Quantitative Methods and Decision Sciences

Step 1: Annual usage value (AUV) = annual demand × unit cost

Step 2: Rank by AUV and compute cumulative percentages

Step 3: Classification

Step 4: ABC chart. Plot cumulative % of items on the X-axis and cumulative % of annual usage value on the Y-axis, joining the points (0, 0), (10, 43.17), (20, 67.15), (30, 81.53), (60, 94.05) and (100, 100). The curve rises steeply to 30% of items (class A) and then flattens through class B (to 60% of items) and class C (the last 40%).

Interpretation: Only 30% of the items (G, F, A) account for about 82% of the investment, so they need tight control, frequent review and accurate records. Class B items need moderate control, while the 40% class C items, only about 6% of value, can be managed with simple controls and bulk ordering.

Assignment problem (minimization)

Cost matrix:

[ \begin{array}{c|ccccc} & A & B & C & D & E\ \hline 1 & 20 & 16 & 22 & 33 & 12\ 2 & 14 & 18 & 30 & 14 & 24\ 3 & 11 & 12 & 14 & 8 & 13\ 4 & 31 & 22 & 16 & 17 & 31\ 5 & 26 & 18 & 16 & 21 & 18 \end{array} ]

Step 1: Row reduction

Subtract each row minimum from that row.
Row minima are (12,14,8,16,16).

[ \begin{array}{c|ccccc} & A & B & C & D & E\ \hline 1 & 8 & 4 & 10 & 21 & 0\ 2 & 0 & 4 & 16 & 0 & 10\ 3 & 3 & 4 & 6 & 0 & 5\ 4 & 15 & 6 & 0 & 1 & 15\ 5 & 10 & 2 & 0 & 5 & 2 \end{array} ]

Step 2: Column reduction

Column minima are (0,2,0,0,0). Subtract 2 from column B.

[ \begin{array}{c|ccccc} & A & B & C & D & E\ \hline 1 & 8 & 2 & 10 & 21 & 0\ 2 & 0 & 2 & 16 & 0 & 10\ 3 & 3 & 2 & 6 & 0 & 5\ 4 & 15 & 4 & 0 & 1 & 15\ 5 & 10 & 0 & 0 & 5 & 2 \end{array} ]

Now assign the zero entries so that each row and each column gets exactly one zero:

[ \begin{aligned} 1 &\to E\ 2 &\to A\ 3 &\to D\ 4 &\to C\ 5 &\to B \end{aligned} ]

Minimum cost

[ \begin{aligned} 1-E &= 12\ 2-A &= 14\ 3-D &= 8\ 4-C &= 16\ 5-B &= 18 \end{aligned} ]

[ \text{Total minimum cost} = 12+14+8+16+18 = \boxed{\text{Rs }68} ]

So the optimal assignment is:

[ \boxed{1\to E,\quad 2\to A,\quad 3\to D,\quad 4\to C,\quad 5\to B} ]

and the minimum transportation/assignment cost is Rs 68.

Each hour is one sample of 5 observations (the five rows), so there are 6 samples.

$\bar{\bar{X}} = \frac{69.0+66.6+65.0+88.6+87.8+83.8}{6} = \frac{460.8}{6} = 76.8$

$\bar{R} = \frac{35+42+36+39+38+28}{6} = \frac{218}{6} = 36.33$

Mean ($\bar{X}$) chart

$CL = 76.8$ $UCL = \bar{\bar{X}} + A_2\bar{R} = 76.8 + 0.577 \times 36.33 = 76.8 + 20.96 = 97.76$ $LCL = \bar{\bar{X}} - A_2\bar{R} = 76.8 - 20.96 = 55.84$

Range (R) chart

$CL = 36.33$ $UCL = D_4\bar{R} = 2.115 \times 36.33 = 76.84$ $LCL = D_3\bar{R} = 0 \times 36.33 = 0$

Comment: All six sample means (65.0 to 88.6) lie between 55.84 and 97.76, and all six ranges (28 to 42) lie between 0 and 76.84. No point falls outside the control limits, so the process appears to be in statistical control. The means do rise noticeably from the 4th hour, which is worth watching, but it is not a signal of lack of control on these limits.

Given:

• Monthly demand = 1500 units
• Annual demand, (D = 1500 \times 12 = 18{,}000) units
• Ordering cost, (S = \text{Rs } 240) per order
• Unit cost = Rs 25
• Annual holding cost rate = 20%

Holding cost per unit per year: [ H = 20% \times 25 = \text{Rs } 5 ]

(a) Economic lot size (EOQ): [ EOQ = \sqrt{\frac{2DS}{H}} ] [ = \sqrt{\frac{2 \times 18000 \times 240}{5}} ] [ = \sqrt{1{,}728{,}000} ] [ = 1314.53 \approx 1315 \text{ units} ]

(b) Total minimum cost: [ \text{Total minimum inventory cost} = \sqrt{2DSH} ] [ = \sqrt{2 \times 18000 \times 240 \times 5} ] [ = \sqrt{43{,}200{,}000} ] [ = \text{Rs } 6572.67 ]

So, the total minimum inventory cost (ordering + holding) is approximately Rs 6,572.67.

If the purchase cost is also included: [ \text{Purchase cost} = 18000 \times 25 = \text{Rs } 450{,}000 ] [ \text{Total annual cost} = 450{,}000 + 6572.67 = \text{Rs } 456{,}572.67 ]

Answer:

• EOQ = 1315 units
• Total minimum inventory cost = Rs 6,572.67
  (including purchase cost: Rs 456,572.67)

Product structure tree (BOM):

A
├── B (5)
│   ├── D (3)
│   └── E (2)
│       ├── F (3)
│       └── H (5)
└── C (3)
    ├── D (4)
    └── F (3)

Bill of materials file:

Requirement for 1,000 units of A:

Assuming no on-hand inventory and no scrap:

• B = $1{,}000 \times 5 = 5{,}000$
• C = $1{,}000 \times 3 = 3{,}000$
• D = $(5 \times 3) + (3 \times 4) = 15 + 12 = 27$ per A
  Total D = $27 \times 1{,}000 = 27{,}000$
• E = $5 \times 2 = 10$ per A
  Total E = $10 \times 1{,}000 = 10{,}000$
• F = $(3 \times 3) + (10 \times 3) = 9 + 30 = 39$ per A
  Total F = $39 \times 1{,}000 = 39{,}000$
• H = $10 \times 5 = 50$ per A
  Total H = $50 \times 1{,}000 = 50{,}000$

Primal (maximisation): Max $Z = 4x_1 + 3x_2 + 5x_3$

Subject to

$3x_1 + 4x_2 + 5x_3 \ge 4$ $4x_1 + x_2 + 5x_3 \le 9$ $2x_1 + 3x_2 + x_3 = 7$ $x_1, x_2, x_3 \ge 0$

Rules used: a maximisation primal gives a minimisation dual. Each constraint gives one dual variable ($y_1, y_2, y_3$), the primal right-hand sides become the dual objective coefficients, and the primal objective coefficients become the dual right-hand sides. Because $x_j \ge 0$, every dual constraint is of "$\ge$" type. The sign of each dual variable follows the primal constraint: for a maximisation primal, a "$\ge$" constraint gives $y \le 0$, a "$\le$" constraint gives $y \ge 0$, and an "=" constraint gives $y$ unrestricted.

Dual:

$\text{Min } Z^* = 4y_1 + 9y_2 + 7y_3$

Subject to

$3y_1 + 4y_2 + 2y_3 \ge 4$ $4y_1 + y_2 + 3y_3 \ge 3$ $5y_1 + 5y_2 + y_3 \ge 5$

$y_1 \le 0,\quad y_2 \ge 0,\quad y_3 \text{ unrestricted in sign}$

Value Engineering

Meaning: Value engineering (VE) is a systematic, organized, and creative approach to analyzing the function of a product, service, or system with the aim of achieving that function at the lowest possible total cost without compromising quality, reliability, or performance. It seeks to improve the "value" of a product, where value is the ratio of function to cost:

$\text{Value} = \frac{\text{Function (performance, quality, reliability)}}{\text{Cost}}$

Value can therefore be increased by improving function at the same cost, keeping function constant while reducing cost, or improving function while reducing cost (the ideal outcome).

Origin and Background: Value engineering was developed by Lawrence D. Miles at the General Electric Company (USA) during the Second World War, when shortages of materials and skilled labour forced the search for substitute materials and cheaper methods. Miles developed the "function analysis" technique, which later became the foundation of VE. It was later adopted by government agencies and industries worldwide; in construction and manufacturing it is often applied as the "40-hour VE job plan."

The VE Job Plan (Phases):

1. Information Phase: Collect all relevant data about the product — its functions, costs, quantities, specifications, and customer requirements. The key question is "What is it?" and "What does it do?"
2. Function Analysis Phase: Identify and classify the basic function (the essential purpose for which the item exists) and secondary functions (supporting or aesthetic features). Functions are expressed in two words — a verb and a noun, e.g., "transmit force," "protect surface."
3. Creative Phase: Generate alternative ways of performing each function through brainstorming, without judging feasibility at this stage. Quantity of ideas is encouraged.
4. Evaluation Phase: Screen and evaluate the alternatives on the basis of cost, feasibility, quality, and time. Techniques such as weighted scoring, life-cycle costing, and cost–benefit analysis are used.
5. Development Phase: Develop the selected alternatives into workable proposals, prepare cost estimates, drawings, and specifications, and compare them with the original design.
6. Presentation and Implementation Phase: Present recommendations to management, obtain approval, implement the changes, and monitor the results. Follow-up ensures that the savings are actually realized.

Techniques Used: Function analysis system technique (FAST) diagram, life-cycle costing, value engineering checklist, brainstorming, and cost models.

Benefits:

• Reduces unnecessary cost and eliminates wasteful features.
• Improves product quality, reliability, and performance.
• Encourages innovation and creative problem solving.
• Shortens project duration and improves productivity.
• Increases customer satisfaction and competitive advantage.
• Focuses attention on function rather than on the existing design.

Distinction from Value Analysis: Value analysis (VA) is essentially the same technique applied to an existing product to improve its value, whereas value engineering is applied at the design stage of a new product. In practice the two terms are often used interchangeably.

Conclusion: Value engineering is thus a disciplined, team-based, function-oriented technique that delivers the required function at the lowest life-cycle cost. It is widely applied in manufacturing, construction, and service industries as an effective tool for cost reduction and value improvement.

Transformation model is a model of organizational change (popularized by David Nadler and Michael Tushman) that explains how an organization converts its inputs into outputs through a series of internal transformation processes, all operating within a larger external environment. It views the organization as an open system that continuously interacts with its surroundings and adapts itself to survive and grow.

The model has four main components:

1. Inputs — the resources and pressures the organization receives from its environment: strategy, resources (money, people, technology), and environmental demands (market, legal, social, political).
2. Transformation process — the internal elements that convert inputs into outputs. Nadler and Tushman group these into four internal subsystems:
   • Work — the core tasks and activities performed.
   • People — the skills, attitudes, and motivation of employees.
   • Formal organization — structure, systems, policies, and reward mechanisms.
   • Informal organization — culture, values, and informal relationships.
3. Outputs — the results produced, such as products/services, performance, customer satisfaction, and organizational growth.
4. Feedback — information about outputs that returns to the inputs and transformation process, enabling the organization to correct and improve.

Key idea (congruence): The model holds that organizational effectiveness depends on the congruence (fit) among the internal subsystems and between them and the inputs and environment. Change is needed when this fit breaks down — for example, when a new strategy (input) no longer matches the existing structure or culture. Managers then diagnose the misalignment and redesign the relevant subsystems.

Significance: It provides a systematic diagnostic framework for planned change, helps identify where the real problem lies (task, people, structure, or culture), and treats the organization as a dynamic, open system rather than a closed one. Its main limitation is that it is more a diagnostic tool than a step-by-step guide to implementing change.

Quality refers to the degree to which a product or service meets or exceeds the expectations of customers by consistently satisfying their stated and implied needs. In my words, quality means "fitness for use" — doing the right thing, in the right way, at the first time and every time, so that the customer remains satisfied and the organisation achieves its objectives with minimum waste.

Total Quality Management (TQM) is a management philosophy and organisation-wide approach that seeks continuous improvement in the quality of all processes, products, services and people in order to meet and exceed customer expectations. It treats quality as the responsibility of every member of the organisation, from top management to the lowest worker, and integrates quality into every stage of the production and service delivery system rather than checking it only at the end.

The main features of TQM are:

1. Customer focus — the customer (both internal and external) is the centre of all quality efforts; customer satisfaction is the ultimate measure of quality.
2. Top management commitment and leadership — quality improvement must begin and be sustained from the top, with clear vision, policy and resource support.
3. Employee involvement and empowerment — every employee participates in quality improvement through teamwork, quality circles, suggestion schemes and training.
4. Continuous improvement (Kaizen) — quality is a never-ending journey; processes are constantly reviewed and improved, not merely maintained.
5. Process-centred approach — quality is built into the process rather than inspected into the product; variation in processes is reduced through statistical tools.
6. Total participation — quality is everyone's responsibility, cutting across all departments and functions.
7. Fact-based decision making — decisions rely on data, measurement and statistical quality control tools rather than guesswork.
8. Supplier partnership — long-term, mutually beneficial relationships with suppliers ensure quality inputs.
9. Strategic and systematic approach — quality is made a part of the organisation's strategy, planning and culture.

In short, TQM means managing the whole organisation so that quality becomes everybody's job and a way of life, leading to lower costs, higher productivity, satisfied customers and long-term competitive advantage.