Formulating linear programming (LP) models is a fundamental quantitative technique for accountancy professionals engaged in resource allocation, cost minimization, or profit maximization within Kenyan businesses and organizations. LP models help accountants and financial analysts translate complex business constraints and objectives into mathematical expressions that can be solved efficiently. This chapter focuses on understanding the core terminology used in linear programming and how to develop accurate models that reflect real-world financial and operational scenarios.

3.1 Meaning of terms

In this section, we define and explore key terms essential to formulating linear programming models. Mastery of these terms allows accountancy professionals to construct precise models that optimize financial decisions such as budgeting, production scheduling, and portfolio management.

3.1.1 Decision Variables

Decision variables represent the unknown quantities that an LP model seeks to determine in order to optimize the objective function. These variables correspond to controllable activities or resources in the accounting context, such as units of products to produce, amounts of funds to allocate, or hours of labour to assign.

$$ \text{Decision variables: } x_1, x_2, ..., x_n $$

Worked Examples

Example 1: A retail business wants to decide how many units of product A and product B to stock. Let \(x_1\) be units of product A and \(x_2\) units of product B.

Given: No numerical calculation yet; define variables.

Example 2: A SACCO plans to allocate funds between two loan products: agricultural loans (\(x_1\)) and business loans (\(x_2\)).

Given: Variables represent loan amounts in Ksh million.

Example 3: A county government office wants to determine the hours staff spend on auditing (\(x_1\)) and financial reporting (\(x_2\)).

Given: Variables represent staff hours per week.

3.1.2 Objective Function

The objective function is a mathematical expression representing the goal of the LP model, typically maximization of profit or minimization of cost. It is formulated as a linear combination of decision variables with coefficients that quantify contribution per unit.

$$ \text{Maximize or Minimize } Z = c_1 x_1 + c_2 x_2 + \dots + c_n x_n $$

where \(c_i\) are coefficients representing profit, cost, or any other measurable metric.

Worked Examples

Example 1: A hotel wants to maximize revenue from two room types. Revenue per room: Ksh 5,000 for deluxe (\(x_1\)) and Ksh 3,000 for standard (\(x_2\)).

$$ Z = 5000 x_1 + 3000 x_2 $$

Example 2: A retail business aims to minimize total cost of stocking two products. Cost per unit: Ksh 200 for product A (\(x_1\)) and Ksh 150 for product B (\(x_2\)).

$$ Z = 200 x_1 + 150 x_2 $$

Example 3: A bank wants to maximize net interest income from two loan types. Net income per million Ksh: 0.12 for personal loans (\(x_1\)) and 0.10 for commercial loans (\(x_2\)).

$$ Z = 0.12 x_1 + 0.10 x_2 $$

3.1.3 Constraints

Constraints are linear inequalities or equalities representing the limitations or requirements within which the decision variables must operate. These constraints reflect resource availability, policy limits, or regulatory requirements.

$$ a_{11} x_1 + a_{12} x_2 + \dots + a_{1n} x_n \leq b_1 $$

$$ a_{21} x_1 + a_{22} x_2 + \dots + a_{2n} x_n \geq b_2 $$

$$ a_{31} x_1 + a_{32} x_2 + \dots + a_{3n} x_n = b_3 $$

Worked Examples

Example 1: A cooperative has a budget limit of Ksh 1,000,000 for two loan products \(x_1\) and \(x_2\) with unit costs of Ksh 500,000 and Ksh 400,000 respectively.

$$ 500000 x_1 + 400000 x_2 \leq 1000000 $$

Example 2: A university’s accounting department requires at least 40 hours of auditing work (\(x_1\)) and 60 hours of financial reporting (\(x_2\)) weekly.

$$ x_1 \geq 40 $$

$$ x_2 \geq 60 $$

Example 3: A retail business must stock at least 100 units of product A (\(x_1\)) and 150 units of product B (\(x_2\)).

$$ x_1 \geq 100 $$

$$ x_2 \geq 150 $$

3.1.4 Non-negativity Restrictions

Non-negativity restrictions require all decision variables to be zero or positive, reflecting the reality that negative quantities of physical or financial resources are not possible in accounting contexts.

$$ x_i \geq 0, \quad \forall i = 1, 2, ..., n $$

Worked Examples

Example 1: A SACCO cannot allocate negative funds to loans.

$$ x_1 \geq 0, \quad x_2 \geq 0 $$

Example 2: A retail business cannot stock a negative number of products.

$$ x_1 \geq 0, \quad x_2 \geq 0 $$

Example 3: A county government office cannot assign negative hours to financial activities.

$$ x_1 \geq 0, \quad x_2 \geq 0 $$

Chapter Summary

This chapter introduced the key terms used in linear programming, including decision variables, objective function, and constraints, to establish a foundational understanding. It then outlined the assumptions underlying linear programming models, such as linearity, divisibility, certainty, and non-negativity, which ensure the applicability of the method. The process of formulating a linear programming model was detailed, emphasizing the translation of real-world problems into mathematical expressions involving an objective function and constraints. Methods for solving linear programming problems were explored, focusing on graphical and algebraic techniques to identify optimal solutions. Finally, the chapter highlighted various practical applications of linear programming in resource allocation, production scheduling, and cost minimization within diverse sectors. The combination of theory and application equips learners with the skills to model and solve optimization problems effectively.