Budget allocation problems

An organization’s management requires information about the resources available to achieve the organization’s purpose. Resources are acquired, allocated, and manipulated under the manager’s control. The organization’s purpose is sometimes stated as its vision or goal.

The vision or goal is attained through the achievement of multiple, numerous, and often competing objectives [31] (Richard O. Mason and Swanson E. Burton, 1979).

There are a variety of ways to achieve a systematic and rational allocation of resources that will provide a competitive advantage to an organization. The methodology discussed below is quite flexible and can be adapted to a wide variety of situations and constraints. The methodology consists of the following steps [19] (Ernest H. Forman and Mary Ann Selly, 2001):

Step 1: Identify/design alternatives

Expertise in the art and science of identifying and/or designing alternatives lies in the domain of the decision makers, who have many years of study and experience with which to act on this task. The goal here is to help them attain better measurements and syntheses in order to better capitalize on their knowledge and experience.

Step 2: Identify and structure the organization’s goals and objectives

The main message is that decisions must be made on the basis of achievement of objectives with resource allocation decisions. And so the entire enterprise’s goals and objectives must be addressed. The executives understand these goals and objectives and can best make judgments about the relative importance of the main organizational objectives and, possibly, the sub-objectives.

Step 3: Prioritize the objectives and sub-objectives

The relative importance of the objectives and sub-objectives must be established in order to make a rational allocation of resources. The prioritization of the organization’s objectives during the resource allocation process will lead to another important advantage—in the top management’s quest for excellence, one will be able to respond to shifts in direction brought about by changes in the environment and competitive forces.

Step 4: Measure alternative’s contribution

Having prioritized the organization’s objectives and sub-objectives, the next step is to evaluate how much each proposed activity (or each possible level of funding for each activity) would contribute to each level’s objectives.

Step 5: Find the best combination of alternatives

After prioritizing the organization’s objectives and sub-objectives and rating the contribution of the competing activities, the lowest level objectives, etc., we have ratio scale measures of the relative contribution of each alternative combination to the organization’s overall objectives.

Otherwise, in applied mathematics, the resource allocation (RA) problem is an optimization problem with a single constraint. Given a fixed amount of the resource B (this is the constraint), DM is asked to determine its allocation to n activities in such a way that the objective function under consideration is optimized. The simple structure of the resource allocation problem discussed is generally formulated as (3.3) [35] (Toshihide Ibaraki and Naoki Katoh, 1988):

(RA)

. ,..., 2 , 1 ,

0 subject to

) ,..., , ( maximize

1 2 1

n j

x

B x

x x x f

j n

j j

n











(3.3)

That is, given one type of resource whose total amount is equal to B, DM wants to allocate it to n activities so that the objective value f(x₁,x₂,...,x_n) becomes as large as possible. The objective value may be interpreted as the profit or reward, and it is natural to maximize f. DM will sometimes considers minimization problems such as the cost, time, or loss.

In general, limited resources must be allocated among several activities, and linear programming often solves resource allocation problems. To use linear programming to allocate resources, Wayne L. Winston in 1991 [37] made three vital assumptions:

(a) The amount of a resource assigned to an activity may be any non-negative number.

(b) The benefit obtained from each activity is proportional to the amount of the resource assigned to the activity.

(c) The benefit obtained from more than one activity is the sum of the benefits obtained from the individual activities.

Wayne L. Winston (1991) [37] had considered a generalized resource allocation (GRA) problem. Suppose that the organization has B units of resource available and n activities to which the resource can be allocated. If activity j is implemented at a level xj (assume xj must be a nonnegative integer), then g_j(x_j) units of the resource are used by activity j, and a benefit v_j(x_j)is obtained. The problem of determining the allocation of resources that

maximizes total benefit that is subject to the limited resource available may be written as the following equation (3.4):

(GRA) choice. Consider an event that may or may not occur and suppose that it is part of the problem in deciding between these two possibilities. To model such a binary, variable x is used and let

 vj. Each project is either done or not done; that is, it is not possible to do a fraction of any of the projects. Also, there is a budget of B available to fund the projects. The problem of choosing a subset of the projects to maximize the sum of the values while not exceeding the budget constraint is the 0-1-knapsack (KP) problem. It is written as the following equation (3.4) (George L., Nemhauser, Laurence A. and Wolsey, 1988) [21]:

(KP-RA) problem of deciding what should be put in a knapsack given a weight limitation on how much can be carried.

According to the methodology of resource allocation mentioned above, the organization makes resource decisions in a rational way in order to achieve its vision or goals. The organization must do the following:

(a) Identify/design alternatives.

(b) Identify and structure the organization’s objectives.

(c) Prioritize the objectives and sub-objectives.

(d) Measure alternatives’ contribution.

(e) Find the best combination of alternatives.

Suppose the headquarters (HQ) of an organization has a budget $B available and the budget will be distributed to its follower units (Ui). The problem of determining the allocation of resources is how one should maximize total contribution (or value) that is subject to the limited resource available. That is, a HQ should refer to the five principles above before a decision is made to allocate resources. The operating procedure includes:

(a) The HQ (upper level) draws out concrete resource allocation rules and measures from the organization’s visions or goals.

(b) Based on these rules and measures, the lower level evaluates its sub-objectives and submits its resource requirement proposal (pij). Each proposal must contain cost (cij) and anticipated value (vij).

(c) The HQ examines proposals before finally issuing the optimal allocation of its limited resources.

Based on the procedure above, a two-stage reviewing process is used. Stage 1, the proposals are reviewed by a committee to ensure the significance of the proposal for the organization’s visions or goals. In this stage, some proposals are disqualified. Stage 2, the committee decides whether the qualified proposals are to be funded or not and how much each should receive in funds. The funded proposals must maximize contribution within a limited budget. Fig. 3.1 is a diagram showing the hierarchical structure for resource allocation.

Figure 3.1 Diagram of Hierarchical Structure for Resource Allocation Source: Study

The problem is the 0-1-knapsack problem. Basically, it may be written as an equation (3.6) in order to maximize the total value that is subjected to the limited resource available and to achieve an organization’s objective.

} 1 , 0 {

,..., 2 , 1 , s.t.

,..., 2 , 1 , max

1 1 1























ij k

i i n

j

i ij ij n

j ij ij

x

B B

k i

B x c

k i

x v

(3.6)

vij: The anticipated value from the jth project of the ith unit.

cij: Cost required in the jth project of the ith unit.

x_ij: Decision variable of the jth project of the ith unit.

$B1

HQ

Allocations Proposal

Lower Level

U₁ U₂ U_k

p

11, p12,…, p1n

p

21, p22,…, p2n

p

k1, pk2,…, pkn

$B2

……….

$Bk

……..

………

Upper Level $B

在文檔中 多個低階決策單位二階層規劃應用之研究－以預算分配為例 (頁 29-35)