Operations Management

Total supply = 20 + 30 + 50 = 100 and total demand = 40 + 10 + 50 = 100, so the problem is balanced.

Step 1: Initial solution by Vogel's Approximation Method

1. Column W₂ has the highest penalty (5 − 3 = 2). Allocate 10 to F₂W₂ (cost 3), so W₂ is met and F₂ has 20 left.
2. Row penalties now: F₁ = 9 − 6 = 3, F₃ = 8 − 5 = 3, F₂ = 5 − 4 = 1. Taking F₁, allocate 20 to F₁W₃ (cost 6), so F₁ is exhausted.
3. Row F₃ has the highest penalty (3). Allocate 40 to F₃W₁ (cost 5), so W₁ is met.
4. Remaining: F₂W₃ = 20 and F₃W₃ = 10.

Cost = 20×6 + 10×3 + 20×5 + 40×5 + 10×8 = 120 + 30 + 100 + 200 + 80 = Rs 530.

Step 2: Optimality test (MODI)

There are 5 occupied cells (= m + n − 1). Put $u_3 = 0$ and use $u_i + v_j = c_{ij}$ for occupied cells:

$v_1 = 5,\ v_3 = 8,\ u_1 = -2,\ u_2 = -3,\ v_2 = 6$.

Opportunity cost of each empty cell $= c_{ij} - (u_i + v_j)$:

No value is negative, so the solution is optimal. (The zero for F₃W₂ shows an alternative optimum of the same cost.)

Optimal plan: F₁ → W₃ 20; F₂ → W₂ 10 and W₃ 20; F₃ → W₁ 40 and W₃ 10.

Minimum transportation cost = Rs 530.

Step 1: Row reduction (subtract each row minimum: 7, 8, 7, 7, 7)

Step 2: Column reduction (only column D has a non-zero minimum, 1)

Step 3: Cover all zeros. The minimum number of lines is 4 (columns A and E, rows M₃ and M₄), which is less than 5, so the assignment is not yet possible. The smallest uncovered element is 1. Subtract it from uncovered elements and add it to elements at line intersections.

Step 4: Assign at zeros. Now 5 lines are needed, so an optimal assignment exists. Row M₁ has a single zero (E), which fixes M₁→E. Then M₅→B, M₂→A, M₃→D and M₄→C.

Minimum total cost = 7 + 8 + 8 + 7 + 8 = Rs 38.

(a) Types of Productivity and Importance of Operations Strategy

1. Meaning of Productivity

Productivity is the ratio of output produced to the inputs used in producing it. It measures how efficiently an organisation converts resources (labour, capital, material, energy, machinery) into goods and services.

$\text{Productivity} = \frac{\text{Output}}{\text{Input}}$

Higher productivity means more output from the same inputs, or the same output from fewer inputs — which lowers cost per unit and strengthens competitiveness. Productivity is a relative measure: it must be compared over time (this year vs last year) or against other firms/industry benchmarks to be meaningful.

2. Types of Productivity

Productivity is classified according to how many inputs are included in the denominator.

(i) Partial Productivity (Single-Factor Productivity)

Output is related to one input only. It is the most widely used type because it is simple and easy to compute.

$\text{Labour productivity} = \frac{\text{Output}}{\text{Labour hours (or labour cost)}}$

$\text{Machine productivity} = \frac{\text{Output}}{\text{Machine hours}}$

$\text{Material productivity} = \frac{\text{Output}}{\text{Material used (units or Rs)}}$

$\text{Capital productivity} = \frac{\text{Output}}{\text{Capital employed}}, \qquad \text{Energy productivity} = \frac{\text{Output}}{\text{Energy used}}$

Limitation: it can be misleading, because a rise in labour productivity may simply be the result of installing better machines, not better labour effort. It also ignores the substitution of one input for another.

(ii) Multifactor Productivity (Total Factor Productivity)

Output is related to a group of inputs, usually labour and capital, or labour, capital and energy.

$\text{Multifactor productivity} = \frac{\text{Output}}{\text{Labour} + \text{Capital} + \text{Energy (in Rs)}}$

It is also called total factor productivity when labour and capital are the only inputs taken. It is more meaningful than partial productivity because it captures the combined effect of the major factors and reflects the efficiency of their joint use.

(iii) Total Productivity

Output is related to all inputs used — labour, capital, material, energy, and other expenses such as overheads.

$\text{Total productivity} = \frac{\text{Total output (in Rs)}}{\text{Labour} + \text{Capital} + \text{Material} + \text{Energy} + \text{Other inputs (in Rs)}}$

It gives the most complete picture of a firm's overall performance, but it is difficult to compute because all inputs must be converted into a common unit (usually money) and price changes over time distort the result.

Illustration. A firm produces 1,000 units in a month. Labour cost = Rs 25,000; capital cost = Rs 15,000; material = Rs 5,000; energy and other = Rs 5,000.

• Labour productivity (if 250 labour hours) = 1,000 ÷ 250 = 4 units per labour hour
• Multifactor productivity = 1,000 ÷ (25,000 + 15,000) = 0.025 units per Rs
• Total productivity = 1,000 ÷ 50,000 = 0.02 units per Rs

3. Importance of Operations Strategy

Operations strategy is a long-range plan that states how the operations function will use its resources and capabilities to support the overall business strategy and gain competitive advantage. It translates corporate goals into decisions about products, processes, capacity, facilities, technology, quality, workforce and supply chain. Its importance can be discussed as follows:

1. Links operations with corporate strategy. It ensures that what the factory or service unit does is consistent with the firm's mission, market position and long-term objectives, so operations does not work in isolation.

2. Creates competitive advantage. By choosing the right competitive priorities — cost, quality, delivery speed/reliability, flexibility (and, for services, courtesy and customisation) — operations becomes a source of order-winning capability rather than a mere support activity. Toyota's cost advantage and Dell's speed of delivery are classic examples.

3. Guides major resource decisions. It provides the framework for decisions on plant location and size, capacity expansion, choice of technology, automation, inventory policy, supplier selection and workforce planning — decisions that are costly and difficult to reverse.

4. Ensures consistency and coordination. A clear strategy aligns operations with marketing (what is promised to customers), finance (funds available) and human resources (skills required), reducing conflicts between functions.

5. Helps make trade-offs explicit. Since no firm can excel on every priority at once, operations strategy decides where to excel and where merely to be adequate (qualifiers), preventing the firm from trying to do everything and doing nothing well.

6. Improves productivity and reduces cost. A well-designed strategy eliminates waste, balances capacity with demand, standardises processes and raises output per unit of input, directly improving profitability.

7. Provides a basis for performance measurement. Targets such as unit cost, defect rate, lead time, on-time delivery and capacity utilisation flow from the strategy, making control and corrective action possible.

8. Builds long-term competitiveness and survival. In a globalised, competitive market (including Nepal's post-LPG liberalisation environment), firms that align operations capabilities with market requirements grow and survive, while those without a strategy lose market share.

Conclusion

Productivity — partial, multifactor and total — is the basic measure of how well resources are used. Operations strategy gives direction to those resources so that productivity gains are converted into competitive advantage. The two are closely linked: strategy decides what capabilities to build, and productivity measurement shows how well those capabilities are performing.

(b) Operational Issues in Product Life Cycle and Types of Layout

1. Product Life Cycle (PLC) — Concept

The product life cycle is the sequence of stages through which a product passes from its introduction to its withdrawal from the market. The four stages are:

1. Introduction — the product is launched; sales are low, costs are high, and profit is usually negative.
2. Growth — sales rise rapidly, competitors enter, and profit begins to increase.
3. Maturity — sales peak and stabilise, competition is intense, and profit starts to decline.
4. Decline — sales fall as substitutes appear or tastes change; the product is eventually withdrawn.

2. Operational Issues in the Product Life Cycle

Operations management must respond differently at each stage, because volume, variety, cost, quality, and delivery requirements change over time.

A. Introduction Stage

• Product design and flexibility: The design is still unstable; operations must keep the process flexible to accommodate frequent design changes.
• Low volume, high unit cost: Small batch production raises unit cost; the firm must decide whether to use a job or batch process.
• Capacity planning: Demand is uncertain, so capacity must be built cautiously to avoid idle plant.
• Quality control: Early defects damage reputation; strict quality checks and pilot runs are essential.
• Supplier development: Reliable suppliers must be developed for new materials and components.

B. Growth Stage

• Capacity expansion: Demand rises fast; operations must add capacity in time to avoid stock-outs and lost sales.
• Process standardization: The process shifts from flexible to more standardized to gain economies of scale.
• Productivity and cost reduction: Learning-curve effects should be exploited to lower unit cost.
• Quality assurance: Volume increases the risk of defects; statistical quality control becomes necessary.
• Distribution and inventory: Wider distribution networks and higher inventory levels are required to meet rising demand.

C. Maturity Stage

• Cost competitiveness: Price competition is severe; operations must focus on cost efficiency, waste reduction, and value engineering.
• Productivity improvement: Automation, work study, and method improvement are used to cut costs.
• Quality and reliability: Products must be defect-free and consistent to retain market share.
• Product proliferation: Many models and variants increase variety, complicating scheduling and inventory.
• Delivery speed and service: Competition shifts to delivery reliability and customer service.

D. Decline Stage

• Capacity reduction: Excess capacity must be trimmed to reduce fixed costs.
• Product rationalization: Unprofitable models and variants are dropped; the product line is pruned.
• Inventory control: Stocks must be liquidated carefully to avoid obsolescence losses.
• Cost minimization: Operations focus on minimum-cost production until withdrawal.
• Reuse and disposal: Decisions on spare parts, service support, and disposal of obsolete stock arise.

E. Cross-Cutting Operational Issues

• Trade-off between volume and variety: As the product moves from introduction to maturity, volume rises and variety falls, so the process must shift from job/batch to mass/continuous.
• Technology and automation decisions: When to automate is a key issue, since early automation reduces flexibility.
• Supply chain coordination: Supplier and distributor relationships must be realigned at each stage.
• Forecasting: Demand forecasting becomes more accurate in maturity but difficult in introduction and decline.
• Productivity measurement: Operations must continuously measure and improve productivity across all stages.

3. Types of Layout

Layout is the arrangement of machines, equipment, personnel, and facilities within a production area so that the flow of materials and people is efficient. The main types are:

A. Process (Functional) Layout

Similar machines and operations are grouped together by function — for example, all lathes in one section, all drilling machines in another. The product moves from one department to another according to its sequence of operations.

Example: A job shop or a general hospital — the X-ray department, laboratory, and pharmacy are separate functional units, and patients move between them.

B. Product (Line) Layout

Machines and workstations are arranged in the exact sequence of operations required to make the product. The product moves in a straight line from one station to the next.

Example: An automobile assembly line or a bottling plant, where the car or bottle moves continuously along the line.

C. Fixed Position Layout

The product remains stationary at one place, and men, machines, and materials are brought to it. This is used when the product is too heavy or bulky to move.

Example: Construction of a bridge, shipbuilding, or aircraft manufacturing.

D. Group (Cellular) Layout

Machines are grouped into small cells, each cell producing a family of similar parts. It combines the flexibility of process layout with the efficiency of product layout.

Example: A cell making all types of gears — lathe, milling, and drilling machines grouped in one cell.

E. Combination (Hybrid) Layout

In practice, most plants use a combination of the above types — for example, a product layout for assembly and a process layout for fabrication.

4. Comparison of Major Layouts

Conclusion

Operations management must adapt its process, capacity, quality, inventory, and supply chain decisions to each stage of the product life cycle. The choice of layout — process, product, fixed position, cellular, or hybrid — depends on the volume and variety of production, and must be aligned with the stage of the product in its life cycle to achieve cost efficiency and customer satisfaction.

(a) Mean ((\bar X)) and Range ((R)) Control Charts

For each sample of size (n=5), compute:

[ \bar X_j=\frac{\sum X}{n}, \qquad R_j=X_{\max}-X_{\min} ]

Computations

Grand Mean and Average Range

[ \bar{\bar X}=\frac{413.6+405.2+403.4+405.4+420.4}{5} =\frac{2048.0}{5}=409.6 ]

[ \bar R=\frac{16+15+19+16+40}{5} =\frac{106}{5}=21.2 ]

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1. Mean Control Chart ((\bar X)-chart)

[ \text{CL}=\bar{\bar X}=409.6 ]

[ \text{UCL}=\bar{\bar X}+A_2\bar R =409.6+0.577(21.2) ]

[ =409.6+12.2324=421.8324 \approx 421.83 ]

[ \text{LCL}=\bar{\bar X}-A_2\bar R =409.6-12.2324=397.3676 \approx 397.37 ]

So,

[ \boxed{\text{UCL}=421.83,\quad \text{CL}=409.6,\quad \text{LCL}=397.37} ]

Plotting Points for (\bar X)-chart

All sample means lie between (397.37) and (421.83).
Therefore, the mean chart shows the process is in control.

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2. Range Control Chart ((R)-chart)

[ \text{CL}=\bar R=21.2 ]

[ \text{UCL}=D_4\bar R =2.115(21.2) ]

[ =44.838 \approx 44.84 ]

[ \text{LCL}=D_3\bar R =0(21.2)=0 ]

So,

[ \boxed{\text{UCL}=44.84,\quad \text{CL}=21.2,\quad \text{LCL}=0} ]

Plotting Points for (R)-chart

All ranges lie between (0) and (44.84).
Therefore, the range chart also shows the process is in control.

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Comment on Results

Since all points on both the mean chart and the range chart fall within their respective control limits, the manufacturing process is statistically in control. There is no evidence of assignable causes of variation.

However, the fifth sample has:

• Mean (\bar X_5=420.4), close to the upper control limit (421.83)
• Range (R_5=40), close to the upper control limit (44.84)

This suggests a possible upward shift in the process mean and increased variability in the last sample. Although it is not outside the control limits, the process should be carefully monitored. If the next sample shows a similar pattern or exceeds the control limits, corrective action should be taken.