Cheat sheet · Operations

One A4 page per half, two columns, no chrome. Ctrl+P (Windows) / ⌘P (Mac).

Operations · Pre mid-sem

Definitions to write verbatim

Transformation process — inputs (land, labour, capital, information) through conversion (alter, inspect, store, transport; plan, organise, coordinate, control) to outputs (goods, services), with value added in the middle.
Feedback — measurements taken at various points in the process. Control — comparing feedback against previously established standards to decide on corrective action.
Goods — physical items: raw materials, parts, subassemblies, final products. Services — activities providing some combination of time, location, form or psychological value.
Operations management — the business function that plans, organises, coordinates and controls the resources needed to produce a company's goods and services.
POM criteria of performance — efficiency, effectiveness, customer satisfaction. Efficiency internal, doing things right; effectiveness external, doing the right things.
Productivity — a measure of the effective use of resources, usually the ratio of output to input.
Plant layout (Moore) — a plan of an optimum arrangement of facilities including personnel, operating equipment, storage space, material handling equipment and all other supporting services, with the design of the best structure to contain them.
Forecast — a statement about the future value of a variable of interest, such as demand. Two aspects: expected level and accuracy.

Productivity formulas

Productivity = Output / Input
Labour productivity = Quantity produced / Labour hours = Output at standard price / Wages paid
Multifactor = Output / (Labour + Material + Overhead)
TFP = Production at standard price ------------------------------------------- Labour + Materials + Overhead + k(capital)
Materials productivity = Production output (Rs) / (Raw materials + Packaging + Supplies)
Growth % = (Current - Previous) / Previous x 100
Process yield = Good output / Raw material input
Trap: overhead given as a multiple applies to labour cost only. Convert workers x hours into labour hours. Use usable output.

Location formulas

Composite score = SUM (weight x score)
Total cost = FC + v Q Total profit = Q(R - v) - FC
Crossover: FC1 + v1 Q = FC2 + v2 Q Q = (FC1 - FC2) / (v2 - v1)
x-bar = SUM(xi wi) / SUM(wi) y-bar = SUM(yi wi) / SUM(wi)
Euclidean d = sqrt((x2-x1)^2 + (y2-y1)^2) Rectilinear d = |x2-x1| + |y2-y1|
Load-distance = SUM (load x distance)
Decimal degrees = degrees + minutes / 60 d = 69 sqrt(dLong^2 + dLat^2) miles d = 111 sqrt(dLong^2 + dLat^2) km
Delivered cost = production cost + transport cost Network total = SUM fixed(open plants) + SUM delivered x units
Four methods: factor rating (many non-money factors), cost-profit-volume (costs differ, volume uncertain), centre of gravity (one facility, known demand points), transportation model (many plants, capacities).

Layout and line balancing

Seven principles: integration; minimum distance; cubic space utilisation; flow (no backtracking); maximum flexibility; safety, security and satisfaction; minimum handling.
Four layout types: product (repetitive, high volume); process (intermittent, job shop or batch); fixed position (product stays still); combination (hospitals, supermarkets, shipyards). Cellular sits inside process layout.
Takt time = Total work time available / Units required Workers required = Total operation time / Takt time
Cycle time = Operating time per day / Desired output rate Output rate = Operating time / Cycle time N(min) = SUM task times / Cycle time (round UP)
Balance delay % = (N x CT - SUM t) / (N x CT) x 100 Efficiency % = SUM t / (N(actual) x CT) x 100
Min cycle time = longest single task Max output = Available time / longest task
RPW(task) = own time + SUM times of ALL following tasks
Layout cost = SUM(i) SUM(j) Xij Cij
CRAFT: Armour and Buffa; an improvement heuristic; six inputs (initial layout, flow data, cost per unit distance, number of departments, fixed departments, areas); swaps only on common border or equal area; not optimal; depends on the starting layout.

Forecasting formulas

Naive: F(t) = A(t-1) MA = SUM(Di) / n WMA = SUM(Wi Di), SUM(Wi) = 1.00
F(t) = F(t-1) + alpha (D(t-1) - F(t-1)) = alpha D(t-1) + (1 - alpha) F(t-1) alpha = 2 / (n + 1)
Error = Actual - Forecast MAD = SUM |A-F| / n MSE = SUM (A-F)^2 / n MAPE = SUM (|A-F| / A) / n x 100
F(t) = a + b t b = (n SUM(ty) - SUM(t) SUM(y)) / (n SUM(t^2) - (SUM t)^2) a = (SUM(y) - b SUM(t)) / n
Seasonal relative = season average / overall average Deseasonalise: value / relative Forecast: trend x relative
Four features of forecasts: the past causal system persists; forecasts are not perfect; group beats item; accuracy falls with the horizon.
Four time-series components: trend (years, population and technology); seasonal (within a year, weather and custom); cyclical (multiple years, business cycle); random (short, no repetition).
Six steps: purpose, horizon, data, technique, forecast, monitor the errors. Choosing: cost, accuracy, data availability, software availability, time, horizon.

Numbers from the class problems, for checking

Productivity: 5,000 units at $30, labour 500 h at $25, materials $5,000, overhead 2x labour gives 3.5294. Wrapping paper 2,000 rolls on Rs 160 + 50 + 320 gives 3.7 rolls per Rs. 25 to 23 units/hour is -8 %.
Factor rating: photo processing 70.6 against 82.7; three-location problem 87.02 / 82.62 / 80.90; unnormalised weights 1,518 / 1,842 / 1,660.
Cost-volume: Athens/Brussels crossover 1,000, Brussels/Lisbon 2,500. Four plants: B up to 4,999, C to 11,111, A above; D dominated. Rustproofing indifference 230 cars. Jamshedpur/Patna/Ranchi at 8,000 units: Ranchi at Rs 32,20,000.
Centre of gravity: ready-mix (238, 444); seven plants (341.0, 350.4). Load-distance sites 125,063 / 99,789 / 77,555, choose site 3. Road version 10,800 / 14,950 / 12,850.
Line balancing: mirrors takt 48 s, workers 2.92. Wing component CT 12 min, N(min) 5.42 to 6, efficiency 90.3 %. Green House CT 72 s, 2 stations, 80.6 %. Dress line CT 120 s, bottleneck D, max output 36/hour, 5 stations, 67.5 %. Water pumps RPW: 6 stations, 67.8 %.
Process layout: three departments 7,600. Six departments $570 to $480. CRAFT five departments 205, best swap 4 and 5 gives 163.
Forecasting: orders series 3-month 110, 5-month 91, weighted 103.4. Error table MAD 2.6, MSE 7.8, MAPE 2.25 %. Five-week trend y = 143.5 + 6.3t. Cell phones y = 699.4 + 7.51t.

Traps that cost marks

Overhead multipliers apply to labour cost, not the running total.
Divide by labour hours (workers x hours), not by workers or by elapsed hours.
Do not rescale factor-rating weights that fail to sum to 1.00; the ranking cannot change.
Check for a dominated location before computing any crossover.
Centre of gravity uses the same denominator for both axes, and lands on a search area, not a site.
Cycle time can never be below the longest single task. Efficiency uses N(actual).
N(min) is a lower bound; precedence often forces an extra station.
Convert degrees and minutes to decimals before squaring.
MAPE divides by the actual, never the forecast. Keep the sign in the error column so bias is visible.
SUM(t squared) is not (SUM t) squared. Compute b before a.

Formula cards

Value added

Value added = Value of outputs - Cost of inputs

Any time you are asked what a transformation process is worth. Take every input the process consumed, including labour.

Value of outputs: Units produced x selling price, or the amount billed
Cost of inputs: Materials, labour, energy and facility charges consumed

Value added per unit

Value added per unit = Value added / Units produced

To compare two days, two shifts or two plants fairly, because it removes the effect of how busy the period was.

Value added: Output value minus input cost, for the period
Units produced: Good output for the same period

Productivity

Productivity = Output / Input

The parent of every measure in this chapter. Decide what counts as output (usable output only) and which inputs go in the denominator before you divide.

Output: Usable goods or services produced, in units or at standard price
Input: Labour, material, energy, capital, in units or in money

Labour productivity

Labour productivity = Quantity produced / Labour hours = Output at standard price / Wages paid

When the question names workers, hours or wages. Convert workers and hours into labour hours before dividing.

Labour hours: Number of workers x hours each worked
Wages paid: Total wage bill for the same period

Multifactor productivity

MFP = Output / (Labour + Material + Overhead)

When more than one input is priced. Overhead stated as a multiple applies to labour cost alone unless the question says otherwise.

Output: Units, or units x standard price if the answer is to be unitless
Overhead: Often given as a multiple of labour cost

Total factor productivity

TFP = Production at standard price --------------------------------------------- Labour + Materials + Overhead + k(capital invested)

When capital invested and a charge rate k are given. It is multifactor productivity with capital added.

k: Charge rate applied to capital, e.g. 0.10
Capital invested: Value of the capital employed in the period

Materials productivity

Materials productivity = Production output (Rs) --------------------------------------------- Raw materials + Packaging materials + Supplies

When the question isolates the material side, typically in a process industry.

Production output: Value of output in rupees
Supplies: Consumables that are not raw material or packaging

Productivity growth

Growth % = (Current productivity - Previous productivity) / Previous productivity x 100

Comparing two periods. The denominator is always the earlier period. Keep the sign.

Current: This period's productivity, on the same measure
Previous: Last period's productivity, on the same measure

Process yield

Process yield = Good output / Raw material input

Where there is scrap, and as the service-sector stand-in for productivity (cars rented / cars available).

Good output: Usable output only, scrap excluded
Raw material input: Quantity of material fed in, or capacity offered in a service

Landed cost per unit

Landed cost = Factory cost + Freight + Duty - Per-unit incentives

Screening countries or regions before any of the four evaluation methods. Check what base the duty percentage applies to.

Factory cost: Ex-works cost of making one unit
Duty: Rate x the stated base, usually factory cost or CIF value
Incentives: Only those expressed per unit; lump sums are applied to the annual total

Capacity gap

Years of headroom = (Capacity - Current demand) / Annual demand growth Decide by = Year the gap opens - Build lead time

Deciding whether a location decision is due now. Use a year-by-year table when growth is a percentage rather than a fixed increment.

Capacity: Units the existing facility can produce per year
Annual demand growth: Units per year, or apply the rate to a running base
Build lead time: Years from decision to first output

Factor rating composite score

Composite score = SUM (weight x score) over all factors

When several factors matter and most cannot be priced. Weights usually sum to 1.00; if they do not, use them as given.

weight: Relative importance of the factor, usually summing to 1.00
score: This location's rating on that factor, usually 0 to 100

Total cost at a location

Total cost = FC + v x Q

Comparing locations on cost at a known volume. Add any other cost the choice changes, such as system transport cost.

FC: Fixed cost for the period
v: Variable cost per unit
Q: Quantity or volume of output for the period

Crossover volume

FC1 + v1 x Q = FC2 + v2 x Q Q = (FC1 - FC2) / (v2 - v1)

To find the volume where the cheaper location changes, and so to state the superiority ranges.

FC1, FC2: Fixed costs of the two locations
v1, v2: Variable costs per unit of the two locations

Total profit at a location

Total profit = Q(R - v) - FC

When a selling price is given. (R - v) is the contribution per unit; set two sites equal to get the indifference volume.

R: Revenue or dealer price per unit
v: Variable cost per unit at that site
FC: Fixed cost for the period at that site

Centre of gravity

x-bar = SUM(xi x wi) / SUM(wi) y-bar = SUM(yi x wi) / SUM(wi)

Placing a single facility to serve known demand points with known volumes. Assumes cost is directly proportional to distance and volume shipped.

xi, yi: Grid coordinates of demand point i
wi: Weight or volume shipped to demand point i

Euclidean distance

d = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )

Straight-line travel, and the default in load-distance problems that give coordinates rather than road distances.

(x1, y1): Coordinates of the first point
(x2, y2): Coordinates of the second point

Rectilinear distance

d = |x2 - x1| + |y2 - y1|

Movement along aisles, streets or a shop floor. The measure the class CRAFT spreadsheet uses between department centroids.

|x2 - x1|: Absolute horizontal separation
|y2 - y1|: Absolute vertical separation

Load-distance score

Score(site i) = SUM over j of ( load at j x distance from i to j )

Ranking a shortlist of real candidate sites. Lowest score wins; a cost per unit distance scales all scores equally.

load at j: Demand or tonnage carried to demand point j
distance: Road, Euclidean or rectilinear, as the question states

Great-circle approximation

d = 69 x sqrt( dLong^2 + dLat^2 ) miles d = 111 x sqrt( dLong^2 + dLat^2 ) km Decimal degrees = degrees + minutes / 60

When locations are given as latitude and longitude. Convert to decimal degrees first.

dLat: Difference of latitudes, in decimal degrees
dLong: Difference of longitudes, in decimal degrees

Delivered cost

Delivered cost (i to j) = production cost at plant i + transport cost from i to j

The first thing to build in any network-design problem. Allocate on this, never on production cost alone.

production cost at i: Cost of making one unit at plant i
transport cost i to j: Cost of moving one unit from plant i to market j

Network total cost

Total cost = SUM(fixed cost of open plants) + SUM(delivered cost i-to-j x units shipped i-to-j)

Costing any candidate network. Fixed cost is counted once per open plant, whatever volume it carries.

fixed cost: Facility cost incurred only if the plant runs
units shipped: The decision variable in each cell of the tableau

Penalty per unit

Penalty = second-cheapest delivered cost - cheapest delivered cost

When a plant is over capacity. Move markets off it in ascending order of penalty until the row fits.

cheapest: Lowest delivered cost for that market, from the overloaded plant
second-cheapest: Next lowest delivered cost, from a plant with spare capacity

Plant closure test

Net saving = fixed cost avoided - SUM(units moved x penalty per unit) Feasible only if remaining capacity >= total demand

Whenever total capacity comfortably exceeds total demand. Run the feasibility check first.

fixed cost avoided: The closed plant's facility cost
penalty per unit: Extra delivered cost at the receiving plant

Floor and cubic space

Floor area = length x width Cubic space = floor area x clear height

Sizing departments and testing the cubic space utilisation principle. Add 15 to 25 % to floor area for aisles and services in a real layout.

clear height: Usable height under the roof structure
floor area: Departmental area, before aisle allowance

Material handling cost of a layout

Annual handling cost = volume x distance per unit x cost per unit-distance

Comparing two layouts. Work with the distance saved rather than costing both layouts in full.

volume: Units moved per year
distance per unit: Total travel of one unit through the layout
cost per unit-distance: Handling charge per unit per metre

Takt time

Takt time = Total work time available / Units required

Staffing a work cell to demand. It is the drumbeat the customer sets.

Total work time available: Scheduled production time in the period
Units required: Demand for the same period

Workers required

Workers required = Total operation time required / Takt time

After takt time, to staff the cell. A fractional answer is rounded up when staffing.

Total operation time: Sum of all operation times for one unit, from the work balance chart
Takt time: Available time per unit required

Cycle time and output rate

Cycle time = Operating time per day / Desired output rate Output rate = Operating time per day / Cycle time

Pacing a line. Check that no single task exceeds the cycle time before going further.

Operating time per day: Minutes or seconds the line runs
Desired output rate: Units required per day

Theoretical minimum stations

N(min) = Sum of task times / Cycle time (round up)

Before assigning tasks, to know the floor. Precedence often forces one or two more.

Sum of task times: Total work content of one unit
Cycle time: Maximum time per station per unit

Balance delay and efficiency

Balance delay % = (N(actual) x CT - Sum of task times) / (N(actual) x CT) x 100 Efficiency % = 100 % - Balance delay = Sum of task times / (N(actual) x CT) x 100

After the assignment, to judge it. Always use the actual number of stations.

N(actual): Stations the assignment really used
CT: Cycle time

Ranked Positional Weight

PW(task) = own time + SUM of the times of ALL following tasks Rank descending by PW, then assign the highest-ranked available task that fits

A line-balancing heuristic that favours tasks with the most work behind them. Section B works two problems on it; Section A only names it.

own time: The task's own processing time
following tasks: Every task downstream of it, not only its immediate successors

Maximum output of a line

Minimum cycle time = longest single task time Maximum output = Operating time / Longest task time

When asked how fast a line could possibly run, or whether a target rate is feasible at all.

Longest task time: The bottleneck task; it cannot be split across stations
Operating time: Available time in the period

Process-layout cost

Minimise SUM(i) SUM(j) Xij x Cij

Scoring any process layout. Cij is normally the handling rate times the centroid-to-centroid distance.

Xij: Number of loads moved from department i to department j
Cij: Cost to move one load between departments i and j
n: Total number of work centres or departments

Rectilinear centroid distance

d(i,j) = |xi - xj| + |yi - yj|

Building the distance matrix in a layout problem, exactly as the class CRAFT spreadsheet does with ABS formulas.

(xi, yi): Centroid of department i in the layout
(xj, yj): Centroid of department j

Adjacency cost rule

Cost of a flow = loads x rate, rate = 1 unit if departments are adjacent, 2 units if they are more than one apart

The simplified rule the class uses for the six-department factory, when only adjacency is given rather than distances.

loads: Number of moves between the two departments per period
rate: 1 for adjacent, 2 for non-adjacent, in the units the question gives

Naive forecast

F(t) = A(t-1)

As a zero-cost benchmark, and when the series has no trend or seasonality worth modelling. Every other method must beat it to justify itself.

F(t): Forecast for period t
A(t-1): Actual demand in the most recent period

Forecast error

Error = Actual - Forecast

Step 6 of the process, every period. Keep the sign in the table; drop it only when computing absolute measures.

Actual: Demand that actually occurred in the period
Forecast: What was predicted for that period, before it happened

Simple moving average

MA = SUM(Di) / n

A stable series with no trend or seasonality. Larger n means more smoothing and less responsiveness.

n: Number of periods in the moving average
Di: Actual demand in period i, the n most recent periods

Weighted moving average

WMA = SUM(Wi x Di), with SUM(Wi) = 1.00

When recent periods deserve more weight than older ones. Check the weights sum to 1.00 and that the largest is on the most recent period.

Wi: Weight for period i, between 0 and 1
Di: Demand in period i

Exponential smoothing

F(t) = F(t-1) + alpha ( D(t-1) - F(t-1) ) = alpha x D(t-1) + (1 - alpha) x F(t-1)

When only the last forecast and last actual are available, and the level moves but does not trend. State the starting forecast you assume.

alpha: Smoothing constant between 0 and 1; the fraction of the last error fed back
F(t-1): Previous period's forecast
D(t-1): Previous period's actual demand

Alpha and moving-average length

alpha = 2 / (n + 1)

To justify a chosen alpha, or to convert a familiar moving-average length into a smoothing constant.

n: Equivalent number of periods in a moving average
alpha: Smoothing constant with roughly the same behaviour

Mean absolute deviation (MAD)

MAD = SUM |Actual - Forecast| / n

Comparing methods when all misses cost roughly in proportion to their size. Expressed in the units of the series.

n: Number of periods for which a forecast existed
|Actual - Forecast|: Absolute error in one period

Mean squared error (MSE)

MSE = SUM (Actual - Forecast)^2 / n

When one large miss is much more damaging than several small ones. Units are squared, so use it for comparison, not description.

(Actual - Forecast)^2: Squared error in one period
n: Number of error terms

Mean absolute percentage error (MAPE)

MAPE = [ SUM ( |Actual - Forecast| / Actual ) / n ] x 100

Comparing forecast quality across products or series of different sizes. Each ratio is taken on the actual, never the forecast.

Actual: The denominator of each period's ratio
n: Number of error terms

Seasonal relative

Relative = average demand in that season / overall average demand Deseasonalise: data point / relative Seasonal forecast: trend estimate x relative

When a series repeats within the year. Deseasonalise before fitting a trend, then re-season the extrapolated forecast.

season: The repeating slot: a month, a quarter, a day of the week
relative: Above 1 for a peak season, below 1 for a trough; the set should average about 1

Linear trend equation

F(t) = a + b t

When a plot shows a trend. Unlike averaging methods it extrapolates, so it can forecast several periods ahead.

a: Value of F(t) at t = 0, the fitted intercept
b: Slope: change in the forecast per period
t: Number of periods from t = 0

Least-squares estimates

b = ( n x SUM(t y) - SUM(t) x SUM(y) ) / ( n x SUM(t^2) - (SUM(t))^2 ) a = ( SUM(y) - b x SUM(t) ) / n

Fitting the trend line. Set out columns t, y, ty and t squared, total them, compute b first.

n: Number of periods in the fit
y: Value of the time series in that period
SUM(t^2): Sum of squared periods, not the square of the sum

Operations · Post mid-sem

Definitions to write verbatim

Inventory — a stock or store of goods. Independent-demand items — items ready to be sold or used.
Six functions: meet anticipated demand; smooth production requirements; protect against stockouts; take advantage of order cycles; hedge against price increases; take quantity discounts.
Reorder point — the quantity on hand at which the item is reordered. Safety stock — stock held in excess of expected demand because of variable demand or lead time. Service level — the probability demand will not exceed supply during lead time.
MRP — a computer-based information system that translates master schedule requirements for end items into time-phased requirements for subassemblies, components and raw materials, working backwards using lead times.
Low-level coding — restructuring the bill of materials so that multiple occurrences of a component all coincide with the lowest level at which it occurs.
Johnson's rule — a technique for minimising makespan for a group of jobs on two machines or work centres; also minimises total idle time.
Project — a unique, one-time operation designed to accomplish a specific set of objectives in a limited time frame.
Supply chain management — a set of approaches used to effectively integrate suppliers, manufacturers, warehouses and stores for the 7 Rs, in order to minimise system-wide cost while satisfying the service level.

Inventory formulas

H = carrying charge % x unit price TC = (Q/2)H + (D/Q)S EOQ = sqrt(2 D S / H) TC at EOQ = sqrt(2 D S H), halves equal
N = D/Q Cycle = Q/D (x working days for days) ROP = d x LT, d = D / working days
I_max = (Qp / p)(p - u) Qp = sqrt(2 D S / H) x sqrt(p / (p - u)) TC = (I_max / 2)H + (D/Q)S Run time = Qp/p, Cycle time = Qp/u
TC(discount) = (Q/2)H + (D/Q)A + P D
ROP = expected LT demand + z sigma(dLT) Demand variable: sigma(dLT) = sigma(d) sqrt(LT) Lead time variable: sigma(dLT) = d sigma(LT) Both: sqrt(LT sigma(d)^2 + d^2 sigma(LT)^2)
z values: 0 at 50 %, 1.28 at 90 %, 1.65 at 95 %, 1.96 at 97.5 %, 2.33 at 99 %, 3.09 at 99.9 %.
Annual value = annual demand x unit cost Cumulative % = cumulative value / total value x 100
ABC: A is 10 to 20 % of items and 60 to 70 % of value; C is 50 to 60 % of items and 10 to 15 % of value. Related: XYZ (stock value), VED (criticality), FSN (movement), SDE (supply).

MRP and aggregate planning

Six MRP rows: gross requirements; scheduled receipts; projected on hand; net requirements; planned-order receipts; planned-order release.
Net = Gross - Projected on hand - Scheduled receipts Release period = Receipt period - lead time Child gross = Parent RELEASE x quantity per parent
Lot-size: lots = ROUND UP(net / lot size) Surplus = lots x lot size - net, carried forward
Demand options: pricing, promotion, back orders, new demand. Supply options: hire and lay off, overtime and slack time, part-time workers, inventories, subcontracting.
Strategies: level (steady output, inventory absorbs); chase (match demand period by period); hybrid.
Beginning(t) = Ending(t-1) Ending = Beginning + Output - Forecast Average = (Beginning + Ending)/2 <- cost THIS row Negative balance = BACKLOG, not negative inventory
Monthly output = output per day x production days Level daily rate = total demand / total production days
Six steps: demand per period; capacities per period; policies; unit costs; alternative plans and costs; select, otherwise return to step 5. Trial and error does not guarantee an optimum.

Scheduling and project management

Five priority rules: FCFS, SPT, EDD, CR, Rush. SPT minimises average flow time and average jobs in the shop; EDD minimises maximum lateness.
Flow time = running total of processing times Tardiness = max(0, flow - due date) Avg flow = SUM flow / n Avg tardiness = SUM late / n Avg jobs = SUM flow / SUM processing CR = (due - now) / processing time
Johnson: shortest time anywhere; at centre 1 schedule first, at centre 2 schedule last; remove and repeat inwards. Conditions: times known and constant; independent of sequence; same two-step route; complete at centre 1 before centre 2.
Start at centre 2 = max(finish at centre 1, centre 2 free) Idle at centre 2 = initial wait + internal gaps
Forward: ES = largest EF of predecessors, EF = ES + t Backward: LF = smallest LS of successors, LS = LF - t Slack = LS - ES = LF - EF (zero = critical)
t-e = (t-o + 4 t-m + t-p) / 6 sigma^2 = ((t-p - t-o) / 6)^2 Path sigma = sqrt(SUM path variances) z = (target - expected) / path sigma
PERT vs CPM: uncertain vs known times; event- vs activity-oriented; probabilistic vs deterministic; time target vs time-cost trade-off.
Crashing: only zero-slack activities; recompute the critical path after every crash, because it moves.

Logistics and supply chain

7 Rs: right product, customer, quantity, condition, place, time, cost.
1PL own cargo; 2PL subcontracted carrier doing what the client instructs; 3PL specialist providing end-to-end management of distribution, storage, transport, fulfilment, plus IT as a service.
SLA contains: purpose; services rendered; responsibilities of both parties; KPIs tracked; methodology for tracking; non-fulfilment consequences.
RATER: Reliability, Assurance, Tangibles, Empathy, Responsiveness Perceived < Minimum -> Negative Min <= Perceived <= Desired -> Satisfactory Perceived > Desired -> Positive (may be over-resourced)
Four cycles: customer order, replenishment (pull) | manufacturing, procurement (push). Three flows: goods forward, finance back, information both ways.
Push long-term forecast, demand certain, high inventory. Pull actual demand, demand uncertain, high lead time. Grocery and books push; computers and furniture pull. Strategies: make-to-stock, assemble-to-order, make-to-order, design-to-order.
Value density = rupee value / weight In-transit = demand per period x transit periods Cycle stock = shipment size / 2 Cost = inventory x carrying/unit/period + demand x freight
Modes: rail low value density and uncertain lead time; road dearer, door to door; water cheapest and slowest; air high value density, expensive, reliable. Structures: direct, via warehouse, milk run.

Numbers from the class problems, for checking

EOQ: D 9,000, S 40, H 0.18 gives EOQ 2,000, N 4.5, TC Rs 360, ROP 240, cycle 66.7 days; +20 % costs Rs 6, -40 % costs Rs 48. Tyres: EOQ 300, 32 orders, 9-day cycle, TC Rs 48,000.
EPQ: paper cones Qp 94,868 (95,000), run every 9.5 days. Toy wheels Qp 2,400, I_max 1,800, TC $1,800, cycle 12 days, run time 3 days.
Discount: cleanser EOQ 70 at $18 costs $14,968; at 80 $14,154; at 100 $13,354, so order 100. Pistons: feasible EOQ 447 at Rs 60; 1,000 at Rs 50 costs Rs 9,04,650 and wins.
ABC: ten items total Rs 75,910; A = item 8 (10 % of items, 52.7 %); B = 3, 6, 1 (30 %, 40.8 %); C = six items (60 %, 6.5 %). Boutique: three items = 15 % of items, 71 % of cost.
MRP: A needs B(2), C(3); B released weeks 2 and 6, C weeks 3 and 7 with 200 on hand. Three-level E: R and S released week 2 for 500 units of E in week 8; cumulative lead time 6 weeks.
Aggregate: skateboards level plan $4,700 (3,600 + 600 + 500). Practice 1 Rs 78,600; Practice 2 Rs 95,500. Jackets: constant workforce Rs 58,850, outsourcing Rs 52,576.
Sequencing: six jobs, total 41 days. FCFS 20.00 / 9.00 / 2.93; SPT 18.00 / 6.67 / 2.63; EDD 18.33 / 6.33 / 2.68; CR 22.17 / 9.67 / 3.24. Johnson: D-E-C-F-A-B, makespan 51, idle 4.
Projects: four paths 18 / 20 / 14, critical 20 weeks. Seven activities: critical path 1-2-5-6, 20 weeks, slacks 0/2/6/6/0/2/0. Practice network: critical 1-3-4-5, 17 weeks, slacks 7/7/8/0/0/0.
Printers: air 150 units of inventory, sea 600. High-end air Rs 47,538 beats sea Rs 55,154; low-end sea Rs 32,077 beats air Rs 41,769; standard is within Rs 1,000, so unpriced factors decide.

Traps that cost marks

Convert a percentage carrying charge into H in rupees per unit per year before touching Q.
Use the working year the question gives, never 365, and daily demand for the reorder point.
EPQ uses I_max / 2, not Q/2. Run time uses p; cycle time uses u.
In a discount problem the P x D term is compulsory, and each cheaper price is tested at its lowest qualifying quantity.
Only the demand-variable safety-stock formula has a square root. Round safety stock up.
ABC ranks on annual value, never on unit cost or volume alone.
MRP: net off inventory at every level, and explode from the parent's release row.
Aggregate plans charge inventory on average inventory; a negative balance is a backlog.
Re-sort the due dates with the jobs when you change sequencing rule. Early jobs have zero tardiness.
PERT: variances add, standard deviations do not. Slack must agree computed both ways.
Transport comparison: count in-transit and cycle stock, and convert the annual carrying rate to the demand period.

Formula cards

Annual holding cost

Annual holding cost = (Q / 2) x H

Whenever stock runs down evenly from Q to zero. Q/2 is the average inventory over the cycle.

Q: Order quantity in units
H: Holding cost per unit per year; convert from a percentage of price first

Annual ordering cost

Annual ordering cost = (D / Q) x S

D/Q is the number of orders placed a year. Ordering cost never depends on how big each order is.

D: Annual demand in units
S: Ordering cost per order, or setup cost per run

Holding cost from a percentage

H = carrying charge % x unit price

The first line of almost every inventory numerical. Do it before touching Q.

carrying charge %: Annual carrying cost as a fraction of inventory value, e.g. 0.09
unit price: Cost of one unit

Economic order quantity

EOQ = sqrt( 2 D S / H )

Buying an independent-demand item under the six basic assumptions. Convert a percentage carrying charge into H first.

D: Annual demand in units
S: Ordering cost per order
H: Holding cost per unit per year

Total annual inventory cost

TC = (Q / 2) H + (D / Q) S At the EOQ: TC = sqrt( 2 D S H ) and the two halves are equal

Costing any order quantity, optimal or not. Use the equal-halves property to check an EOQ.

Q: Order quantity being costed
(Q/2)H: Annual holding cost
(D/Q)S: Annual ordering cost

Orders per year and cycle length

N = D / Q Cycle length = Q / D (in years) = Q / D x working days (in days) = working days / N

Immediately after computing EOQ. The exam almost always asks for at least one of these.

N: Optimum number of orders a year; may be fractional
working days: The year length the question gives, not 365

Reorder level under certainty

ROP = d x LT, where d = D / working days

When lead time and demand are both certain. The stock at ROP covers exactly one lead time.

d: Demand per day
LT: Lead time in days

Maximum inventory (EPQ)

I_max = (Qp / p) x (p - u)

The first thing to compute once Qp is known. Read it as run length times net rate of build-up.

Qp: Run size (economic production quantity)
p: Production or delivery rate, per day
u: Usage rate, per day, in the same unit as p

Economic production quantity

Qp = sqrt( 2 D S / H ) x sqrt( p / (p - u) )

When the item is produced in batches while being used continuously, and p exceeds u.

D: Annual demand; derive it from the daily usage rate if needed
S: Setup cost per production run
H: Carrying cost per unit per year

Total cost (EPQ)

TC = (I_max / 2) H + (D / Q) S

Costing a production batch. Note I_max/2 rather than Q/2; the two halves are equal at the optimum.

I_max / 2: Average inventory over the cycle
(D / Q) S: Annual setup cost: runs a year times setup cost

Run time, cycle time, runs a year

Run time = Qp / p Cycle time = Qp / u Runs/year = D / Qp Idle time = cycle time - run time

Whenever a duration is asked for. Run time uses p; cycle time uses u.

p: Production rate per day
u: Usage rate per day

Total cost with quantity discounts

TC = (Q / 2) H + (D / Q) A + P D

Whenever the unit price depends on the order quantity. All three terms, every time.

H: Holding cost per unit per year; recompute it if it is a fraction of price
A: Ordering cost per order, written S in the basic EOQ slides
P: Unit price applying to the quantity being tested
D: Annual demand in units

Holding cost as a fraction of price

H = carrying fraction x P

When the question gives a carrying-cost fraction rather than a rupee amount. H then changes at every price break, and so does the EOQ.

carrying fraction: Annual carrying cost as a fraction of unit value, e.g. 0.15
P: The price being tested

Discount decision procedure

1. EOQ = sqrt(2 D A / H) 2. Feasible? Is EOQ inside the range for that price? 3. TC at the feasible EOQ, all three terms 4. TC at the lowest qualifying quantity of each cheaper price above 5. Choose the lowest TC

Every quantity discount problem. Never test quantities below the feasible EOQ.

Feasible: The EOQ lies within the quantity range that earns the price used to compute it
lowest qualifying quantity: The smallest order that still earns that lower price

Reorder point under certainty

ROP = d x LT

Demand and lead time both constant and known. No safety stock is required.

d: Demand rate per period (day, week)
LT: Lead time in the same periods as d

Reorder point under uncertainty

ROP = expected demand during lead time + z x sigma(dLT) Safety stock = z x sigma(dLT)

Whenever demand or lead time varies. z comes from the target service level.

z: Number of standard deviations for the service level: 1.65 at 95 %, 2.33 at 99 %
sigma(dLT): Standard deviation of demand during lead time

Sigma of lead-time demand: demand variable

sigma(dLT) = sigma(d) x sqrt(LT) ROP = d-bar x LT + z x sigma(d) x sqrt(LT)

Demand varies, lead time is constant. The square root comes from variances adding over independent days.

sigma(d): Standard deviation of demand per period
LT: Constant lead time, in the same periods

Sigma of lead-time demand: lead time variable

sigma(dLT) = d x sigma(LT) ROP = d x LT-bar + z x d x sigma(LT)

Lead time varies, demand is constant. No square root: one late period costs a full period's demand.

d: Constant demand per period
sigma(LT): Standard deviation of lead time, in the same periods as d

Service level

Service level = 100 % - stockout risk

To translate a management risk tolerance into a z value, per replenishment cycle rather than per year.

stockout risk: Probability demand exceeds supply during one lead time

Annual value

Annual value = annual demand (units) x unit cost

The first step of every ABC analysis. Compute it for every item before sorting.

annual demand: Units consumed or sold in a year
unit cost: Cost of one unit

Cumulative percentage of value

Cumulative % = (cumulative annual value / total annual value) x 100

After sorting descending. The cut points between A, B and C are read off this column.

cumulative annual value: Running total down the sorted list
total annual value: Sum across all items

Typical class shares

A: 10-20 % of items, 60-70 % of value B: the middle C: 50-60 % of items, 10-15 % of value

As a guide when cutting the sorted list. Follow the bend in the Pareto curve and state the percentages you used.

% of items: Share of the number of line items
% of value: Share of total annual value

Net requirement

Net requirement = Gross requirement - Projected on hand - Scheduled receipts

Row 4 of every MRP record. Do it at every level, not only at the top.

Gross requirement: Demand for the part in the period, ignoring stock
Projected on hand: Expected inventory at the beginning of the period
Scheduled receipts: Orders already placed, arriving at the start of the period

The explosion

Gross requirement of child = Planned-order RELEASE of parent x quantity per parent Planned-order release period = Planned-order receipt period - lead time

Moving from one level of the product structure tree to the next. Use the release row of the parent.

quantity per parent: The bracketed number on the tree, per one parent unit
lead time: That item's own lead time, in periods

Lot-size ordering

Lots ordered = ROUND UP ( net requirement / lot size ) Receipt = lots x lot size Surplus = receipt - net requirement, carried forward

When a fixed batch is imposed by the supplier or the machine. Always round up and always carry the surplus forward.

lot size: The fixed order multiple
surplus: Added to projected on hand for the next period

Cumulative lead time

Cumulative lead time = sum of the lead times of the sequential phases

To set the minimum horizon of the master schedule, and to know how early the deepest component must be released.

sequential phases: From ordering raw material through to completing final assembly

Level output rate

Level rate per period = total demand over the horizon / number of periods Level rate per day = total demand / total production days

Sizing a level plan. Check it against total demand before costing anything, or the plan may be infeasible.

total demand: Sum of the period forecasts across the whole horizon
production days: Working days in each period; they usually differ

Inventory and backlog balance

Output - forecast, cumulated = ending inventory If the cumulative balance is negative, it is a BACKLOG, not negative inventory

Building the inventory rows of an aggregate plan. Cumulate; never recompute each period from scratch.

ending inventory: Stock at the end of the period, never below zero
backlog: Unmet demand carried into the next period, charged per unit per period

Inventory rows

Beginning(t) = Ending(t-1) Ending(t) = Beginning(t) + Output(t) - Forecast(t) Average(t) = ( Beginning(t) + Ending(t) ) / 2

Every aggregate plan. If Ending would be negative, set it to zero and put the shortfall in the backlog row.

Output: Regular time plus part time plus overtime plus subcontract
Average: The row the carrying charge is applied to

Total cost of an aggregate plan

Total = Regular units x regular rate + Overtime units x overtime rate + Part-time units x part-time rate + Subcontract units x subcontract rate + Hire/layoff cost + Total AVERAGE inventory x carrying rate + Backlog units x backorder rate

Costing any plan. Give a per-period total row as well, so an error can be localised.

carrying rate: Cost per unit per period, applied to average inventory
backorder rate: Cost per unit per period of unmet demand; usually much higher than the carrying rate

Production-days output

Monthly output = output per day x production days in the month Level daily rate = total demand / total production days

When the problem gives working days per month. A constant daily rate gives an uneven monthly output.

output per day: Units the workforce produces in one working day
production days: Working days in that particular month

Flow time, lateness and tardiness

Flow time (single centre) = running total of processing times Lateness = Flow time - Due date (can be negative) Tardiness = max(0, Lateness)

Every single-work-centre sequencing table. Set negative lateness to zero before averaging.

Flow time: Arrival to completion, including waiting
Due date: Days from now, as the question states it

The three averages

Average flow time = SUM flow times / n Average tardiness = SUM days late / n Average number of jobs = SUM flow times / SUM processing times

Comparing two priority rules. The third is a work-in-process measure and its denominator never changes with the sequence.

n: Number of jobs
SUM processing times: Also the makespan at a single work centre

Critical ratio

CR = (Due date - Current time) / Processing time

Smallest CR goes next. Recompute for every remaining job after each completion; below 1 means the job cannot make its date.

Current time: The clock now, which advances as jobs finish
Processing time: Work remaining on that job

Johnson's rule

1. List job times at both work centres 2. Find the shortest time anywhere - at centre 1: schedule that job FIRST - at centre 2: schedule that job LAST 3. Remove the job 4. Repeat, working towards the centre

Two work centres, same two-step route for every job. Minimises makespan and total idle time.

shortest time: The single smallest number remaining in the whole table
ties: Broken arbitrarily

Two-centre Gantt timing

Start at centre 2 = max( finish at centre 1, centre 2 becomes free ) Makespan = finish of the last job at centre 2 Idle at centre 2 = initial wait + sum of internal gaps

Reading makespan and idle time off a two-row chart after applying Johnson's rule.

initial wait: Time before the first job reaches centre 2
internal gaps: Periods when centre 2 is free but no job has arrived

Forward pass

ES = largest EF among the activities feeding this activity's start event EF = ES + duration Project duration = largest EF in the network

First pass through any network. An event occurs only when every activity feeding it is complete, so take the largest.

ES: Earliest start
EF: Earliest finish

Backward pass and slack

LF = smallest LS among this activity's successors LS = LF - duration Slack = LS - ES = LF - EF

Second pass, starting from the project duration. Compute slack both ways as a check.

LS: Latest start without delaying the project
LF: Latest finish without delaying the project
Slack: Zero on the critical path

Path slack

Slack of a path = critical path length - that path's length

Small networks where every path can be listed. The longest path is the critical path and the project duration.

critical path length: Longest total duration from start to finish

PERT expected time and variance

t-e = ( t-o + 4 t-m + t-p ) / 6 sigma^2 = ( ( t-p - t-o ) / 6 )^2

When activity durations are uncertain and three estimates are given. The most likely time does not appear in the variance.

t-o: Optimistic time, under optimal conditions
t-m: Most likely time, the most probable duration
t-p: Pessimistic time, under worst conditions

Probability of on-time completion

Path variance = SUM of variances of activities on the critical path Path sigma = sqrt( path variance ) z = ( target date - expected project length ) / path sigma

After a PERT network is timed. Add variances, never standard deviations, and look z up in a normal table.

target date: The completion date being tested
expected project length: Sum of expected times along the critical path

Value density

Value density = rupee value of the product / its weight

Choosing a transport mode. High value density favours air; low value density favours rail or water.

rupee value: Unit cost or price of the item
weight: Shipping weight of the item

Inventory created by a transport mode

In-transit (pipeline) stock = demand per period x transit periods Cycle stock = shipment size / 2 Total inventory = in-transit + cycle

Comparing two modes. A slower mode with bigger lots creates inventory on both counts.

transit periods: Transport plus customs time, in the same period as demand
shipment size: Lot size the mode imposes

Cost of a transport mode per period

Cost = total inventory x carrying cost per unit per period + demand per period x freight per unit Carrying per unit per week = annual % x unit cost / 52

The mode comparison itself. Convert the annual carrying rate to the period the demand is quoted in.

annual %: Inventory carrying cost as a fraction of item cost, e.g. 0.20
freight per unit: Quoted rate for that mode

RATER classification rule

Perceived < Minimum -> Negative Minimum <= Perceived <= Desired -> Satisfactory Perceived > Desired -> Positive

Scoring a service against the three baselines. Positive can also mean over-resourced.

Minimum: Lowest adequate service level
Desired: The highest level, what the customer hopes to receive
Perceived: The customer's perception of current service

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