Thursday, February 27, 2014


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Thursday, September 26, 2013

Break-Even Analysis

The objective of most business enterprises is to make as much profit as possible.  The purpose of break-even analysis is to determine the number of units of a product (the volume) to produce that will equate total revenue with total cost.  At this point, referred to as the break-even point, profit is zero.  The break-even point gives the manager a point of reference in determining how many units will be needed to ensure a profit.

The three components of break-even analysis are volume, cost and profit.

Volume is the level of production by a company.  Volume can be expressed as the number of units (quantity) produced and sold, as the dollar volume of sales or a percentage of total capacity available,

Two types of costs are typically incurred in the production of a product: fixed costs and variable costs.  Fixed costs are generally independent of the volume of units produced,  That is, fixed costs remain constant regardless of how many units of product are produced within a given range.  The costs of items such as the following, taken together, result in total fixed costs: Rent on plant and equipment, taxes, insurance, management and staff salaries, advertising, interest on investment, depreciation on plant and equipment, heat and light, janitorial services.

Variable costs are determined on a per-unit basis.  Thus, total variable costs depend on the number of units produced.  The costs of the following items are variable costs: raw materials and resources, direct labor, packaging, material and product handling, maintenance, freight

Total variable costs are function of the volume and the variable cost per unit.  This relationship can be expressed mathematically as :
 total variable cost = VCv
where: Cv = variable cost per unit; v = volume (number of units)

The total cost of an operation is computed by summing total fixed cost and total variable cost, as follows:
total cost = total fixed cost + total variable cost or TC = Cf +VCv where Cf = fixed cost

The third component in our break-even model is profit.  Profit is the difference between total revenue and total cost.  Total revenue is the volume multiplied by the price per unit.

Total revenue = vp where p (price per unit)

Total profit = total revenue - total cost

break - even volume = Cf/p - Cv

break-even sales volume = pv

break-even volume as percentage of capacity = v/k

1.  The Texas Electronics Company produces calculators.  The annual fixed cost of producing calculators is $280,000.  The variable cost of producing a calculator is $8.  The company sells the calculators for $26.  Given an annual volume of 60,000 calculators, determine the total cost, total revenue, profit, break-even volume, and break-even sales volume.  If production capacity is 80,000 calculators, determine the break-even volume as a percentage of capacity.

2. The Willow Furniture produces tables.  The fixed monthly cost of production is $8,000, and the variable cost per table is $65.  The tables sell for $180 apiece.  For a monthly volume of 300 tables, determine the total cost, total revenue, and profit, determine the monthly break-even volume for the Willow Furniture Company operation. Determine the break-even sales volume.

3.  The Retread Tire Company recaps tires.  The fixed annual cost of the recapping operation is $60,000.  The variable cost of the recapping a tire is $9.  The company charges $25 to recap a tire.  For an annual volume of 12,000 tires, determine the total cost, total revenue, and profit.  Determine the annual break-even volume.  Determine the break-even sales volume.  If the maximum operating capacity of the Retread Tire Company is 8,000 tires annually, determine the break-even volume as a percentage of capacity.

4.  The Rolling Creek Textile Mill produces cotton denim.  The fixed monthly cost is $21,000, and the variable cost per yard of denim is $0.45.  The mill sells a yard of denim for $1.30.  For a monthly volume of 18,000 yards of denim, determine the total cost, total revenue, and profit.  Determine the monthly break-even volume. If the maximum operating capacity of the Rolling Creek Textile Mill  is 25,000 yards of denim per month, determine the break-even volume as a percentage of capacity.

Tuesday, September 17, 2013

Inventory Model

An inventory is a stock or store of goods,  It may be thought of as a resource or a list of some category of materials, machines, people, money or information for some organizational unit at some time.  Proper inventory management is an essential function of all business operations. Most of the problems in inventory fall into one of the following categories A. The proper quantity of inventory to order at any given time (How much to order? B. the proper time to order the quantity (When to order?).  Four basic costs are associated with inventories: purchase cost; holding or carrying cost; ordering costs and shortage cost.  Annual cost of inventory is the sum of the annual ordering cost and annual carrying cost.

1. Holding or carrying cost relates to physically holding items in storage.  This cost is proportional to the amount of inventory and the time over which it is held.  They include interest, insurance, taxes, depreciation, obsolescence, deterioration, spoilage, pilferage, breakage, and warehousing cost (heat, light, rent, security).

2.  Ordering or set up costs are costs associated with ordering and receiving inventory.  This cost is incurred whenever an inventory is replenished.  These costs include determining how much is needed, typing up invoices, inspecting goods upon arrival for quality and quantity, moving the goods to temporary storage.

3.  Purchase or Direct Production cost is for vendor supply environments or direct production cost in case of items produced by user.  In either situation the unit cost may be constant for all replenishment quantities, or it may vary with the quantity purchased or produced.

4.  Shortage costs result when demands excess the supply of inventory on hand.  The cost can include the opportunity cost of not making a sale, loss of customer goodwill, lateness charges, and similar costs. Furthermore, if the shortage occurs in an item carried for internal use, the cost of lost production or downtime is considered a shortage cost.


1.  Suppose that MYCOLA company has a beverage product that has a constant annual demand rate of 7200 cases.  A case of soft drink costs P288.  Ordering cost is P200 per order and inventory carrying cost is charged at 25% of the cost per unit.  Identify the following aspects of the inventory policy: a) economic order quantity; b) annual carrying cost; c) total annual cost

2.  Electronic village stocks and sells a particular brand of personal computer.  It costs the store P450 each time it places an order with the manufacturer for the personal computers.  The annual cost of carrying the PCs in inventory is P170.  The store manager estimates that annual demand for the PCs will be 1200 units.  Determine the optimal order quantity and the total minimum inventory cost.

3. The Docs Company purchases a component used in the manufacture of automobile generators directly from the supplier.  Docs's generator production operation which is operated at a constant rate, will require 10,000 components per month throughout the year (120,000 units annually).  Assume ordering cost at P2500 per order, unit cost is P40,000 per component, and annual inventory holding costs are charged at 20%.  Answer the following: a) WHat is the EOQ for this componet?; b) What are the total annual inventory holding and ordering costs associated with the recommended EOQ?

4.  Mimi Corporation has an annual sales of 9000 CD's.  Each inventory item has a value of P200.  Ordering cost is P400 per order.  Carrying cost is 25% of average inventory value.  Find the following: a) optimal number of order per year; b) prior order quantity; c) annual carrying and ordering costs; d) total annual cost.

5.  The annual inventory requirement at the Q and A Enterprises is 14400 units.  Price is P60 per unit.  Ordering cost is P800 per order.  Carrying cost is 25% of average inventory.  Find the following a) economic order quantity; b) annual carrying cost; c) annual ordering cost; d) total annual cost.

6.  The K and V Company is panning to stock its product which is classified as an inventory item.  The company has developed the following information: annual usage is 16,200 units, cost of the inventory per unit is P730; ordering cost is P220 per order; carruing cost is 28% of inventory value per year.  Determine the optimum number of units per order.

7.  The Atlantic Paper company produces paper from wood pulp, which it purchases from the Adirondack Lumber Products Company.  Atlantic needs 450,000 pounds of wood pulp per year (365 days) to meet its customers' demand for paper.  Each order of pulp costs Atlantic P700, and it costs 0.30 per pound per year to carry a pound of pulp in inventory.  It takes 8 days for Atlantic to receive an order from the Adirondack Company.  Determine the following: a) economic order quantity; b) minimum total annual inventory cost.

Monday, September 9, 2013


Forecasting is the art and science of predicting future events.  It may involve taking historical data and projecting them into the future with some sort of mathematical model.

The three categories of Forecasting Time Horizons are:
1. Short-range forecast this has a time span of up to one year but is generally less than three months.
2.  Medium-range forecast that generally spans from three months up to three years.
3.  Long-range forecast three years or more is its time span.

Types of Forecasts
1. Economic forecasts  model of this type are valuable in helping business prepare medium-to-long range forecasts.
2.  Technological forecasts are concerned with the rates of technological progress.Such forecasts can be critical in such high technology industries such as nuclear power, aerospace, oil and computing.
3.  Demand forecast is a projection of a company's sales for each time period in the planning horizon.

Quantitative Forecasting Techniques
1. Moving Averages  are useful if we can assume that market demands will stay fairly steady over time.
2.  Weighted Moving Averages when there is a trend or pattern, weights can be used to place more emphasis on recent values.
3.  Exponential Smoothing is a forecasting method that is easy to use and efficiently handled by computers, although it is a type of moving average technique, it involves very little record keeping of past data.
4.  Regression Analysis it is the most common quantitative causal forecasting model.

Practice Exercises
1. The sales at Ed's Catering are shown in the middle column of the following table.  A three-month moving average appears on the right.
Month                             Actual Sales    Three-month moving average Three-month weighted moving average
January                                             15
February                                           18
March                                               21
April                                                 19
May                                                 24
June                                                 23
July                                                  21
August                                             26
September                                       29
October                                           31
November                                       30
December                                       29

2. Period                Actual Demand               Forecast (0.10)                   Forecast (0.40)
1                                      42
2                                      40
3                                      43
4                                      40
5                                      41
6                                      39
7                                      46
8                                      44
9                                      45
10                                    38
11                                    40

3. Year                                      Local Payroll                           Sales
1992                                                 2                                        1
1993                                                 3                                        3
1994                                                 2.5                                     4
1995                                                 2                                        2
1996                                                 2                                        1
1997                                                 3.5                                     7

4. Hours of study                      Exam score
           25                                      93
           12                                      57
           18                                      55
           26                                      90
           19                                      82
            20                                     95
            23                                     95
            15                                     80
            22                                     58
            8                                       61

5.  Sales              Profit
          7                0.15
          2                 0.10
         6                  0.13
         4                  0.15
        14                 0.25
        15                 0.27
        16                 0.24
        12                 0.20
        14                 0.27
        20                  0.44
        15                 0.34
          7                 0.17

6.  Month                  Delivered Orders per month     three-month moving      three-month weighted
January                                     120
February                                   90
March                                       100
April                                          75
May                                           110
June                                            50
July                                             75
August                                        130
September                                  110
October                                       90

7.  Attendance and Promotional Expenditure Data
Year                     Promotional Expenditures                             Home Attendance
1975                                     5.7                                                        7
1976                                     5.5                                                       10
1977                                     6.5                                                         9
1978                                     9.0                                                        12
1979                                    6.9                                                          8
1980                                     8.1                                                         14
1981                                     9.5                                                         15
1982                                    10.2                                                        17
1983                                     8.2                                                         16
1984                                     10.6                                                       18


Thursday, August 15, 2013


Many situations do, in fact, involve several decision makers who compete with one another to arrive at the best outcome.  These types of competitive decision-making situations are the subject of game theory.  Anyone who has played such games as card games or board games is familiar with situations in which competing participants develop plans of action in order to win.  Game theory encompasses similar situations in which competing decision makers develop plans of action in order to win.

Types of Game Situations

Competitive game situations can be subdivided into several categories.  One classification is based on the number of competitive decision makers, called players, involved in the game.  A game situation consisting of two players is referred to as two-person game.  When there are more than two players, the game situation is known as an n-person game.

Games are also classified according to their outcomes in terms of each player's gains and losses.  If the sum of the players' gains and losses equals zero, the game is referred to as zero-sum game.  In a two-person game, one player's gains represent at another's losses.  For example, if one player wins 100, then the other player loses 100; the two values sum to zero.  Alternatively, if the sum of the players' gains and losses does not equal zero, the game is known as a non-zero sum game.

The two-person, zero-sum game is the one most frequently used to demonstrate the principles of game theory because it is the simplest mathetimatically.

Example of competitive situations that can be organized into two-person, zero-sum games include
1. a union negotiating a new contract with management
2. two armies participating in a war game
3. two politicians in conflict over a proposed legislative bill, one attempting to secure its passage and the other attempting to defeat it
4. a retail firm trying to increase its market share with a new product and a competitor attempting to minimize the firm's gains
5. a contractor negotiating with a government agent for a contract on a project

Game Strategies

A strategy is a plan of action to be followed by a player.  Each player in a game has two or more strategies, only one of which is selected for each playing of a game.

1. A Pure Strategy when each player in the game adopts a single strategy as an optimal strategy, then the game is a pure strategy game.  The value of a pure strategy game is the same for both the offensive player and the defensive player.  In contrast, in a mixed strategy game, the players adopt a mixture of strategies if the game is played many times.
A pure strategy game can be solved according to the minimax decision criterion.  According to this principle, each player plays the game in order to minimize the maximum possible losses.  The offensive player will select the strategy with the largest of the minimum payoffs (called the maximin strategy), and the defensive player will select the strategy with the smallest of the maximum payoffs (called the minimax strategy). 

2.  Dominant Strategies. Dominance occurs when all the payoffs for one strategy are better than the corresponding payoffs for another strategy.

3.  A mixed strategy occurs when each player selects an optimal strategy and they do not result in an equilibrium point( the same outcome) when the maximin and mnimax decision criteria are applied.

Practice exercises

1. Payoff Table for Camera Companies
Camera Company 1 Strategies                                  Camera Company 2 Strategies
                                                                           A                      B                       C
1                                                                         9                       7                        2
2                                                                        11                      8                        4
3                                                                         4                       1                        7

2.  Two fast foods chains, MacBurger and Burger Doodle, dominate the fast food market.  MacBurger is currently the market leader, and Burger Doodle has developed three marketing strategies, encompassing advertising and new product lines, to gain a percentage of the market now belonging to MacBurger.  The following payoff table shows the gains for Burger Doodle and the losses for MacBurger given the strategies of each company.

Burger Doodle Strategies                            MacBurger Strategies
                                                                       A                        B                      C
1                                                                      4                        3                      6
2                                                                     -2                        5                      1
3                                                                      3                        2                      4

Determine the mixed strategy for each company and the expected market share gains for Burger Doodle and losses for Macburger.

3.  Consider the following payoff table for a mixed strategy game between two players

Player 1 Strategies                                      Player 2 Strategies
                                                                     A                         B                     C
1                                                                    50                      60                    30
2                                                                    10                       32                   25
3                                                                    20                       55                    4
Determine the mixed strategy for each player and the expected gains and losses  that result.

4.  Given the following payoff table for a mixed strategy game between two players, determine the strategy and the gains and losses for each player.

Player 1 Strategies                              Player 2 Strategies
                                                                  A                    B               C                   D
1                                                                40                 30               20                  80
2                                                                90                 50               60                  65
3                                                                80                 75               52                  90
4                                                                60                 40               35                  50

5.  Consider the following payoff table for two game players.

Player 1 Strategies                                Player 2 Strategies
                                                                 A                   B                 C                  D
1                                                               6                   25                18                 10
2                                                              12                  14                19                 11
3                                                               20                 15                  7                   9
4                                                              15                  30                 21                 16

Tuesday, August 6, 2013


A decision is the act of deciding what single act among all alternatives is to be taken into account.  Successful decision making comprises a number of steps to process:

1.  Clearly define the problem.
2.  Identify the possible alternatives and their outcomes.
3.  List the profit for each combination of alternatives and outcomes.
4.  Select one mathematica; decision theory model.
5.  Apply the model and make your decision.

The success or failure that persons, companies, and management experience depends on the decision they make.  Making use of quantities in decision-making helps a great deal in minimizing mistakes and failures.  Some methods of computation should therefore be learned.

Decision theory is a general approach to decision making that is useful in many different aspects of operations management.  It provides a framework for analysis of decision.  It includes a number of different techniques that can classify according to the degree of uncertainty.

Decision under Certainty - when one knows with certainty which of the possible future conditions will actually happen, he simply chooses the alternative with the highest payoff under the state of nature.

Decision under Uncertainty - from certainty, the opposite extreme is uncertainty where there is no available information on how likely the various states of nature are.  Under this condition, there are four possible decision criterions, namely :

a) Maximin - takes into account only the worst possible outcome for each alternative.  This approach establishes a guaratee minimum.
b) Maximax - takes into account only the best possible outcome for each alternative.
c) Laplace - takes into account only the best average possible outcome for each alternative.
d) Minimax Regret - determines the worst regret for ach alternative, and chooses the alternative with the best worst.


Another useful technique for analyzing a decision situation is a decion tree.  A decision tree, like the probability tree is a graphical diagram consisting of nodes and branches.  However, rather than determining the probability of each branch as in a probability tree, in a decision tree the user computes the expected value of each outcome and makes a decision based on these expected values.  The primary benefit of a decision tree is that it provides an illustration (or picture) of the decision-making process.  This makes it easier to correctly compute the necessary expected values and to understand the process of making the decision.

The decision tree has two types of nodes: a square represents a decision point and a circle stands for a chance event.  The tree is read left to right, and analyzed from right to left (starting with the last  decision that might be made).  For each decision, choose the alternative that will yield the greatest return.  If chance events follow a decision, choose the alternative that has the highest expected value or the lowest expected loss. 

1.  T. Bone Puckett, a corporate raider, has acquired a textile company and is contemplating the future of one of its major plants located in South Carolina.  Three alternative decisions are being considered: (1) expand the plant and produce lightweight, durable materials for possible sales to the military, a market with little foreigh competition; (2) maintain the status quo at the plant, continuing production of textile goods that are subject to heavy foreign competition; or (3) sell the plant now.  If one of the first two alternatives is chosen, the plant will still be sold at the end of a year.  The amount of profit that could be earned by selling the plant in a year depends on foreign market conditions, including the status of a trade embargo bill in Congress.  The following payoff table describes this decision situation.

Decision                                                            States of Nature
                     Good Foreign Competitive Conditions               Poor Foreign Competitive Condition

Expand                            800,000                                                         500,000
Maintain status quo         1,300, 000                                                    -150, 000
Sell now                           320,000                                                         320, 000

a.  Determine the best decision using the following decision criteria.
1. Maximax
2.  Maximin
3.  Minimaz regret
4.  Hurwicz (alpha =0.3)
5.  Equal likelihood

b.  Assume that it is now possible to estimate a probability of 0.70 that good foreign competitive conditions will exist and a probability of 0.30 that poor conditions will exist.  Determine the dest decision using the expected value and expected opportunity loss.

c.  Compute the expected value of perfect information.
d.  Develop a decision tree for this decision situation, with expected values at the probability nodes.

2.  A farmer in Iowa is considering either leasing some extra land or investing in savings certificates at the local bank.  If weather conditions are good next year, the extra land will allow the farmer to have an excellent harvest.  However, if weather conditions are bad, the farmer will lose money.  The savings certificates will result in the same return regardless of the weather conditions.  The return for each investment given each type of weather condition is hown in the following payoff table.

Decision                                                Weather
                                         Good                    Bad
Lease land                        90,000               -40,000
Buy savings certificate    10,000                 10,000

Hurwicz (alpha = 0.4)

3.  The owner of the Burger Doodle Restaurant is considering two ways to expand operations: operning a drive-up windom or serving breakfast.  The increase in profits resulting from these proposed expansions depends on whether a competitor opens a franchise down the street.  The possible profits from each expansion in operations given in both future competitive situations are shown in the following payoff table.

Decision                                         Competitor
                                      Open                        Not Open
Drive-up window           -6,000                       20,000
Breakfast                         4,000                          8,000

Hurwicz (alpha = 0.10)

4.  Stevie Stone, a bellhop at the Royal Sundown Hotel in Atlanta, has been offered a management position.  Although accepting the offer would assure him a job if there were a recession, if good economic conditionsprevailed he would actually make less money as a manager than as a bellhop (because of the large tips he gets as a bellhop).  His salary during the next five years for each job given each economic condition is shown in the following payoff table.

Decision                                        Economic Conditions
                                             Good                   Recession
Bellhop                               120,000                    60,000
Manager                               85,000                     85,000

Hurwicz (alpha = 0.4)

5.  Consider the following payoff table for three alternative investments, A, B, and C, under two future states of the economy, good and bad.

Investment                                    Economic Conditions
                                             Good                   Bad
A                                           70,000               25,000
B                                           120,000            -60,000
C                                            40,000               40,000

Hurwicz (0.3)

6.  A farmer in Georgia must decide which crop to plant next year on his land: corn, peanuts, or soybeans.  The return from each crop will be determined by whether a new trade bill with the USSR passes the Senate.  The profit the farmer will realize from each crop given the two possible results on the trade bill is shown in the following payoff table.

Crop                                           Trade Bill
                                      Pass                 Fail
Corn                            35,000             8,000
Peanuts                       18,000             12,000
Soybeans                     22,000            20,000

Hurwicz (0.3)

7.  The owner of the Columbia Construction Company must decide among building a housing development, constructing a shopping center, or leasing all the company's equipment to another company.  The profit that will result from each alternative will be determined by whether material costs remain stable or increase.  The profit from each alternative given the two possibilities for material costs is shown in the following payoff table.

Decision                                  Material Costs
                                    Stable              Increase
Houses                        70,000              30,000
Shopping center        105,000              20,000
Leasing                       40,000              40,000

Hurwicz (0.2)

8.  An investor is considering investing in stocks, real estate, or bonds under uncertain economic conditions.  The payoff table of returns for the investor's decision situation is shown below.

Investment                                      Economic Conditions
                                       Good                     Stable                   Poor
Stocks                            5,000                     7,000                   3,000
Real estate                     -2,000                   10,000                  6,000
Bonds                             4,000                      4,000                  4,000

Hurwicz (0.3)

9.  A concessions manager at the Tech vs A and M football game must decide whether to have the vendors sell sun visors or umbrellas.  There is a 30% chance if rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast in College Junction, where the game is to be held.  The manager estimates the following profits will result from each decision given each set of weather conditions.

Decision                                           Weather Conditions
                                          Rain                 Overcast                Sunshine
                                           0.30                   0.15                        0.55
Sun visors                           -500                   -200                      1,500
Umbrellas                           2000                     0                         -900

10.  The Blitzkrieg Banking House in Berlin speculates in the money market.  The status of the U.S dollar determines the return from investments in other currencies.  The banking house will invest in the dollar, the yen, or other currencies.  The banking house will invest in the dollar, the yen, or the mark.  The return from each is shown in the following payoff table.

Currency                                   Value of the Dollar
                                         Increases (0.3)               Remains stable(0.50)           Declines(0.20)
Dollar                               210,000                                   0                                  -170,000
Yen                                    -10,000                              20,000                                 80,000
Mark                                 -40,000                              35,000                                 150,000


Sunday, August 4, 2013


The Assignment is a special type of transportation model where the resources (rows) are collected to the destinations (columns) on a one-to-one correspondence so as to minimize cost.  This may refer, in particular, to teams assigned to projects; jobs assigned to workers; machines assigned to laborers; etc.  It should be observed that the number of rows is equal to the number of columns in the Assignment Table to ensure that no row cannot be assigned to a column, or vice-versa on a one-to-one correspondence.

1.  The Atlantic Coast Conference has four basketball games on a particular night.  The conference office wants to assign four teams of officials to the four games in a way that will minimize the total distance traveled by the officials.  The distances in miles for each team of officials to each game location are shown below:

Officials                                        Game Sites
                                     Raleigh               Atlanta                  Durham                 Clemson
A                                       210                     90                          180                      160
B                                       100                     70                          130                      200
C                                       175                     105                        140                      170
D                                         80                       65                        105                     120

2.  A metal parts shop  has three operators and three machines: a drill press, a lathe, and a grinder.  Each operator is qualified to operate each machine.  The following table shows the time ( in minutes) required by each operator to produce a part on each machine.

Operator                                          Machine
                                       Press                      Lathe                  Grinder
1                                       22                           18                           35
2                                       41                           30                           28
3                                       25                           36                           18

Determine the optimal assignment that will minimize the total machine time.

3.  A plant has four operators to be assigned to four machines.  The time (minutes) required by each worker to produce a product on each machine is shown below.

Operator                                        Machine
                                        A                     B                    C                      D
1                                     10                     12                   9                       11
2                                      5                      10                   7                       8
3                                     12                     14                   13                      11
4                                      8                      15                    11                     9

4.  A shop has four machinists to be assigned to four machines.  The hourly cost of having each machine operated by each machinist is as follows.

Machinist                                        Machine
                                        A                    B                        C               D
1                                     12                   11                       8               14
2                                     10                     9                       10             8
3                                     14                    8                         7               11
4                                       6                    8                        10              9

5.  The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions.  Given their various previous contacts, the salespersons are able to cover the regions in different amounts of time.  The amount of time (days) required by each salesperson to cover each city is shown below.

Salesperson                                      Region
                                   A                B                    C                    D               E
1                                17               10                   15                   16             20
2                                12                9                    16                    9              14
3                                11                16                   14                   15            12
4                                14                10                   10                    18            17
5                                13                12                     9                   15             11

6.  The Bunker Manufacturing firm has five employees and six machines, and wants to assign the employees to the machines so as to minimize cost.  A cost table showing the cost incurred by each employee on each machine is given below.

Employee                                      Machine
                                A              B             C             D              E           F
1                              12             7            20            14              8          10
2                              10             14          13            20              9          11
3                               5               3             6             9               7           10
4                               9               11           7             16             9           10
5                              10              6            14             8              10          12

7.  Given the following cost table for an assignment problem, determine the optimal assignment and compute total minimum cost.  Identify all alternative solutions if there are multiple optimal solutions,

Operator                                        Machine
                               A                B               C            D
1                            10                2                 8            6
2                             9                 5                 11          9
3                             12              7                 14          14
4                             3                 1                  4            2

8.  An electronics firm produces electronic components, which it supplies to various electrical manufacturers.  Quality control records indicate that different employees produce different numbers of defective items.  The average number of defects produced by each employee for each of six components is given in the following table.

Employee                                 Component
                             A            B        C        D         E             F
1                           30           24       16      26        30            22
2                           22           28        14     30        20             13
3                           18           16        25     14        12             22
4                           14            22       18      23        21            30
5                           25            18        14      16        16           28
6                           32            14        10      14       18           20

9.  A dispatcher for the Citywide Taxi Company presently has six taxicabs at different locations and five customers who have called for service.  The mileage from each taxi's present location to each customer is shown below.

Cab                                        Customer
                               1               2                3                     4               5
A                          7                2                4                    10              7
B                          5                1                5                      6              6
C                          8                7                6                      5              5
D                          2                5                2                      4              5
E                           3               3                 5                      8              4
F                           6               2                 4                     3               4

10.  The Southwest Athletic Conference has six basketball officials who must be assigned to three conference games, two to each game.  The conference office wants to assign the officials so that the total distance they travel will be minimized.  The distance each official would have to travel to each game is given in the following table:

Official             Game
                           Austin                Houston                Lubbock
1                              20                      45                           10
2                              40                      90                           70
3                               60                     70                            30
4                               30                      60                           40
5                               70                      15                           50
6                               80                      25                           35

11.  A university department head has five instructors to be assigned to four different courses.  All of the instructors have taught the courses in the past and have been evaluated by the students.  The rating for each instructor for each course is given in the following table

Instructor               Course
                                  A           B            C             D
1                               80           75           90           85
2                               95           90           90           97
3                              85            95           88            92
4                               93           91            80           84
5                               91           92            93           88