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MAS – LECTURE NOTES
ARMIN GLENN ARANETA, CPA QUANTITATIVE METHODS
QUANTITATIVE METHODS(Operations Research) – various application of mathematics in business or any complex system. OPERATIONS RESEARCH – specifically, the discipline of applying quantitative methods oriented to planning. NETWORK MODELS NETWORK MODELS involve project scheduling techniques that are designed to aid the planning and control of large-scale projects having many interrelated activities.
Uses Uses of Ne N etwo twor k A nalys nalysii s: 1) Planning 2) Measuring progress to schedule 3) Evaluating changes to schedule
4) Forecasting future progress 5) Predicting and controlling costs
Sam Sample Applica Applicattions: ions: 1) Building construction 2) Book publishing projects 3) New product planning
4) Feasibility studies 5) Research and developments 6) Auditing
Project Scheduling Techniques: 1) Gantt or bar charts 2) Program evaluation and Review Techniques (PERT) 3) Critical Path Method (CPM) Program evaluation and Review Techniques (PERT) – developed to aid managers aid managers in controlling large-scale, complex problems.
Pert Diagram – a a probabilistic diagram of the interrelationship of a complex series of activities; a free-form network showing each activity as a line between events. discrete moments in time representing the start or finish of an activity; they consume no E vents vents – discrete resources. Acti Acti vities ities – tasks to be accomplished; they consume resources, (including time) and have a duration over time.
T ype ypes of of Act A ctii vity: a) Ser Ser ies ies – an activity that cannot be performed unless another activity is undertaken. b) Parallel – can be performed simultaneously. Critical Path – longest path through the network. Expected time (te) – the the average time an activity would require if it were repeated a large number of times. te =
to
+ 4tm + t p 6
where: to = optimistic time tm = most likely time t p = pessimistic time
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Slack time – the amount of time that can be added to an activity without increasing the total time required on the critical path; the length of time an activity can be delayed without forcing a delay for the entire project. Critical Path method (CPM) – like PERT, it is a network technique, but unlike PERT. It uses deterministic time and cost estimates; its advantages include cost estimates plus the concept of crash efforts and costs.
Crash time – time to complete an activity assuming that all variable resources were devoted to the task (overtime, extra crew, etc.) EXERCISES IN NETWORK ANALYSIS 1. PERT NETWORK. A company is faced with the following PERT network situation (time in days):
1 – 5 – 9
B START 2 – 4 – 6
FINISH 4 – 8 – 12
A
D
2 – 6 – 10
E
1 – 3 – 5
C
3 – 4 – 5
Required: a) Calculate te (expected time) for each activity. For each activity, the estimates are to, tm, and t p, in that order. b) Calculate the total time for each path and identify the critical path. 2. York New City Building Corporation uses the critical path method to monitor construction jobs. The company is currently 2 weeks behind schedule on Job 143, which is subject to a P10, 500 per week completion penalty. Path A-B-C-F-G-H-I has a normal completion time of 20 weeks, and critical path A-D-E-F-G-H-I has a normal completion time of 22 weeks. The following activities can be crashed.
Activities BC DE EF
Cost to Crash 1 week P 8, 000 10, 000 8, 800
Cost to Crash 2 weeks P 15, 000 19, 600 19, 500
York New City Building desires to reduce the normal completion time of Job 143 and, at the same time, report the highest possible income for the year. York New City Building should crash: a. Activity BC 1 week and activity EF 1 week c. Activity EF 2 weeks b. Activity DE 1 week and activity BC 1 week d. Activity DE 1 week and activity EF 1 week
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LEARNING CURVES LEARNING CURVES describe the efficiencies arising from experience, because with experience comes increased productivity. This productivity increases with production size, but at a decreasing rate as diagrammed below:
Productivity
Experience The time required to perform a given task becomes progressively shorter, but this is applicable only to the early stages of production or any new task. The curve is expressed as a percentage of reduced time (usually between 60% and 80%0 to complete a task of each doubling of cumulative production. Hence, the time required is reduced by 20% to 405 each time cumulative production is doubled. Assumptions: 1) The cumulative average time per unit is reduced by a certain percentage each time production doubles. 2) Incremental unit time (time to produce the last unit) is reduced when production doubles. EXERCISES IN LEARNING CURVES
1. A particular manufacturing job is subject to an estimated 50% learning curve. The first unit required 30 labor hours to complete. Required: a. What is the cumulative average time per unit after four units are completed? b. What is the total time required to produce 2 units? c. How many hours are required to produce the second unit? 2. CPA Company recently completed and sold an order of 50 units that had the following costs: Direct materials P1, 500 Direct labor (1, 000 hours @ P8.50) 8, 500 Variable overhead (1, 000 hours @ P4.00)* 4, 000 Fixed overhead** 1, 400 Total P15, 400 *Applied on the basis of direct labor hours **Applied at the rate of 10 percent of variable cost The company has now been requested to prepare a bid for 150 units of the same product. 2a. If an 80% learning curve is applicable, CPA Company’s total cost on this order would be estimated at: a. P26, 400 c. P37, 950 b. P31, 790 d. P37, 500 2b. If CPA Company had experienced a 70% learning curve, the bid for the 150 units would:
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a. show a 30% reduction in the total direct labor hours required with no learning curve. b. include increased fixed overhead costs. c. include 2, 560 total direct labor hours at 8.50 per hour. d. include 6.40 direct labor hours per unit at p8.50 per hour. 3. Accountant Corporation is preparing a bid to produce engines. The company has experienced the following costs: Cumulative Total Cumulative Costs Units Produced Materials Labor 10 P60, 000 P120, 000 20 120, 000 192, 000 40 240, 000 307, 200 At Accountant Corporation, variable overhead is applied on the basis of 100% of direct labor cost. Based on historical costs, the company knows that the production of 40 engines will incur P100, 000 of fixed overhead costs. The bid request is for an additional of 40 units; all companies submitting bids are allowed to charge a maximum of 25% above full cost for each order. Required: a. Compute for the rate of learning on the engine contract. b. The maximum bid price that Accountant Corporation can submit for the 40 units. c. To ensure that the company will not lose money on the project, what should the minimum bid price for the project be?
PROBABILITY ANALYSIS PROBABILITY is important to management decision-making becaus e of the unpredictability of future events.
Decision-making under conditions of risk possible future states of nature is known.
–
occurs when the probability distribution of the
Decision-making under conditions of uncertainty – occurs when the probability distribution of possible future states of nature is not known and must be subjectively determined. The probability of an event varies from 0 to 1 (or 0 to 100%) a. 0 probability – the event cannot occur. b. Probability of 1 (or 100%) – the event is certain to occur. BASIC TERMS USED WITH PROBABILITY: 1) Two events are mutually exclusive if they cannot occur simultaneously. 2) The joint probability of two events is the probability that both will occur. 3) The conditional probability of two events is the probability that one will occur given that the other has already occurred. 4) To events are independent if the occurrence of one has no effect on the probability of the other. Expected Value The expected value of an action is found by multiplying the probability the probability of each outcome by its pay-off and summing the products.
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EXERCISES IN PROBABILITY
1. In planning its budget for 2014. Future Company prepared the following pay-off probability distribution describing the relative likelihood of monthly sales volume levels for its product: Monthly Sales Volume Probability 200 5% 300 10% 500 60% 800 20% 1, 000 5% Required: a. What is the expected monthly sales volume, considering the given probabilities? b. If the product’s contribution margin per unit is P2, what is the expected value of the monthly contribution margin for this product? 2. A computer store sells four computer models designated as Y143, A143, T143 and S143. The store manager has made random number assignments to represent customer choices based on past sales data. The assignments are shown below. Model Random Numbers Y143 0 – 1 A143 2 – 6 T 143 7 – 8 S 143 9 Required: a. What is the probability that a customer will select model Y143, A143, T143 and S143? b. In running a simulation of the computer demand, the following numbers are drawn in sequence: 2, 8, and 6. The simulation indicates that the third customer will purchase? 3. The probabilities shown in the table represent the estimates of sales for a new product: Sales in units Probability 0 – 200 15% 201 – 400 45% 401 – 600 25% 601 – 800 15% What is the best estimate of the expected sales of the new product? a. 480 c. 400 b. 380 d. 800 4. A company uses two major material inputs in its production. To prepare its manufacturing operations budget, the company has to project the cost changes of these material inputs. The cost changes are independent of one another. The purchasing department provides the following probabilities associated with projected cost changes: Cost Change 3% increase 5% increase 10% increase
Material 1 0.3 0.5 0.2
Material 2 0.5 0.4 0.1
What is the probability that there will be a 3% increase in the cost of both Material1 and Material 2? ; 5%?; 10%? Joint probability of occurrence of two independent events = product of their individual probabilities.
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5. A firm will either produce product A or B. the total costs (TC) for both products can be estimated by the equations: Product A: TC = P300, 000 + (P23 x Sales Volume) ` Product B: TC = P100, 000 + (P29 x Sales Volume) The firm believes there is a 20% chance for the sales volume of each product to equal 10, 000 units and an 80% chance they will both equal 20, 000 units. The selling price of product A is P42, and the selling price of product B is P40. The expected profit from producing product A equals? Product B equals?
LINEAR PROGRAMMING
LINEAR PROGRAMMING – a technique used to optimize an objective function (maximize revenue or profit function, or minimize a cost function), subject to constraints (such as scarce resources, minimum/maximum levels of production, performance, etc.)
In business, linear programming is used for planning resource allocations (to make optimum use of limited resources). Conditions calling for the use of linear programming include: 1) Specification of a cost or revenue objective formula. 2) The limited resources must be subject to alternative uses. 3) The alternative uses of the limited resources must be specified.
EXERCISES IN LINEAR PROGRAMMING ] 1. Anim Company has excess capacity on two machines, 24 hours on Machine 142 and 16 hours on Machine 143. To use this excess capacity, the company has two products, known as Product A and Product M that must use both machines in manufacturing. Both have excess product demand, and the company can sell as many units as it can manufacture. The company’s objective is to maximize profits.
Product A has incremental profit of P6 per unit, and each unit utilizes 2 hours of time on Machine 142 and then 2 hours of time on Machine 143. Product M has an incremental profit of P7 per unit, and each unit utilizes 3 hours of time on Machine 142 and then 1 hour of time on Machine 143. Let A be the number of units for product A, M be the number of units for Product M, and P be the company’s profit. Required: a. Objective function involving maximization of the compan y’s contribution margin. b. Constraint function for Machine 142. c. Constraint function for Machine 143. d. The optimal mix of products that must be produced by Anim Company. 2. The Click Five Inc. manufactures Product J and Product E which are processed as follows: Type D Machine Type B Machine Product J 6 hours 4 hours Product E 9 hours 5 hours The contribution margin is P12 for Product J and P7 for Product E. The available time daily for processing the two products is 120 hours for Machine T ype D and 80 hours for Machine Type B. How would the restriction (constraint) for Machine Type B be expressed? a. 4J + 5E c. 6J + 9E < 120 b. 4J + 5E < 80 d. 12J + 7E
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3. The Hall of Fame Company makes toys A and B, each of which needs two processes: cutting and wrapping. The contribution margin is P6 for Product A and P5 for Product B. the table below shows the maximum number of units (constraint) of each product that may be processed in two departments.
Product A Product B
Maximum Capacities (In Product Units) CUTTING WRAPPING 60 80 60 40
Considering the constraint in processing, which combination of Products A and B maximizes the total contribution margin? a. 40 units of A and 20 units of B c. 80 units of A and 0 units of B b. 60 units of A and 0 units of B d. 0 units of A and 40 units of B 4. A firm decides the mix of production of Product X and Product Z. there are only two resources used in the two products, resources A and B. data related to the two products are given in the following table:
Resource A Resource B Unit Profit
Product X 3 2 P8
Product Z 7 1 P6
What is the appropriate objective function to max imize profit? a. 3X + 7Y c. 8X + 6Y b. 2X + Y d. 5X + 8Y
I f we are facing the ri ght direction, all we have to do is keep on walking.” -END-
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