Operations Research and Production Management

EOQ model

The economic order quantity (EOQ) is the lot size that minimises the sum of annual ordering cost and annual inventory holding cost for one item with known, constant demand. Let D be annual demand, S the ordering cost per order and H the holding cost per unit per year. The total annual cost is C(Q)=\frac{DS}{Q}+\frac{HQ}{2}, because the average stock is Q/2. The ordering term falls as Q rises, while the holding term rises linearly. Setting dC/dQ=0 gives the Wilson formula Q^*=\sqrt{\frac{2DS}{H}}. At this point the ordering cost and the holding cost are equal, each being C^*/2, so C^*=\sqrt{2DSH}. Because the cost curve is asymmetric in Q, a wrong lot size is judged by the ratio C'/C^*=\frac{1}{2}\left(k+\frac{1}{k}\right), where Q'=kQ^*. A lot size twice the optimum therefore raises the minimum cost by 25 percent, not by 100 percent. The EOQ is proportional to \sqrt{DS/H}, so the ratio of the EOQs of two items can be found directly from their demands, ordering costs and holding costs. The time between orders is T=Q^*/D=\sqrt{\frac{2S}{HD}}. If S and D are both increased by the same percentage, the effects cancel and T is unchanged. When applying the formula, D and H must refer to the same time unit, and the result is a number of units. When the unit price depends on the order quantity, the holding cost is I=iP, where i is the carrying rate as a fraction of the price band's price P, and the total cost is TC=PD+\frac{DS}{Q}+\frac{IQ}{2}. For each price band, compute the EOQ with that band's I. If the EOQ lies inside its band it is feasible; otherwise the band's cost is evaluated at the end of the band nearest to the EOQ. Compare the total costs of all candidates and choose the smallest, since a lower price usually outweighs a smaller lot. The basic EOQ model assumes constant deterministic demand, zero lead time (instantaneous replenishment), a constant unit price, a constant holding cost rate and no shortages. Stockouts are not permitted because the cost function contains only ordering and holding costs. Shortage or backorder models are extensions of the basic model.

Critical path method

The critical path method (CPM) plans a project as a network of activities with known, deterministic durations and precedence relations. A forward pass computes the earliest start ES and earliest finish EF=ES+t of each activity. An activity with several predecessors can start only after the largest of their earliest finish times. The completion time is therefore the longest chain through the network, and the forward pass gives it at the final node. A backward pass then computes the latest finish LF and latest start LS from the project end, and the total float is TF=LS-ES=LF-EF. Activities with zero total float form the critical path, which is the longest path from start to finish. The path with the most activities is not necessarily critical; the path with the greatest total duration is. When one activity duration is unknown, write the longest path through that activity as a function of the unknown and set it equal to the given project duration. Each branch of the network must be compared, because the maximum over predecessors may come from a different branch. In a probabilistic PERT-style evaluation, the project duration variance is the sum of the variances of the activities on the critical path only, and the standard deviation is the square root of that sum. Activities off the critical path do not enter the calculation. Total float is the delay an activity can absorb without delaying project completion, while free float is the delay it can absorb without delaying the earliest start of any successor. Free float never exceeds total float. Zero total float means the activity is critical, so its start cannot be delayed without delaying completion. A dummy activity has zero duration, uses no resources and only shows logical dependence. An activity with positive duration, such as painting, is a real activity and not a dummy.

Job sequencing rules

Sequencing rules decide the order in which jobs are processed on a single machine. Flow time is completion time minus release time, lateness is L_i=C_i-d_i, and tardiness is T_i=\max(0,C_i-d_i). The shortest processing time (SPT) rule processes jobs in increasing order of processing time. When all jobs are available at time zero, SPT minimises the average flow time and the average number of jobs in the system. The earliest due date (EDD) rule processes jobs in increasing order of due date and minimises the maximum lateness. Average tardiness is the sum of the tardiness of all jobs divided by the number of jobs, so an on-time job contributes zero. To count tardy jobs, find each job's completion time on a common time base and compare it with its due date. A job is tardy when its completion time exceeds its due date. If due dates are given as clock times, convert them to elapsed time from the common start before comparing. For a two-machine flow shop in which every job visits machine 1 and then machine 2, Johnson's rule minimises the makespan. Find the smallest processing time among the unscheduled jobs. If it occurs on machine 1, place that job at the front of the sequence; if it occurs on machine 2, place it at the back. Remove the job and repeat. Front jobs appear in increasing machine 1 time and back jobs in decreasing machine 2 time. The makespan is then obtained by the usual two-machine completion-time recurrence.

Break-even production quantity

The break-even point is the volume at which total revenue equals total cost. With fixed cost F, selling price p and variable cost v per unit, the contribution per unit is c=p-v, the break-even quantity is Q_{BE}=F/c, and the break-even revenue is pQ_{BE}. For several products sold in a fixed ratio, group them into a bundle that matches the sales ratio. Compute the bundle revenue and the bundle contribution, then divide the fixed cost by the bundle contribution to obtain the number of bundles. Multiply by the bundle revenue to obtain the break-even revenue. If a plant produces at a constant rate of r units per day, the daily contribution is rc. The number of days needed to recover the fixed cost is F/(rc), and it is rounded as the problem requires. Setup or plant costs are treated as fixed costs recovered from the contribution of production. In a make-or-buy decision, equate the in-house cost F+vD with the purchase cost bD, where b is the purchase price per unit. The crossover demand is D^*=F/(b-v). When v<b, in-house production is cheaper for demand above D^*. When profit is stated as a fraction of revenue, write profit as pQ-F-vQ and set it equal to that fraction of pQ. This equation gives the unknown selling price, from which the contribution and the break-even quantity follow. The answer can be checked by confirming that revenue minus total cost equals the stated profit.

Linear programming product mix

A product-mix linear programme (LP) maximises a linear profit \sum_j c_jx_j subject to linear resource constraints and nonnegativity. With two decision variables, the feasible region is a convex polygon, and if an optimum exists it occurs at a corner point. The corner-point method evaluates the objective at each vertex, which is formed by intersecting two constraint boundaries, and then selects the largest value. Vertices that violate any constraint are discarded. A corner where two resource constraints are both fully used is a common candidate for the optimum. At optimality, complementary slackness holds. A constraint with positive slack has a zero dual value, and a primal variable that is positive makes its dual constraint hold as an equation. If all primal variables are positive, every dual constraint is binding, so the dual values can be found by solving those equations. Strong duality then makes the primal and dual objective values equal. The shadow price of a resource is the rate of change of the optimal objective per unit increase in its right-hand side, valid within the feasible range. It is nonzero only for a binding constraint. To identify binding resources, substitute the optimal solution into each constraint and compare the left side with the right-hand side. A resource with slack has zero shadow price.

Straight-line depreciation

Straight-line depreciation charges the same amount each year: D=\frac{C-S}{n}, where C is the initial cost, S the salvage value and n the useful life. The book value after k years is B_k=C-kD, which equals the salvage value when k=n. Given any three of C, S, D and n, the fourth follows from the formula. For example, the useful life is n=(C-S)/D. To find the book value at a given year, multiply the annual charge by the number of elapsed years and subtract the result from the initial cost, as in B_k=C-kD. Two assets with different costs, salvage values and lives can have equal book values at the same date, so each book value must be computed from its own annual charge rather than assumed from the purchase prices.

Production line cycle time

In a serial production line, each workstation receives the output of the previous one. The cycle time of the line is set by the slowest station, called the bottleneck, so CT=\max_i t_i. In steady state, the output rate is the reciprocal of the cycle time. When CT is in seconds, the output is 3600/CT units per hour. Over any period, the number of units produced is the period length divided by the cycle time, ignoring start-up effects. In line balancing, the cycle time cannot be smaller than the longest indivisible task. The minimum number of stations is at least the total work content divided by the cycle time, rounded up. To check feasibility for a chosen cycle time, assign tasks to stations in sequence without exceeding that cycle time at any station.

Transportation problem initial solution

A transportation problem is balanced when total supply equals total demand. If it is unbalanced, add a dummy source or a dummy destination with zero unit costs to absorb the surplus or the shortage. The dummy does not change the cost of the real shipments. The Northwest-corner method starts at the top-left cell, allocates the smaller of the remaining supply and remaining demand, and then moves right or down. It ignores unit costs and produces a basic feasible solution, but the total cost is often high. Vogel's approximation method (VAM) computes, for each row and each column, a penalty equal to the difference between its two smallest unit costs. It then allocates as much as possible to the cheapest cell in the row or column with the largest penalty, removes the satisfied row or column, and repeats. VAM usually gives a better initial solution than the Northwest-corner rule. When the supply and demand constraints of the transportation model are written as inequalities, the model may ship less than the demand. Since unit shipping costs are nonnegative, the cheapest feasible plan is to ship nothing, and the optimal value is zero. Equality constraints must be used if full delivery is required.

Exponential smoothing forecast

Simple exponential smoothing forecasts the next period as a weighted average of the latest actual value and the previous forecast: F_{t+1}=\alpha A_t+(1-\alpha)F_t, with smoothing constant 0<\alpha\le 1. A larger \alpha reacts faster to changes in demand, while a smaller \alpha smooths more. To forecast several periods ahead, apply the recursion period by period, starting from a known forecast, since each new forecast uses the previous forecast and the most recent actual. Simple exponential smoothing needs only the latest actual value, the previous forecast and \alpha. This is why it requires the least amount of data among the common forecasting methods. Linear regression, ARIMA and econometric models need long histories or several explanatory variables to estimate their parameters.

Forecast error measures

Forecast accuracy is judged from the error e_t=A_t-F_t, where A_t is the actual value and F_t the forecast. Mean absolute deviation (MAD) is the average of |e_t|. Mean squared error (MSE) is the average of e_t^2. Mean absolute percent error (MAPE) is the average of |e_t|/A_t, expressed as a percentage. Bias is the mean of e_t, and the tracking signal is the cumulative error divided by MAD. Terms such as mean sum product error do not name a standard forecast error measure, so they are not valid measures.

Weighted factor scoring

Weighted factor scoring is a method for comparing alternatives, such as candidate sites, when several criteria matter and they differ in importance. Each criterion receives a weight, usually a fraction, and the weights are chosen so that they sum to 1. Each alternative is then scored on every criterion using a common scale, where a higher score means better performance.

The overall score of an alternative is the weighted sum S_j = \sum_i w_i s_{ij}, and the alternative with the highest total is preferred. A common mistake is to take a simple average of the scores, which ignores importance. A criterion with weight 0.4 must count four times as much as one with weight 0.1. Before summing, check that the weights add to 1, and reverse the scale for any criterion where a lower value is better. For example, with weights 0.6 and 0.4 and scores 70 and 90, the total is 0.6(70) + 0.4(90) = 78.

Facility layout types

A facility layout is the arrangement of machines, workstations and material-handling paths, and its type depends mainly on the volume and variety of production. Three basic types are used.

• Process (functional) layout — machines of the same function are grouped together, so the layout is flexible for many different products. It suits batch and job production, where routings vary.
• Product (line) layout — machines are arranged in the sequence of operations for one standard product, which gives short flow paths. It suits mass or flow production.
• Fixed-position layout — the product stays in one place and workers and equipment move to it. It is used for very large or bulky items such as ships and aircraft.