Operations Research and Operations Management

A linear programme has decision variables, a linear objective to maximise or minimise, linear constraints and non-negativity. Take maximise Z = 3x + 5y subject to x ≤ 4, 2y ≤ 12, 3x + 2y ≤ 18, x, y ≥ 0. The feasible region is a convex polygon and the optimum lies at a corner point: the corners are (0, 0), (4, 0), (4, 3), (2, 6) and (0, 6), with Z = 0, 12, 27, 36 and 30, so the optimum is x = 2, y = 6, Z = 36. If the objective line is parallel to a binding edge there are multiple optima; if the region is unbounded in an improving direction the problem is unbounded; if no point satisfies all constraints it is infeasible.

The simplex method moves from one basic feasible solution (corner) to an adjacent better one. Each ≤ constraint gets a slack variable, each ≥ constraint a surplus and an artificial variable, and each = constraint an artificial variable, handled by the Big-M or two-phase method. For a maximisation the entering variable is the one with the most positive net contribution C_j − Z_j; the leaving variable follows the minimum ratio test (right-hand side divided by the positive entries of the entering column). The tableau is optimal when every C_j − Z_j ≤ 0. A zero C_j − Z_j for a non-basic variable signals alternative optima, a tie in the ratio test signals degeneracy, an entering column with no positive entry signals unboundedness, and an artificial variable left positive in the final tableau signals infeasibility.

Every LP has a dual. For a primal 'maximise c·x subject to Ax ≤ b, x ≥ 0' the dual is 'minimise b·u subject to Aᵀu ≥ c, u ≥ 0': one dual variable per primal constraint, and the roles of objective coefficients and right-hand sides swap. Weak duality: any feasible dual objective bounds the primal. Strong duality: at optimality the two objectives are equal. Complementary slackness: a constraint with slack at the optimum has a zero dual price, and a positive dual price means its constraint is binding. If the primal is unbounded the dual is infeasible. The optimal dual values are the shadow prices — the gain in Z per unit increase in a right-hand side. In the example, x ≤ 4 has slack (x = 2) so its price is 0, and the other two prices are 1.5 and 1: check 12 × 1.5 + 18 × 1 = 36. Sensitivity analysis asks how far a cost coefficient can move before the optimal basis changes (range of optimality) and how far a right-hand side can move with the shadow price still valid (range of feasibility); the reduced cost of a non-basic variable is how much its objective coefficient must improve before it enters.

The transportation model ships from m sources to n destinations at least cost. It is balanced when total supply equals total demand; otherwise a dummy row or column with zero cost absorbs the difference. An initial basic feasible solution comes from the north-west corner rule (ignores cost), the least-cost method, or Vogel's approximation (penalties = difference of the two smallest costs in each row and column; allocate where the penalty is largest), which usually starts closest to optimal. A basic solution has m + n − 1 allocations; fewer is degenerate, fixed by placing a tiny ε in an independent empty cell. Optimality is tested by MODI: solve u_i + v_j = c_ij on occupied cells (setting one u = 0), then compute Δ_ij = c_ij − u_i − v_j for empty cells; if all Δ_ij ≥ 0 the solution is optimal, otherwise the most negative cell enters around a closed loop. The assignment model is the special case of n jobs to n agents one-to-one, solved by the Hungarian method: subtract row minima, then column minima, cover all zeros with the fewest lines, and if fewer than n lines are needed subtract the smallest uncovered number from the uncovered cells and add it at intersections; repeat until n lines, then assign on independent zeros. Maximisation problems are converted by subtracting every entry from the largest; unbalanced ones get a dummy row or column.

Integer programming requires some or all variables to be integers (0-1 variables for yes/no decisions such as opening a plant). Rounding the LP optimum can be infeasible or far from optimal. Branch and bound solves the LP relaxation, branches on a fractional variable (x ≤ ⌊x⌋ and x ≥ ⌈x⌉), and prunes any branch whose relaxation bound is no better than the best integer solution found; Gomory's cutting-plane method instead adds constraints that cut off the fractional optimum but no integer point. The LP relaxation's optimum is always at least as good as the integer optimum, which is why it serves as the bound.

Markovian queues have Poisson arrivals (exponential inter-arrival times, rate λ) and exponential service (rate μ); Kendall notation (a/b/c):(d/e) names arrival and service distributions, servers, capacity and discipline. For the single-server M/M/1 queue with ρ = λ/μ < 1: P0 = 1 − ρ, P(n) = (1 − ρ)ρⁿ, P(n ≥ k) = ρᵏ, L = ρ/(1 − ρ), L_q = ρ²/(1 − ρ), W = 1/(μ − λ) and W_q = λ/[μ(μ − λ)]. Little's law L = λW (and L_q = λW_q) holds for any stable queue. With λ = 8 per hour and μ = 10 per hour: ρ = 0.8, L = 4, L_q = 3.2, W = 0.5 h = 30 min and W_q = 0.4 h = 24 min. Multi-server M/M/c queues follow the same logic with c servers sharing the load.

Discrete event simulation models a system whose state changes only at instants — arrivals, service completions, breakdowns. A simulation clock jumps from event to event taken from a time-ordered future event list; entities carry attributes; statistics accumulate as the clock advances. Random inputs are generated from uniform random numbers U in (0, 1) by the inverse-transform method: for an exponential variable with rate λ, x = −(1/λ) ln(1 − U), equivalently −(1/λ) ln U. Monte Carlo sampling from an empirical distribution maps random-number ranges to values through the cumulative probabilities. Results are estimates: several independent replications, a warm-up period discarded, and confidence intervals are needed before a conclusion is drawn.

Depreciation spreads an asset's cost, less salvage, over its life. With first cost P, salvage S and life n: straight line D = (P − S)/n every year (P = Rs 1 00 000, S = Rs 10 000, n = 9 gives Rs 10 000 a year and a book value of Rs 70 000 after three years); declining balance charges a fixed rate on the opening book value each year, so the charge falls year by year (double-declining uses twice the straight-line rate); sum-of-the-years'-digits charges (remaining life/SYD) × (P − S), SYD = n(n + 1)/2 — for n = 5, SYD = 15 and the first year's charge is (5/15) × 90 000 = Rs 30 000; and the sinking-fund method sets aside equal annual amounts that, with interest, accumulate to P − S. The accelerated methods write off more early, which matches how value is often lost and defers tax.

Break-even analysis separates fixed cost F from variable cost v per unit. At selling price p, profit = Q(p − v) − F, so the break-even quantity is Q_BE = F/(p − v), where p − v is the contribution per unit; the break-even sales value is F divided by the profit-volume ratio (p − v)/p, and the margin of safety is actual sales minus break-even sales. With F = Rs 60 000, p = Rs 50 and v = Rs 30, Q_BE = 60 000/20 = 3000 units. Activity-based costing replaces a single overhead rate (say per direct labour hour) with cost pools for the activities that actually cause overhead — machine set-ups, purchase orders, inspections, material moves — each with its own cost driver and rate: if set-ups cost Rs 50 000 for 100 set-ups, the rate is Rs 500 per set-up, and a product needing 20 set-ups carries Rs 10 000 of set-up overhead whatever its labour hours. It stops low-volume, complex products from being subsidised by high-volume simple ones.

Causal forecasting relates demand to a driver x by least-squares linear regression y = a + bx, with b = (nΣxy − ΣxΣy)/(nΣx² − (Σx)²) and a = ȳ − b x̄. For (1, 2), (2, 4), (3, 5), (4, 8): n = 4, Σx = 10, Σy = 19, Σxy = 57, Σx² = 30, so b = (228 − 190)/(120 − 100) = 1.9 and a = (19 − 19)/4 = 0, giving a forecast of 9.5 at x = 5. Time-series methods use the demand history alone. A simple moving average of the last N periods smooths noise but lags a trend; a weighted moving average puts more weight on recent data. Exponential smoothing F_t+1 = F_t + α(D_t − F_t) = αD_t + (1 − α)F_t weights past data geometrically: with F_t = 100, D_t = 120 and α = 0.2, F_t+1 = 104. A larger α responds faster but passes on more noise. Trend is handled by double (Holt's) smoothing or by fitting a line to time; seasonality by seasonal indices (the ratio of a season's average to the overall average), deseasonalising, forecasting and then re-seasonalising. Accuracy is measured by MAD, MSE and MAPE, and bias by the tracking signal = running sum of errors/MAD.

Aggregate production planning sets total output, workforce and inventory by period over a medium horizon (typically months) for a product family. Its three pure strategies are a level plan (constant output, inventory absorbs demand swings), a chase plan (output follows demand through hiring, firing or overtime) and a mixed plan; the costs traded are regular and overtime production, hiring and lay-off, inventory holding, backorders and subcontracting. The master production schedule disaggregates the plan into quantities of specific end items by week; its time fences freeze the near term, and the available-to-promise quantity tells sales what can still be committed. MRP explodes the MPS through the bill of materials: for each item, net requirement = gross requirement − on-hand inventory − scheduled receipts (+ safety stock), and planned orders are released one lead time ahead (lead-time offsetting), lot-sized lot-for-lot or otherwise. Gross 500, on hand 120 and scheduled receipts 80 give a net requirement of 300. MRP-II (manufacturing resource planning) closes the loop with capacity requirements planning, shop-floor control and the financial plan; ERP extends the same integrated database to the whole enterprise — sales, purchasing, finance, human resources and the supply chain.

Routing fixes the path and sequence of operations a part follows; scheduling fixes when each operation happens on which machine; dispatching releases the work and decides which waiting job goes next. Priority rules on a single machine: SPT (shortest processing time first) minimises mean flow time, mean waiting time and average work-in-process; EDD (earliest due date first) minimises maximum lateness; FCFS is fair but performs poorly; critical ratio = time remaining to due date/processing time remaining, smallest first; Moore's algorithm minimises the number of late jobs. For n jobs through two machines in the same order, Johnson's rule minimises the makespan: find the smallest time anywhere; if it is on machine 1 schedule that job as early as possible, if on machine 2 as late as possible; remove it and repeat. Jobs A (5, 2), B (1, 6), C (9, 7), D (3, 8), E (10, 4) give the order B-D-C-E-A and a makespan of 30.

In a push system production is scheduled in advance from forecasts (MRP is the classic push system) and work is pushed to the next stage whether or not it is ready. In a pull system consumption downstream authorises production upstream, typically through kanban cards, capping work-in-process. Lean manufacturing, from the Toyota Production System, eliminates the seven wastes (muda) — overproduction, waiting, transport, over-processing, inventory, motion and defects — through value-stream mapping, 5S, set-up reduction (SMED), one-piece flow, levelled production (heijunka), mistake-proofing (poka-yoke) and continuous improvement. Agile manufacturing stresses the ability to respond quickly and economically to unpredictable changes in demand and product variety, through reconfigurable resources, modular products and virtual partnerships; lean is strongest with stable demand, agile with volatile demand. Logistics plans and controls the flow and storage of goods and information from origin to consumption; distribution covers warehousing, transport modes, cross-docking and network design; supply chain management coordinates suppliers, manufacturers, distributors and retailers as one system. The bullwhip effect — order variability amplifying as it moves upstream, from order batching, forecast updating, price fluctuation and shortage gaming — is its classic problem, reduced by sharing demand information, smaller batches and vendor-managed inventory.

Inventory functions: cycle stock (ordering in lots), safety stock (buffer against uncertainty), anticipation stock (seasonal build-up), pipeline stock (in transit) and decoupling stock (letting stages run independently). Costs: ordering or set-up cost per order (S), holding or carrying cost per unit per year (H — capital, storage, insurance, obsolescence, often a percentage of the unit price), shortage or stock-out cost, and the purchase cost itself. Classifications: ABC by annual consumption value (a few A items take most of the value and get tight control); VED by criticality (vital, essential, desirable); FSN by movement (fast, slow, non-moving); HML by unit price; SDE by availability (scarce, difficult, easy).

With D = 10 000 units a year, S = Rs 200 and H = Rs 4, Q* = √(2 × 10 000 × 200/4) = 1000 units, ordering and holding cost Rs 2000 each, Rs 4000 together. Quantity discounts: compute the EOQ at each price (H being a percentage of price); if an EOQ is feasible in its price range keep it, otherwise move it to the nearest boundary of that range; then compare the total cost including purchase cost at every candidate and take the least. With H = 20 % of price, Rs 20 below 2000 units and Rs 19 at 2000 or more: at Rs 20 the EOQ of 1000 is valid, total Rs 2 04 000; at Rs 19 the EOQ of 1026 is below 2000, so the candidate is 2000, total 1 90 000 + 1000 + 3800 = Rs 1 94 800 — order 2000. The reorder point in a continuous system is ROP = dL + safety stock (d demand per day, L lead time in days).

Perpetual (continuous-review, Q) system: stock is tracked continuously and a fixed quantity Q, usually the EOQ, is ordered whenever stock falls to the reorder point; the order interval varies, and safety stock covers only the lead time — fewer units of safety stock, more record-keeping (the two-bin system is its manual form). Periodic (fixed-interval, P) system: stock is reviewed every T periods and an order brings it up to a target level M, so the order quantity varies; safety stock must cover the review period plus the lead time, so more is needed, but orders for many items from one supplier can be combined.

A project network shows activities on arrows (AOA, with dummy activities where needed to show dependence) or on nodes (AON). The forward pass gives each activity's earliest start ES and earliest finish EF = ES + t (ES = the largest EF of its predecessors); the backward pass from the project end gives the latest finish LF and latest start LS = LF − t (LF = the smallest LS of its successors). Total float = LS − ES = LF − EF is how far an activity can slip without delaying the project; free float is how far it can slip without delaying any successor's earliest start. The critical path is the longest path, made of zero-float activities, and its length is the project duration; there can be more than one. Example: A(3) and B(4) start; C(2) and D(5) follow A; E(3) follows B and C; F(2) follows D and E. Forward: A 0-3, B 0-4, C 3-5, D 3-8, E 5-8, F 8-10, so the project takes 10; backward, B has LS 1 and ES 0, a total float of 1, and both A-D-F and A-C-E-F are critical.

PERT treats activity times as uncertain, with optimistic a, most likely m and pessimistic b estimates from a beta distribution: expected time t_e = (a + 4m + b)/6 and standard deviation σ = (b − a)/6 — for 4, 6 and 14 days, t_e = 7 and σ = 1.67. The project time is taken as normal with mean T_e = Σt_e and variance Σσ² along the critical path (the variances add, not the standard deviations), so the probability of finishing by T is Φ(Z) with Z = (T − T_e)/σ_project. A Gantt chart draws each activity as a horizontal bar against a time scale — clear for tracking progress, but it does not show the dependencies that a network does. Time-cost trade-off (crashing): each activity has a normal time and cost and a shorter crash time at a higher crash cost; its cost slope = (crash cost − normal cost)/(normal time − crash time). To shorten the project, crash the critical activity with the least cost slope, one unit at a time, re-checking the critical path after each step; when several paths are critical, all must be shortened together. Total cost = direct cost (rising as the project is crashed) + indirect cost (falling as it shortens), and the optimum duration minimises their sum.