Agribusiness Management · Functional core · also ABM 537, AEC-508 Linear Programming, AEC-605 Operations Research, Quantitative Techniques for Agribusiness (MANAGE)
Mathematical models for decisions: linear programming, transport and assignment, inventory, queues, decision theory, games, simulation and networks.
In the exams: agribusiness management PhD entrance and fellowship examinations, operations research (heavy); agricultural economics PhD entrance and fellowship examinations (present); agricultural scientist eligibility tests, operations research methods (heavy)
Operations research (OR), also taught as quantitative techniques or management science, is the use of mathematical models to choose among alternatives when resources are limited and outcomes are uncertain. It grew out of Second World War military planning and reached agriculture in the 1950s, when George Dantzig's simplex method (1947) made linear programming practical and agricultural economists such as Earl Heady began using it to plan whole farms. Today a feed mill choosing its cheapest ration, a farmer producer organisation (FPO) routing produce to warehouses, and a sugar mill scheduling cane trucks at its weighbridge are all solving OR problems.
This manual covers the standard OR syllabus of agribusiness and agricultural economics postgraduate programmes: linear programming formulation, the graphical and simplex methods, duality, transportation and assignment models, network analysis (CPM and PERT), inventory models, queuing, decision criteria and decision trees, two-person zero-sum games, simulation, Markov chains, data envelopment analysis and an introduction to dynamic programming.
In examinations such as the ICAR AICE-JRF/SRF (Agri-Business Management, operations research unit) and the ASRB NET (agricultural economics, operations research methods), OR is examined through short numericals (an EOQ, an M/M/1 waiting time, a saddle point, an EVPI, a PERT probability, a transportation starting solution) and conceptual questions on assumptions, duality and special cases. The same core is taught worldwide under names such as quantitative methods or management science. Money values in this manual are in plain 'units of currency' or thousands of them, so that the examples read the same in any country.
A linear programme (LP) chooses non-negative values of decision variables to maximise or minimise a linear objective function subject to linear constraints. In symbols, maximise Z = c1x1 + c2x2 + ... + cnxn subject to a11x1 + ... + a1nxn <= b1, and so on for each resource, with every xj >= 0. The cj are objective coefficients (for a farm, usually gross margin per hectare or per animal), the aij are technical coefficients (how much of resource i one unit of activity j uses), and the bi are resource availabilities (right-hand sides). The model rests on five assumptions. Proportionality: doubling an activity doubles its resource use and its contribution. Additivity: total resource use and total return are sums across activities, with no interaction. Divisibility: activities can take fractional values, such as 2.7 hectares. Certainty: all coefficients are known constants. Non-negativity: activities cannot be negative. When divisibility fails (you cannot buy 0.4 of a tractor) the problem becomes integer programming; when certainty fails badly, risk programming methods such as MOTAD (minimisation of total absolute deviations, introduced by Hazell in 1971) or chance-constrained programming are used; when several goals must be balanced, goal programming is used.
Formulation is where most marks are lost. First, identify decision variables and state their units precisely: x1 = hectares of wheat, not simply 'wheat'. Second, write the objective in the same units as the coefficients: if gross margins are in thousands per hectare, Z is in thousands. Third, list every limiting resource and write one constraint per resource, checking that both sides are in the same unit (labour days, hectare-centimetres of water, quintals, kilograms of protein). Fourth, add policy or market constraints: a rotation rule that pulses must occupy at least 20 percent of land, a contract that caps sales of a crop, a minimum food-security area. Fifth, add non-negativity. Direction of inequality matters: resource availability gives <=, nutrient requirements in a ration give >=, and fixed contracts give =.
A farmer has 10 hectares, 240 days of family labour in the season and 36 hectare-centimetres of irrigation water. Wheat earns a gross margin of 40 thousand per hectare and needs 20 labour days and 4 hectare-centimetres of water per hectare. Mustard earns 50 thousand per hectare and needs 30 labour days and 2 hectare-centimetres. Let W and M be hectares of wheat and mustard. Maximise Z = 40W + 50M subject to W + M <= 10 (land), 20W + 30M <= 240 (labour), 4W + 2M <= 36 (water), W, M >= 0. Solved (Chapter 2), the optimum is W = 6, M = 4, Z = 440 thousand. Land and labour are fully used; 4 hectare-centimetres of water are left over. Mustard has the higher margin per hectare, yet an all-mustard plan earns only 400 thousand because labour runs out at 8 hectares.
The oldest applied LP is the diet problem, first posed by George Stigler in 1945 and solved exactly once the simplex method appeared. Compound feed plants run it as least-cost feed formulation. Decision variables are the kilograms of each ingredient (maize, soybean meal, rice bran, mineral mixture) in 100 kilograms of feed. The objective minimises total ingredient cost. Constraints require the mix to meet at least the minimum crude protein, energy, calcium and phosphorus, to stay below maxima such as crude fibre, to respect inclusion limits for ingredients with anti-nutritional factors, and to sum to exactly 100 kilograms. Nutrient standards vary by species, stage and country. The same structure serves fertiliser blending and dairy product-mix planning.
With two decision variables an LP can be solved on graph paper. Draw each constraint as a line by treating it as an equality, find its intercepts, and shade the side that satisfies the inequality. The region satisfying all constraints and non-negativity is the feasible region; for an LP it is always convex. The fundamental theorem of LP states that if an optimal solution exists, at least one occurs at a corner (extreme) point of the feasible region. So either evaluate Z at every corner, or slide an iso-profit line (a line of constant Z) outward parallel to itself until it last touches the region. In the farm example the corners are (0,0) with Z = 0, (9,0) with Z = 360, (8,2) with Z = 420, (6,4) with Z = 440 and (0,8) with Z = 400. The optimum is (6,4), where land and labour intersect. The corner (8,2) is where the land line meets the water line, and (9,0) is the water intercept on the W axis.
Four special cases appear in every exam. Infeasibility: the constraints have no common region, for example W + M <= 4 together with W + M >= 6; the model is telling you the requirements are inconsistent. Unboundedness: the feasible region extends indefinitely in the direction of improvement, so Z can grow without limit; in practice it means a constraint has been forgotten. Alternative (multiple) optima: the iso-profit line is parallel to a binding constraint, so every point on that edge is optimal; in the simplex tableau this shows as a zero net evaluation (Cj - Zj = 0) for a non-basic variable at the optimum. Degeneracy: more constraints pass through a corner than are needed to define it, so a basic variable takes the value zero; in the simplex it shows as a tie in the minimum ratio test, and in rare cases it can cause cycling.
The simplex method, published by Dantzig in 1947, moves from one corner of the feasible region to an adjacent, better corner until no improvement is possible. First convert to standard form: add a slack variable to each <= constraint (unused resource), subtract a surplus variable from each >= constraint (amount by which a requirement is over-met), and add an artificial variable to each >= or = constraint so that a starting basis exists. A basic feasible solution has as many basic variables as there are constraints; the others (non-basic) are zero. Artificial variables are driven out either by the Big-M method, which gives them a huge penalty M in the objective, or by the two-phase method, which first minimises the sum of artificials and then optimises the real objective. A positive artificial at the optimum means the problem is infeasible.
For a maximisation problem in the Cj - Zj convention, compute the net evaluation Cj - Zj for each non-basic variable, where Zj is the sum over the basis of (objective coefficient of the basic variable times the tableau entry in column j). The entering variable is the one with the largest positive Cj - Zj. The leaving variable is found by the minimum ratio test: divide each right-hand side by the positive entry in the entering column and choose the smallest ratio; ignore zero and negative entries. The intersection is the pivot element. Divide the pivot row by the pivot, then use row operations to make every other entry in the pivot column zero. Stop when every Cj - Zj <= 0 for maximisation (>= 0 for minimisation). Texts using Zj - Cj reverse these signs. If the entering column has no positive entry, the problem is unbounded. In the farm example the first iteration brings in M (largest coefficient 50); the ratio test gives 10/1 = 10, 240/30 = 8 and 36/2 = 18, so labour's slack leaves. The second iteration brings in W and land's slack leaves, reaching W = 6, M = 4, Z = 440.
Every LP (the primal) has a companion LP (the dual) built from the same data. For a primal in canonical form, maximise c'x subject to Ax <= b, x >= 0, the dual is: minimise b'y subject to A'y >= c, y >= 0. The rules are mechanical. Each primal constraint becomes a dual variable, and each primal variable becomes a dual constraint. Right-hand sides of the primal become objective coefficients of the dual, and vice versa. The coefficient matrix is transposed. Maximisation becomes minimisation and <= constraints become >=. An equality constraint in the primal gives an unrestricted (free) dual variable, and a free primal variable gives an equality dual constraint. The dual of the dual is the primal. For the farm plan, the dual is: minimise 10y1 + 240y2 + 36y3 subject to y1 + 20y2 + 4y3 >= 40 (wheat) and y1 + 30y2 + 2y3 >= 50 (mustard), y >= 0.
The weak duality theorem states that for any feasible primal solution x and any feasible dual solution y of a maximisation primal, c'x <= b'y: every dual value is an upper bound on every primal value. The strong duality theorem states that if either problem has a finite optimum, so does the other, and the two optimal objective values are equal. If the primal is unbounded, the dual is infeasible. Complementary slackness links the two optima: if a primal constraint has positive slack at the optimum, its dual variable is zero; if a dual variable is positive, its primal constraint is binding. Likewise, if a primal activity is in the plan at a positive level, its dual constraint holds with equality, meaning the imputed cost of the resources it uses exactly equals its return. For the farm, solving the dual gives y1 = 20, y2 = 1, y3 = 0 with objective 10(20) + 240(1) + 36(0) = 440, the same as the primal. Water has slack, so its dual value is zero.
The optimal dual variable of a resource constraint is its shadow price (also called imputed value, marginal value product or opportunity cost): the increase in the optimal objective from one more unit of that resource, other things fixed. In the optimal simplex tableau they appear as the Zj values under the slack columns. In the farm example one more hectare of land raises gross margin by 20 thousand, one more labour day by 1 thousand, and more water by nothing. This is the number a farmer needs when deciding whether to lease in land (worth it if rent is below 20 thousand per hectare, within the valid range) or hire labour (worth it if the wage is below 1 thousand per day). The dual constraints also give an economic test: an activity is profitable only if its return at least covers the imputed value of the resources it uses. For wheat, 20 + 20(1) + 4(0) = 40, exactly its gross margin, so it is in the plan. A resource that is not fully used is a free good at the margin and has a zero shadow price. Shadow prices are valid only over a range of the right-hand side; beyond that range the optimal basis changes and the shadow price must be recalculated.
Sensitivity analysis asks how far the data can change before the optimal plan changes. Objective-coefficient ranging gives, for each activity, the interval of gross margins over which the current plan stays optimal (the plan stays the same but Z changes). Right-hand-side ranging gives, for each resource, the interval over which the shadow price stays constant (the basis stays the same, though the values of basic variables change). Reduced costs tell how much a non-basic activity's margin must improve before it enters. Varying a crop's price step by step (parametric programming) traces a normative supply curve.
The transportation problem finds the least-cost way to ship a commodity from m sources, each with a supply si, to n destinations, each with a demand dj, when the unit cost cij of each route is known. It is an LP with m x n variables xij and m + n constraints, but its special structure allows a much faster tableau method. A problem is balanced when total supply equals total demand. If supply exceeds demand, add a dummy destination with zero costs to absorb the surplus (the dummy allocation is stock left at source); if demand exceeds supply, add a dummy source (the dummy allocation is unmet demand, and a penalty cost can be put on it if shortages are costly). A route that cannot be used, such as a road closed in the monsoon, gets a very large cost M. A maximisation version (for example maximising the margin from sending produce to different markets) is solved by converting profits to opportunity losses, subtracting each profit from the largest profit, and then minimising. A basic feasible solution has m + n - 1 positive allocations; fewer than that is a degenerate solution. The model routes milk from chilling centres to plants, cane from zones to mills and grain from procurement to consumption regions; spatial equilibrium models (Samuelson, 1952; Takayama and Judge) extend it by letting supply and demand respond to price.
Three starting methods are taught. The north-west corner rule allocates as much as possible to the top-left cell, then moves right or down as supply or demand is exhausted, ignoring costs entirely; it is quick but usually expensive. The least-cost (matrix minimum) method allocates as much as possible to the cheapest remaining cell each time. Vogel's approximation method (VAM) computes for each row and column a penalty equal to the difference between its two lowest remaining costs, picks the row or column with the largest penalty, and allocates as much as possible to its cheapest cell; it usually gives a solution at or near the optimum. Example: three FPO collection centres supply 35, 50 and 40 tonnes of onion to three warehouses needing 45, 20 and 60 tonnes. Costs per tonne (in currency units) are: centre 1 to warehouses 8, 6, 10; centre 2: 9, 12, 13; centre 3: 14, 9, 16. The north-west corner rule costs 1,510, the least-cost method 1,410, and VAM 1,320, which here is already optimal: centre 1 sends 35 to warehouse 3; centre 2 sends 45 to warehouse 1 and 5 to warehouse 3; centre 3 sends 20 to warehouse 2 and 20 to warehouse 3. That solution has five positive cells, equal to m + n - 1 = 5.
The modified distribution (MODI, or u-v) method assigns a row number ui and a column number vj so that ui + vj = cij for every occupied cell, starting by setting one of them (commonly u1) to zero. For each empty cell compute the opportunity cost (improvement index) dij = cij - (ui + vj). For a minimisation problem, if every dij >= 0 the solution is optimal; if some dij = 0 there are alternative optima. If any dij is negative, bring the most negative cell into the solution: trace a closed loop of horizontal and vertical steps through occupied cells, alternately add and subtract, and shift the largest quantity that keeps all allocations non-negative (the smallest allocation in a minus position). The stepping-stone method computes the same indices by tracing a loop for every empty cell, which is slower. MODI needs exactly m + n - 1 occupied cells; if the solution is degenerate, place a tiny quantity epsilon in an independent empty cell (one that does not form a loop) and proceed.
The assignment problem assigns n agents to n tasks, one to one, to minimise total cost or time (or maximise total return). Examples: harvesters to fields, extension officers to blocks, trucks to routes. It is a special case of the transportation problem in which every supply and every demand equals 1, and every variable is 0 or 1. Because only n of the 2n - 1 basic cells are positive, the transportation method would be highly degenerate, so a special algorithm is used. The totally unimodular structure of the constraint matrix guarantees that the LP relaxation has an integer optimum, so the 0-1 condition does not need to be imposed separately.
The Hungarian method (Kuhn, 1955, building on the work of the Hungarian mathematicians König and Egerváry) rests on the fact that adding or subtracting a constant from every element of a row or column does not change the optimal assignment. Step 1: subtract the smallest element of each row from every element in that row. Step 2: subtract the smallest element of each column from every element in that column. Step 3: cover all zeros with the minimum number of horizontal and vertical lines. If the number of lines equals n, an optimal assignment exists among the zeros. Step 4: otherwise, find the smallest uncovered element, subtract it from every uncovered element, add it to every element at the intersection of two lines, leave other covered elements unchanged, and return to Step 3. Step 5: make the assignment by choosing a row or column with a single zero, assigning it, crossing out the other zeros in its row and column, and repeating. Read the cost from the original matrix.
Four tractors (T1 to T4) must plough four fields (F1 to F4). Hours required are: T1: 10, 12, 19, 11; T2: 5, 10, 7, 8; T3: 12, 14, 13, 11; T4: 8, 15, 11, 9. Row reduction subtracts 10, 5, 11 and 8. Column reduction then subtracts the column minima of the reduced matrix. After covering and adjusting, the optimal assignment is T1 to F2 (12 hours), T2 to F3 (7), T3 to F4 (11) and T4 to F1 (8), a total of 38 hours. Summing the reduced matrix, or giving each tractor its cheapest field (which sends T2 and T4 both to F1), are the usual errors.
If the number of agents and tasks differ, add dummy rows or columns of zeros to make the matrix square; the agent assigned to a dummy task stays idle. If an assignment is not allowed (a heavy tractor cannot enter a waterlogged field), put M in that cell. For maximisation, such as assigning salespeople to territories to maximise sales, subtract every element from the largest element in the matrix (or multiply by -1) and minimise.
A project, such as building a pack-house, commissioning a milk chilling plant or setting up a drip-irrigation scheme, is a set of activities with precedence relations. It is drawn as a network. In activity-on-arrow (AOA) diagrams, arrows are activities and nodes are events; dummy activities, drawn as dashed arrows with zero time, are used to show a dependency correctly or to avoid two activities sharing the same start and end nodes. In activity-on-node (AON) diagrams, nodes are activities and no dummies are needed. The critical path method (CPM) was developed around 1957 by DuPont and Remington Rand for plant maintenance projects with known durations. The programme evaluation and review technique (PERT) was developed in 1958 for the US Navy's Polaris missile programme, where durations were uncertain.
In the forward pass, each activity's earliest start (ES) is the largest earliest finish (EF) of its predecessors, and EF = ES + duration. The project duration is the largest EF. In the backward pass, each activity's latest finish (LF) is the smallest latest start (LS) of its successors, and LS = LF - duration. Total float (slack) = LS - ES = LF - EF: the time an activity can be delayed without delaying the project. Free float = the ES of the earliest successor minus the activity's own EF: delay that does not affect any successor. Activities with zero total float form the critical path, the longest path through the network; any delay on it delays the whole project. Example for a small cold-store project (weeks): A site preparation 4; B civil works 6, after A; C equipment order 3, after A; D insulation and fitting 5, after B and C; E electrical connection 6, after C; F trial run 2, after D and E. Forward pass: A 0-4, B 4-10, C 4-7, D 10-15, E 7-13, F 15-17. Project time is 17 weeks. Critical path A-B-D-F. E has total float 2 weeks (LS 9 minus ES 7), and C also has 2 weeks, but the float is shared along the path C-E: using it on C removes it from E.
PERT uses an optimistic time a, a most likely time m and a pessimistic time b for each activity, based on a beta distribution. The expected time is te = (a + 4m + b)/6 and the variance is ((b - a)/6)^2. The expected project time is the sum of te along the critical path, and, assuming activity times are independent and invoking the central limit theorem, the project time is approximately normal with variance equal to the sum of the critical activities' variances. The probability of finishing by a target date T is P(Z <= (T - Te)/sigma). If, in the cold-store example, the critical activities have variances 0.444, 1.0, 0.444 and 0.111 weeks squared, the total is 2.0 and sigma is 1.414 weeks; the probability of finishing within 19 weeks is P(Z <= 2/1.414) = P(Z <= 1.41), about 0.92. Two cautions: only critical-path variances are summed (not all activities), and the standard deviation, not the variance, goes into the Z formula.
Many activities can be shortened by spending more, for example by hiring an extra crew. The cost slope of an activity is (crash cost - normal cost)/(normal time - crash time): the extra cost per unit of time saved. To shorten a project at least cost, crash the critical activity with the lowest cost slope, one period at a time, recomputing the critical path after each step because new paths can become critical; when several paths are critical, they must be shortened together. Crashing stops when the saving in indirect costs (supervision, rent, penalty for late completion, lost market season) no longer exceeds the extra direct cost.
Three further network problems are standard. The shortest-route problem finds the least-distance or least-cost path from one node to another; Dijkstra's algorithm labels nodes permanently in order of distance from the origin, and is used for routing collection vehicles. The minimal spanning tree connects all nodes with the least total link length, with no cycles; it is found greedily by Kruskal's or Prim's algorithm and is used to design pipeline, irrigation channel or rural road networks. The maximal flow problem finds the greatest flow from source to sink through arcs with capacities, as in a canal system or a supply chain; the max-flow min-cut theorem states that the maximum flow equals the capacity of the smallest cut separating source from sink.
Agribusinesses hold inventory of inputs (seed, fertiliser, packaging, veterinary medicines), work in progress and finished goods (milk powder, processed rice). Stock buffers uncertainty, captures discounts and covers seasonality, such as grain bought at harvest and milled all year. Ordering (set-up) cost S per order: paperwork, transport, inspection, machine set-up. Holding (carrying) cost H per unit per year: interest on capital, storage, insurance, deterioration, fumigation; it is often expressed as a percentage i of unit price, H = iC. Shortage (stock-out) cost: lost sales and goodwill. Ordering more at a time reduces ordering cost but raises holding cost; the economic order quantity balances the two.
The classic EOQ model, usually credited to Ford Harris (1913) and popularised by R. H. Wilson, assumes constant known annual demand D, instantaneous replenishment, a constant lead time, no shortages and no quantity discounts. Average inventory is Q/2, so annual holding cost is (Q/2)H and annual ordering cost is (D/Q)S. Minimising their sum gives Q* = sqrt(2DS/H). At Q*, annual ordering cost equals annual holding cost, and the minimum total variable cost is sqrt(2DSH). The number of orders per year is D/Q* and the time between orders is Q*/D years. Example: a fertiliser dealer sells 2,400 bags a year, each order costs 300 and holding one bag for a year costs 16. Then Q* = sqrt(2 x 2400 x 300 / 16) = sqrt(90,000) = 300 bags, eight orders a year, one every six to seven weeks, and total variable cost is 4,800 (2,400 ordering plus 2,400 holding). The cost curve is flat near Q*, so rounding to a truck load costs little. When units come from the firm's own plant at a production rate p greater than the demand rate d, the economic production quantity is sqrt(2DS/(H(1 - d/p))).
The reorder point (ROP) is the stock level at which a new order is placed. With certain demand, ROP = daily demand x lead time in days. With uncertain demand, ROP = expected demand during lead time + safety stock, and safety stock = z x sigma(L), where sigma(L) is the standard deviation of demand over the lead time and z is the standard normal value for the chosen cycle service level (the probability of not stocking out in a replenishment cycle). If daily demand is independent with standard deviation sigma(d) and lead time is L days, sigma(L) = sigma(d) x sqrt(L). Higher service levels need disproportionately more safety stock because z rises steeply beyond 95 percent.
The fixed-order-quantity or Q system (continuous review) tracks the stock position continuously and orders the fixed quantity Q, often the EOQ, whenever the position falls to the reorder point. The order interval varies with demand. Safety stock protects only the lead time. The fixed-order-interval or P system (periodic review) checks stock every P periods and orders enough to raise the position to a target level T; the order quantity varies. Safety stock must protect the review period plus the lead time, so T = expected demand over (P + L) + z x sigma(d) x sqrt(P + L), and a P system usually carries more safety stock than a Q system for the same service level. The P system suits items ordered together from one supplier or reviewed on a fixed route, such as a village input shop replenished weekly by a distributor's van; the Q system suits expensive, critical items. Inventory position is always on-hand plus on-order minus backorders, not on-hand alone.
With quantity discounts, compute the EOQ at each price; if it is infeasible for that price band, move it to the lowest quantity in the band; compare total annual cost including purchase cost DC for each candidate and choose the lowest. For perishables and single-season items (cut flowers, fresh vegetables for a retail chain, day-old chicks, a festival stock), the single-period or newsvendor model applies: stock the quantity at which the cumulative probability of demand equals the critical ratio Cu/(Cu + Co), where Cu is the cost of understocking (margin lost per unit short) and Co the cost of overstocking (loss per unsold unit after salvage). ABC analysis ranks items by annual value: typically a small share of items (A) accounts for most of the value and deserves tight control, while C items get simple rules; the exact percentages vary by firm. VED classifies items as vital, essential or desirable, and is used with ABC for spares and medicines.
A queue forms whenever customers arrive faster, at least for a while, than they can be served: tractors at a cane-mill weighbridge, farmers at a procurement centre, trucks at a cold-store dock, animals at a veterinary clinic. A queuing system is described by its arrival process (rate lambda, the mean number of arrivals per unit time), the service process (rate mu, the mean number served per busy server per unit time), the number of servers c, the queue discipline (first come first served is usual; priority rules apply in clinics), the system capacity and the size of the calling population. Kendall's notation writes A/B/c, where A and B describe the inter-arrival and service time distributions (M for Markovian, that is exponential; D for deterministic; G for general) and c is the number of servers. Extended forms add capacity, population and discipline, as in (M/M/1):(infinity/infinity/FCFS). Poisson arrivals and exponential inter-arrival times are the same assumption stated two ways: if arrivals are Poisson with rate lambda, the time between arrivals is exponential with mean 1/lambda.
For a single server with Poisson arrivals, exponential service, infinite queue and population and FCFS, define the utilisation (traffic intensity) rho = lambda/mu. A steady state exists only if rho < 1; if arrivals equal or exceed service capacity, the queue grows without limit. The standard results are: probability the system is empty P0 = 1 - rho; probability of n in the system Pn = (1 - rho) rho^n; probability of more than k in the system rho^(k+1); average number in the system L = lambda/(mu - lambda) = rho/(1 - rho); average number in the queue Lq = rho^2/(1 - rho) = lambda^2/(mu(mu - lambda)); average time in the system W = 1/(mu - lambda); average waiting time in the queue Wq = lambda/(mu(mu - lambda)). Example: cane trucks arrive at a weighbridge at 10 per hour and it can weigh 12 per hour. Then rho = 0.833, L = 10/(12 - 10) = 5 trucks, Lq = 4.17 trucks, W = 1/2 hour = 30 minutes, Wq = 25 minutes, and the weighbridge is idle 16.7 percent of the time. Note how sensitive waiting is to utilisation: at 11 arrivals per hour W doubles to 60 minutes.
Little's law, L = lambda W and Lq = lambda Wq, holds for any stable queue regardless of distributions, and is the most useful check on queuing calculations: if 10 trucks arrive per hour and each spends half an hour in the system, on average 5 are in the system. Also W = Wq + 1/mu. For c identical parallel servers (M/M/c), the system is stable if lambda < c mu; utilisation per server is lambda/(c mu). The formulas involve the Erlang C expression for the probability that an arrival must wait; for c = 2 and rho = lambda/(2 mu), P0 = (1 - rho)/(1 + rho) and Lq = 2 rho^3/(1 - rho^2). One pooled queue feeding several servers gives shorter waits than separate queues.
The purpose of queuing analysis is a decision: how many servers, how fast, what schedule. Total expected cost per unit time = service cost (number of servers x cost per server, or cost of a faster server) + waiting cost (expected number in the system or queue x cost of one customer waiting per unit time). For perishable loads, and for cut sugarcane that loses recoverable sugar, waiting costs are high. Compare total cost for alternative designs and choose the lowest. Where arrivals are not Poisson, service is not exponential or the system has several stages (weighbridge, unloading, empty weighing), analysts use simulation (Chapter 11) rather than formulas. Token systems that smooth arrivals also cut waiting.
Decisions are made under certainty (outcomes known; use LP and similar models), under risk (states of nature have known probabilities; use expected value) or under uncertainty (probabilities unknown; use decision criteria). A payoff table lists acts (alternatives the manager controls) in rows and states of nature (events the manager does not control, such as rainfall, price or disease incidence) in columns, with the payoff of each combination in the cells. Before applying any criterion, remove dominated acts: an act that is never better and sometimes worse than another is never chosen by a rational decision maker. Running example (payoffs in thousands): a potato trader can store 1,000 tonnes (A1), store 500 tonnes (A2) or sell at harvest (A3). If the off-season price is high, payoffs are 200, 140 and 60; if medium, 80, 100 and 60; if low, -40, 20 and 60.
Maximax (the optimist's criterion) picks the act with the largest maximum payoff: A1, with 200. Maximin (Wald's criterion, the pessimist's) picks the act with the largest minimum payoff: A3, whose worst case is 60, against -40 and 20. Minimax regret (Savage's criterion) builds a regret (opportunity loss) table by subtracting each payoff from the best payoff in its column, then picks the act with the smallest maximum regret. Regrets: A1 has 0, 20, 100; A2 has 60, 0, 40; A3 has 140, 40, 0. Maximum regrets are 100, 60 and 140, so A2 is chosen. The Laplace (equal-likelihood, principle of insufficient reason) criterion averages payoffs across states with equal weights: A1 80, A2 86.7, A3 60, so A2. The Hurwicz criterion weights the best payoff by a coefficient of optimism alpha and the worst by 1 - alpha; alpha = 1 gives maximax and alpha = 0 gives maximin.
When probabilities can be attached to states, the expected monetary value (EMV) of each act is the probability-weighted sum of its payoffs, and the act with the highest EMV is chosen by a risk-neutral decision maker. The expected opportunity loss (EOL) is the probability-weighted regret; the act with the highest EMV always has the lowest EOL. If in the potato example the probabilities of high, medium and low prices are 0.3, 0.5 and 0.2, then EMV(A1) = 0.3(200) + 0.5(80) + 0.2(-40) = 92, EMV(A2) = 96 and EMV(A3) = 60, so store 500 tonnes. The expected payoff with perfect information (EPPI or EVwPI) takes the best payoff in each state: 0.3(200) + 0.5(100) + 0.2(60) = 122. The expected value of perfect information is EVPI = EPPI - maximum EMV = 122 - 96 = 26, which also equals the minimum EOL. EVPI is the most a trader should pay for any forecast; imperfect information is worth less. The expected value of sample information (EVSI) uses Bayes' theorem to revise probabilities after a survey or forecast.
A decision tree represents a sequence of decisions and chance events in time order. Squares are decision nodes, circles are chance nodes, and branch ends carry payoffs. The tree is solved by backward induction (rolling back): at each chance node compute the expected value of its branches; at each decision node choose the branch with the best value and prune the others. Trees are the natural tool when a decision leads to later decisions, for example whether to commission a soil test before deciding on a crop, or whether to run a pilot market launch of a new seed variety before a full launch. Costs of tests or pilots are subtracted along the branch where they are incurred. A risk-averse decision maker maximises expected utility instead; a certainty equivalent below the EMV signals risk aversion.
Game theory, founded by von Neumann and Morgenstern's Theory of Games and Economic Behavior (1944), studies decisions whose outcome depends on the choices of rational opponents. In a two-person zero-sum game, one player's gain is exactly the other's loss. The game is written as a payoff matrix from the row player's point of view: row player A maximises, column player B minimises. A pure strategy is a single choice; a mixed strategy assigns probabilities to pure strategies. The value of the game is the expected payoff to A when both play optimally. A fair game has value zero. It models pricing or promotion rivalry between two dairies or seed firms in a fixed-size market; 'games against nature' are better handled by the decision criteria of Chapter 9, because nature is not a rational opponent.
For each row find the minimum (the worst A can get), and take the largest of these: the maximin value. For each column find the maximum (the worst B can suffer), and take the smallest: the minimax value. Always maximin <= value <= minimax. If maximin equals minimax, the game has a saddle point, an element that is the smallest in its row and the largest in its column. Both players then use pure strategies, and the saddle-point entry is the value of the game. Example: for a matrix whose rows are (3, 5, 4), (2, 1, 0) and (6, 4, 2), the row minima are 3, 0 and 2, so maximin = 3; column maxima are 6, 5 and 4, so minimax = 4; there is no saddle point and the value lies between 3 and 4. A game is strictly determinable if maximin equals minimax, and in that case the value can be zero, positive or negative.
Before solving, reduce the matrix by dominance. For A (maximiser), a row is dominated and can be deleted if every element is less than or equal to the corresponding element of another row. For B (minimiser), a column is dominated if every element is greater than or equal to the corresponding element of another column. A row or column can also be dominated by an average of others. For a 2 x 2 game without a saddle point, with A's payoffs a, b in the first row and c, d in the second, A plays row 1 with probability p = (d - c)/((a + d) - (b + c)), B plays column 1 with probability q = (d - b)/((a + d) - (b + c)), and the value is V = (ad - bc)/((a + d) - (b + c)). Example: payoffs (4, -2) and (-1, 3) have no saddle point (maximin -1, minimax 3). The denominator is (4 + 3) - (-2 - 1) = 10, so p = 4/10 = 0.4, q = 5/10 = 0.5 and V = (12 - 2)/10 = 1. A check: if A plays 0.4 and 0.6, B's expected loss is 0.4(4) + 0.6(-1) = 1 against column 1 and 0.4(-2) + 0.6(3) = 1 against column 2, equal, as it must be.
Games of size 2 x n or m x 2 are solved graphically: plot A's expected payoff against p for each of B's strategies, take the lower envelope, and find its highest point (for A); the two lines crossing there identify the 2 x 2 sub-game to solve. Any finite two-person zero-sum game can be solved as a linear programme, and the row and column players' problems are duals of each other, which is the link between game theory and LP duality. The minimax theorem (von Neumann, 1928) guarantees that every finite zero-sum game has a value in mixed strategies. Non-zero-sum games need Nash equilibrium concepts from microeconomics.
Simulation imitates the operation of a real system over time on a computer, using the probability distributions of its uncertain inputs, so that the manager can observe its behaviour and test policies. It is used when an analytical model is too complex or its assumptions do not hold: a queue with several stages and non-exponential service, an inventory system with random demand and random lead times, a farm business with correlated yield and price risk, or a supply chain exposed to disease outbreaks. Simulation does not optimise by itself; it evaluates the policies the analyst feeds it, so it is often combined with search or with experimental design. Its results are estimates with sampling error, and a badly validated model gives confident but wrong answers.
Monte Carlo simulation samples random values from input distributions. Step 1: define the problem and the variables with random behaviour. Step 2: establish each probability distribution from historical data or expert judgement. Step 3: compute the cumulative probability distribution. Step 4: assign random-number intervals in proportion to the probabilities; with two-digit random numbers 00 to 99, an event with probability 0.20 gets 20 numbers. Step 5: draw random numbers and read off the simulated values. Step 6: run the simulation for many trials and compute the performance measures. Step 7: repeat for alternative policies and compare. Example: daily demand for a dairy booth's curd is 20, 21, 22 or 23 cups with probabilities 0.1, 0.3, 0.4 and 0.2. The cumulative probabilities are 0.1, 0.4, 0.8 and 1.0, so random numbers 00 to 09 map to 20 cups, 10 to 39 to 21, 40 to 79 to 22 and 80 to 99 to 23. A random number of 57 gives a demand of 22 cups.
Verification checks that the computer program does what the modeller intended; validation checks that the model represents the real system, for example by comparing simulated and actual queue lengths or yields. Whole-farm risk simulation uses correlated yield and price distributions to estimate the probability of a cash deficit or loan default. Computer random numbers are pseudo-random: generated by an algorithm but behaving randomly for practical purposes.
A Markov chain is a stochastic process that moves among a finite set of states in discrete steps, where the probability of the next state depends only on the current state, not on the history (the Markov property). The transition probability matrix P has pij = probability of moving from state i to state j in one step; each row sums to 1. If the state probabilities now are the row vector s0, then after one step they are s1 = s0 P and after n steps sn = s0 P^n. Example: two seed brands X and Y. Of farmers buying X this season, 80 percent buy X again next season and 20 percent switch to Y; of Y buyers, 30 percent switch to X and 70 percent stay. If current shares are 50:50, next season's shares are 0.5(0.8) + 0.5(0.3) = 0.55 for X and 0.45 for Y, and the season after 0.575 and 0.425.
A regular chain (some power of P has all positive entries) converges to a unique steady-state (equilibrium) vector pi, which satisfies pi = pi P with the elements of pi summing to 1, and does not depend on the starting shares. For the seed example, pi_X = 0.8 pi_X + 0.3 pi_Y and pi_X + pi_Y = 1, so 0.2 pi_X = 0.3 pi_Y, giving pi_X = 0.6 and pi_Y = 0.4. Brand X's long-run share is 60 percent. An absorbing state is one that, once entered, cannot be left (pii = 1), such as 'loan written off' or 'animal culled'. Absorbing-chain analysis uses the fundamental matrix N = (I - Q)^-1, where Q holds transitions among the non-absorbing states, to find the expected number of steps before absorption and the probability of ending in each absorbing state; banks use it to project how current loan balances will end up repaid or written off.
Data envelopment analysis (DEA) is a non-parametric, linear-programming method for measuring the relative efficiency of comparable decision-making units (DMUs), such as farms, dairy cooperatives, FPOs, bank branches or mandis, that use several inputs to produce several outputs. It builds a piecewise-linear frontier that envelops the data, from the best-performing units, and measures each unit's distance from that frontier. It was introduced by Charnes, Cooper and Rhodes (1978), whose CCR model assumes constant returns to scale (CRS); Banker, Charnes and Cooper (1984) added the BCC model with variable returns to scale (VRS). DEA builds on Farrell's (1957) decomposition of efficiency. Unlike parametric stochastic frontier analysis (SFA), DEA needs no functional form but treats all deviation as inefficiency, so outliers and measurement error distort it.
In the ratio form of the CCR model, each DMU's efficiency is the weighted sum of its outputs divided by the weighted sum of its inputs, with the weights chosen to make that unit look as good as possible, subject to the condition that no DMU's ratio exceeds 1 with the same weights. This fractional programme is converted to an LP (the multiplier form); its dual is the envelopment form, which for an input-oriented model asks: by what proportion theta can this unit's inputs be reduced while a combination of other units produces at least its outputs? A theta of 1 means technically efficient; 0.8 means the same output could be produced with 80 percent of the inputs. Input orientation suits units that control inputs more than outputs; output orientation asks how much output could be expanded with the same inputs. Under CRS the two orientations give the same score. Units in the reference combination are the inefficient unit's peers (benchmarks). Remaining slacks after radial contraction show additional input excess or output shortfall. One LP is solved for each DMU.
Four milk cooperatives use staff (one input) to handle milk (one output). Staff numbers are 10, 20, 15 and 25; daily milk handled is 50, 120, 60 and 100 thousand litres. Output per staff member is 5, 6, 4 and 4. Under CRS, each score is its ratio divided by the best ratio (6): 0.833, 1, 0.667 and 0.667. Under VRS, the frontier joins the smallest and the best unit, and scores are 1, 1, 0.762 and 0.686. Scale efficiency is CRS technical efficiency divided by VRS technical efficiency: 0.833, 1, 0.875 and 0.972. Cooperative 1 is managerially efficient for its size (VRS score 1) but too small to reach the most productive scale; cooperative 4 is close to the right scale but manages its staff poorly.
Dynamic programming (DP), developed by Richard Bellman in the 1950s, solves problems that can be broken into a sequence of stages, each with a decision that changes the state of the system. Its basis is Bellman's principle of optimality: whatever the initial state and decision, the remaining decisions must be optimal with respect to the state that results from the first decision. The components are stages (seasons, crops, periods), states (the resource remaining, the stock level, the age of a machine), decisions, a return function for each stage, and a recursive equation, usually solved backward from the last stage: f_n(s) = best over decisions of [return at stage n + f_(n+1)(new state)]. DP handles non-linear returns and integer decisions naturally, which LP cannot. Its limit is the 'curse of dimensionality': the number of states grows explosively when there are several state variables.
A farmer has 3 units of irrigation water to divide among wheat, mustard and vegetables. Returns (thousands) for 0, 1, 2 and 3 units are wheat 0, 5, 8, 10; mustard 0, 4, 7, 9; vegetables 0, 7, 10, 11. Treat each crop as a stage and the water left as the state. Stage 3 (vegetables): f3(s) = returns for vegetables. Stage 2 (mustard plus vegetables): with 3 units, the options are 0+11, 4+10, 7+7 and 9+0, so f2(3) = 14; similarly f2(2) = 11, f2(1) = 7. Stage 1 (wheat, with 3 units): 0 + 14, 5 + 11, 8 + 7, 10 + 0; the best is 16, with 1 unit to wheat and 2 units left, which stage 2 divides as 1 to mustard and 1 to vegetables. Optimal allocation: one unit each, return 16. Agricultural uses include reservoir release, machine and dairy-cow replacement, grain storage and selling, and the stagecoach (shortest route) problem.
A student should be able to formulate an LP from a paragraph of farm or plant data, solve two-variable problems graphically and small ones by simplex, write the dual and interpret shadow prices, set up and solve transportation and assignment tables by hand, and carry out the one-step calculations that dominate objective papers: EOQ, reorder point, M/M/1 measures, PERT probabilities, EMV and EVPI, saddle points and 2x2 mixed strategies, Markov steady states. The habit to build is to state units for every variable and to check every answer against common sense and Little's law.
A manager five years into a feed mill, dairy, FPO or input company uses these tools with a spreadsheet solver: re-running the least-cost ration when ingredient prices change and reading the sensitivity report to time purchases; routing procurement trucks with a transportation model; setting reorder points and safety stock by ABC class; sizing weighbridges, unloading bays or clinic staff from arrival data; and building a simple decision tree before committing to a new product or a storage position. The value is in collecting clean coefficients and explaining the result to people who never see the model.
A senior manager with twenty-five years of experience decides which problems deserve a model and which do not. They commission network plans for new plants and hold contractors to the critical path; they use DEA or similar benchmarking to compare branches, chilling centres or member cooperatives and to set targets; they insist that inventory and service-level policies reflect the real cost of stock-outs in a perishable chain; and they treat model outputs such as shadow prices and EVPI as inputs to negotiation (what to pay for land, capacity or a forecast) rather than as final answers. They also know when assumptions such as linearity or independence have broken down.
A researcher with long experience builds and tests models for policy and science: whole-farm and regional mathematical programming models (with positive mathematical programming calibration, risk and integer components) to simulate policy reform, water pricing or climate adaptation; spatial equilibrium models of trade; DEA and stochastic frontier studies of farm and cooperative efficiency with bootstrapped confidence intervals; dynamic programming for resource management under uncertainty; and stochastic simulation for risk analysis. The research frontier lies in combining optimisation with econometric estimation and in handling uncertainty honestly, and a senior researcher insists on validation against observed behaviour.
This manual covers the standard syllabus at the level of objective exams and a first graduate course. It does not yet cover integer programming algorithms (branch and bound, cutting planes) beyond mention, goal programming and MOTAD in detail, positive mathematical programming, non-linear and quadratic programming, the M/M/c formulas beyond c = 2, finite-queue and finite-population queuing formulas, the full simplex tableau worked step by step with numbers, sequencing (Johnson's rule), replacement models, stochastic DEA and bootstrapping, or stochastic dynamic programming. Figures (feasible regions, networks, trees) cannot yet be shown, so they are described in words.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions.
These notes are free to read here, with no account.
What the Laisha app adds:
Free for ten days, then a subscription. We would rather write that here than have you find it later.