},$	$\displaystyle t\in[T],$		(28)
		$\displaystyle\alpha+\gamma_{t}\geq f_{B}(X_{t},\widehat{D}_{t}+\Delta_{t})-\pi_{t},$	$\displaystyle t\in[T],$		(29)
		$\displaystyle\alpha,\gamma_{t}\geq 0,$	$\displaystyle t\in[T],$		(30)
		$\displaystyle\pi_{t}\text{ unrestricted},$	$\displaystyle t\in[T],$		(31)

where constraints (26)-(27) specify the maximum operators in the first sum of (17) and the remaining constraints model (22)-(24) and the coefficients $c_{t}$, $t\in[T]$. We now prove that (25)-(31) solves the Adv problem.

Adding linear constraints $X_{t}=\sum_{i\in[t]}x_{i}$, $t\in[T]$, and $\boldsymbol{x}\in\mathbb{X}$ to (25)-(31) and using the fact that $f_{I}$ and $f_{B}$ are linear with respect to $X_{t}$ yield a linear program for the MinMax problem. This leads to the following theorem.

In this section we drop the assumption that the cumulative demand intervals are non-overlapping. We first consider the Adv problem. The following lemma is the key to constructing algorithms in this section, namely it shows that it is enough to consider only $O(T)$ values of the cumulative demand in each interval.

Equality (33) means that, in order to solve Adv, it is enough to examine only $O(T)$ values for every cumulative demand $D_{k}$, $k\in[T]$. This fact allows us to transform, in polynomial time, the Adv problem to a version of the following restricted longest path problem (RLP for short). We are given a layered directed acyclic graph $G=(V,A)$. Two nodes $\mathfrak{s}\in V$ and $\mathfrak{t}\in V$ are distinguished as the source node (no arc enters to $\mathfrak{s}$) and the sink node (no arc leaves $\mathfrak{t}$). Two attributes $(c_{uw},\delta_{uw})$ are associated with each arc $(u,w)\in A$, namely $c_{uw}$ is the length (cost) and $\delta_{uw}$ is the weight of $(u,w)$ and a bound $W$ on the total weights of paths. In the RLP problem we seek a path $p$ in $G$ from $\mathfrak{s}$ to $\mathfrak{t}$ whose total weight is at most $W$ ($\sum_{(u,w)\in p}\delta_{uw}\leq W$) and the length (cost) of $p$ ($\sum_{(u,w)\in p}c_{uw}$) is maximal.

Given an instance of the Adv problem, the corresponding instance of RLP is constructed as follows. We first build a layered directed acyclic graph $G=(V,A)$. The node set $V$ is partitioned into $T+2$ disjoint layers $V_{0},V_{1},\ldots,V_{T},V_{T+1}$ in which $V_{0}=\{\mathfrak{s}\}$ and $V_{T+1}=\{\mathfrak{t}\}$ contain the source and the sink node, respectively. Each node $u\in V_{k}$, $k\in[T]$, corresponds to exactly one possible value of the $k$th component of a cumulative demand vertex scenario (see (33)), denoted by $D_{u}$, $D_{u}\in\mathcal{D}_{k}$, $|V_{k}|=|\mathcal{D}_{k}|$. We also partition the arc set $A$ into $T+1$ disjoint subsets, $A=A_{1}\cup\cdots\cup A_{T}\cup A_{T+1}$. Arc $(u,w)\in A_{1}$ if $u\in V_{0}$ and $w\in V_{1}$; $(u,w)\in A_{T+1}$ if $u\in V_{T}$ and $w\in V_{T+1}$; and arc $(u,w)\in A_{k}$, $k=2,\ldots,T$, if $u\in V_{k-1}$, $w\in V_{k}$ and $D_{u}\leq D_{w}$. Set $W=\Gamma^{d}$. Finally two attributes are associated with each arc $(u,w)\in A$: $c_{uw}$ and $\delta_{uw}\in\{0,1\}$, whose values are determined as follows:

$$(c_{uw},\delta_{uw})=\begin{cases}(\max\{f_{I}(X_{k},D_{w}),f_{B}(X_{k},D_{w})\},1)&\text{if $(u,w)\in A_{k}$, $D_{w}\not=\widehat{D}_{k}$, $k\in[T]$,}\\ (\max\{f_{I}(X_{k},D_{w}),f_{B}(X_{k},D_{w})\},0)&\text{if $(u,w)\in A_{k}$, $D_{w}=\widehat{D}_{k}$, $k\in[T]$,}\\ (0,0)&\text{if $(u,w)\in A_{T+1}$}.\end{cases}$$

(36)

The second atribute $\delta_{uw}$ is equal to 1 if the value of $D_{w}$ differs from the nominal value $\widehat{D}_{k}$, and 0 otherwise, $k\in[T]$. The transformation can be done in $O(T^{3})$ time, since $A$ has $O(T^{3})$ arcs. An example is shown in Figure 2. At the nodes, other than $\mathfrak{s}$ and $\mathfrak{t}$, the possible values of the cumulative demands $D_{u}$ are shown. Observe that $\bigcup_{t\in[T]}\{\widehat{D}_{t}-\Delta_{t},\widehat{D}_{t},\widehat{D}_{t}+\Delta_{t}\}=\{1,2,3,5,6,7,9\}$. Hence $D_{1}\in\mathcal{D}_{1}=\{1,3,5\}$, $D_{2}\in\mathcal{D}_{2}=\{3,5,6,7,9\}$ and $D_{3}\in\mathcal{D}_{3}=\{5,6,7\}$. We can further assume that $D_{2}\neq 9$, due to the constraint $D_{2}\leq D_{3}$.

In general, the restricted longest (shortest) path problem is weakly NP-hard, even for series-parallel graphs (see, e.g., [18]). However, it can be solved in pseudopolynomial time $O(|A|W)$ in directed acyclic graphs, if the bound $W\in\mathbb{Z}_{+}$ and the weights $\delta_{uw}\in\mathbb{Z}_{+}$, $(u,v)\in A$, (see, e.g., [25]). Fortunately, in our case $W=\Gamma^{d}\leq T$ and $A$ has $O(T^{3})$ arcs. We are thus led to the following theorem.