A branch & bound algorithm for the 0-1 mixed integer knapsack problem with linear multiple choice constraints

Abstract

This paper presents a branch and bound (B&B) algorithm for the 0-1 mixed integer knapsack problem with linear multiple choice constraints. The formulation arose in an application to transportation management for allocating funds to highway improvements. Several model properties are developed and utilized to design a B&B solution algorithm. The algorithm solves at each node of the B&B tree a linear relaxation using an adaptation of an existing algorithm for the linear multiple choice knapsack problem. The special relationship between the parent and children subproblems is exploited by the algorithm. This results in high efficiency and low storage space requirements. The worst case complexity of the algorithm is analyzed and computational results that demonstrate its efficiency in the average case are reported.

Scope and purpose statement

Optimal resource allocation is one of the most widely studied areas in mathematical programming. We introduce a single resource allocation model that considers both discrete and continuous activities. The model is a natural extension of the knapsack problem with both binary and continuous variables. It has application in transportation management for allocating funds to highway improvements. We explore in depth the special structure of the problem and we present important theory that arises from its study. After identifying the fundamental properties of the problem, we present an efficient solution procedure that outperforms existing commercial software packages.

Introduction

In this paper, we present an algorithm for the 0-1 mixed integer knapsack problem with linear multiple choice constraints (MIMCK), defined as follows:

$$\text{MIMCK:}\text{Max}\text{∑}\text{k∈S}\text{∑}\text{i∈R}{}_{\mathrm{k}}\text{p}{}_{\mathrm{ki}}\text{x}{}_{\mathrm{ki}}\text{+}\text{∑}\text{k∈S}\text{∑}\text{j∈D}{}_{\mathrm{k}}\text{q}{}_{\mathrm{kj}}\text{y}{}_{\mathrm{kj}}\text{,}$$

$$\text{s.t.}\text{∑}\text{k∈S}\text{∑}\text{i∈R}{}_{\mathrm{k}}\text{c}{}_{\mathrm{ki}}\text{x}{}_{\mathrm{ki}}\text{+}\text{∑}\text{k∈S}\text{∑}\text{j∈D}{}_{\mathrm{k}}\text{d}{}_{\mathrm{kj}}\text{y}{}_{\mathrm{kj}}\text{⩽b,}$$

$$\text{∑}\text{i∈R}{}_{\mathrm{k}}\text{x}{}_{\mathrm{ki}}\text{⩽l}{}_{\mathrm{k}}\text{,}\text{∀k∈S,}$$

$$\text{x}{}_{\mathrm{ki}}\text{⩾0,}\text{i∈R}{}_{\mathrm{k}}\text{,}\text{∀k∈S,}$$

$$\text{y}{}_{\mathrm{kj}}\text{∈{0,1},}\text{j∈D}{}_{\mathrm{k}}\text{,}\text{∀k∈S.}$$

The traditional Knapsack Problem usually deals with either continuous or discrete decision variables. The above knapsack formulation contains both continuous and discrete decision variables. These variables are partitioned into disjoint sets whose indices are contained in set S. For each k∈S, sets R_k and D_k contain the indices of continuous and discrete variables, respectively, associated with set k. The sum of all continuous variables within each set k is restricted to at most l_k by the set of constraints (3), which are called multiple choice constraints. For consistency with the existing literature we use the terms profit coefficients for parameters p_ki and q_kj and cost coefficients for parameters c_ki and d_kj. In our case all these coefficients are positive numbers. The objective function (1) maximizes total profit, while constraint (2) limits the total budget (resource) amount used to a maximum value b. Constraints , restrict the continuous variables to nonnegative values and the discrete variables to (0,1) values, respectively.

The above formulation can be used to model the allocation of funds to highway improvements. In that context, the decision variables of the problem represent continuous and discrete highway improvements. The highways are divided into segments, k, with homogeneous characteristics. Disjoint sets R_k and D_k denote the sets of continuous and discrete improvements, respectively, considered for highway segment k. The length of highway segment k over which continuous improvement i is carried out is represented by the variable x_ki. Parameters p_ki and c_ki represent the profit and cost, respectively, incurred per unit length of application of this continuous improvement. Examples of continuous highway improvements are paving, lighting, and lane widening. The selection or not of a discrete improvement j that is considered at a specific point of highway segment k is represented by the variable y_kj. Such improvements are represented by binary variables, because there are only two options available for each improvement, implementation or not. The smoothing of a dangerous curve and the repair of an overhead bridge are examples of discrete highway improvements. Parameters q_kj and d_kj represent the profit and cost, respectively, incurred if discrete improvement j is implemented on highway segment $\text{k}\text{(y}{}_{\mathrm{kj}}\text{=1)}$. Parameter b represents the available budget for the implementation of the considered improvements. Parameter l_k denotes the length of highway segment k.

The multiple choice constraints are introduced to accommodate the interactions that arise among different continuous improvements. In some cases, the cost incurred to apply two or more distinct continuous improvements over the same part of a highway segment may be higher or lower than the sum of the costs of these improvements considered independently. For example, total cost may decrease when resources are being shared. In a similar manner, the actual profit incurred from the application of two or more continuous improvements over the same part of a highway segment may be lower or higher than the profit computed when these improvements are treated independently. In general, the combined profit is expected to be lower except in the case of synergy between improvements. The procedure described next is used to handle the case in which such interactions are present and motivates the introduction of the multiple choice constraints.

For each combination of continuous improvements within a segment, another variable is defined, representing the length of this segment over which all these individual improvements are applied. The profit and cost of this new variable is equal to the actual profit and cost incurred when all the improvements that are associated with it are implemented. Suppose that three continuous improvements are considered for highway segment k. Then, the following variables are defined: Variable x_k1, x_k2 and x_k3, represents the length of highway segment k over which single improvement 1,2 and 3 is implemented, respectively. Three combined variables are introduced representing segment lengths over which two improvements are implemented, i.e. 1 and 2 (x_k4), 1 and 3 (x_k5), and 2 and 3 (x_k6). Finally, a combined variable is introduced that represents the segment length over which all three improvements are implemented (x_k7). Thus, seven variables should be used for this segment, of which three represent individual improvements and four represent combined improvements. The sum of all seven variables should not exceed the length of highway segment k.

The above described technique for the treatment of interactions among continuous improvements will increase the number of continuous decision variables of the model. Nevertheless, in practical situations this increase remains manageable, since the number of single continuous highway improvements considered in each highway segment is small. In addition, as it is shown later, the algorithm is very efficient in handling continuous variables.

A key assumption in the above formulation is that discrete improvements are associated with different points of a highway. Therefore, no interactions between discrete improvements are present. In addition, we assume that no interactions between discrete and continuous improvements exist. This is a reasonable assumption for discrete improvements that are applied to “idealized” points of a highway. Finally, we assume that setup costs for continuous variables can be ignored.

The remainder of this paper is organized as follows: Section 2 is divided into three parts. The first part presents a brief description of an existing algorithm for the linear multiple choice knapsack problem. After slightly modifying that algorithm to accommodate the needs of our model, we utilize it in the second part of Section 3 towards the development of a B&B algorithm for MIMCK. In the third part we present a numerical example that illustrates the application of the proposed algorithm. In Section 4 we analyze the worst case complexity of the algorithm and we present computational results showing its average case efficiency. Finally, in Section 5 we summarize the conclusions obtained from this work and we point to directions for future research.