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Purchasing Examples

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									Purchasing Examples
On each example worksheet, read the comments at the bottom of the sheet, then
click Tools Solver... to examine the decision variables, constraints, and objective.
To find the optimal solution, click the Solve button.
In this series of models we will look at how the Solver can help make decisions about
purchasing goods, awarding contracts, etc. You can view these purchasing models as
allocating a scarce resource -- namely money -- to various uses in an optimal way.

In the contract awards worksheets Award1 and Award2, we have bids from 3 different
suppliers to supply diskettes at different prices to our facilities in 4 different states. We
want to choose from the suppliers' bids in a way that will minimize our total cost. In
Award2, one supplier has specified a minimum size bid for each state.

In the inventory policy worksheets Invent1 and Invent2 we compare the EOQ (Economic
Order Quantity) with the optimal amounts the Solver suggests we should order. These
inventory policy models can also be found in the Finance Examples workbook.

In the Media worksheet, a company wishes to buy advertising at the lowest possible
cost while still reaching a certain target number of prospects. This type of media buying
decision is a common Solver application.

In the Purchase worksheet, we examine a purchasing/transportation problem where a
company can buy goods at several different places and it needs those goods
delivered to several different locations.
                                            189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


Contract Awards 1
A large software company with 4 separate buildings in different states, has offers from 3 different
floppy disk manufacturers to supply their monthly need of new diskettes. To whom should the
contracts be awarded to minimize cost?

Bids per 1000 diskettes
                    Building 1      Building 2   Building 3      Building 4
Manufacturer 1        $50             $45          $48             $52
Manufacturer 2        $52             $48          $51             $54
Manufacturer 3        $49             $51          $50             $52
Contracts awarded per 1000 diskettes
                    Building 1      Building 2   Building 3      Building 4      Total     Available
Manufacturer 1           5              5             5              5            20               25
Manufacturer 2           5              5             5              5            20               30
Manufacturer 3           5              5             5              5            20               25
Total                   15             15            15             15
Required                20             25            15             15
Total Cost          $3,010



Problem
A large software company with 4 different buildings in different states, needs a large supply of
diskettes on a monthly basis in each of those buildings. The company has 3 different offers from
several floppy disk manufacturers. Which offer or combination of offers should the company
accept in order to minimize cost?


Solution
1) The variables are the number of diskettes to buy from each manufacturer. On worksheet
Award1 these are given the name Contracts.
2) The contracts awarded need to meet the demand of the software company and should not
exceed the number of diskettes available from each manufacturer. This gives
                  Contracts_given >= Contracts_required
                  Total_contracts <= Contracts_available
Besides these constraints, we also have the logical constraint
                  Contracts >= 0 via the Assume Non-Negative option
3) The objective is to minimize cost. In Award1 this cell is given the name Total_Cost.


Remarks
Models like the one discussed here are often used by the government. A common example is the
contracts that are awarded to companies to supply fuel for airbases. Normally, we have further
constraints on the bids from each supplier, such as a minimum number of diskettes in this case.
In the Award2 worksheet we will see how to handle such a constraint.




                                                                    Page 2
                                            189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


Contract Awards 2
A large software company with 4 separate buildings in different states, has offers from 3 different
floppy disk manufacturers to supply their monthly need of new diskettes. To whom should the
contracts be awarded to minimize cost?

Bids per 1000 diskettes
                    Building 1     Building 2     Building 3     Building 4
Manufacturer 1         $50            $45           $48            $52
Manufacturer 2         $52            $48           $51            $54
Manufacturer 3         $49            $51           $50            $52
Contracts awarded per 1000 diskettes
                    Building 1     Building 2     Building 3     Building 4        Total     Available
Manufacturer 1           5              5              5              5             20               25
Manufacturer 2           5              5              5              5             20               30
Manufacturer 3           5              5              5              5             20               25
Total                   15             15             15             15
Required                20             25             15             15
Manufacturer 1 is only interested in contracts of 15000 diskettes or more.
Decisions                0              0              0             0
                         0              0              0             0
                         0              0              0             0
Total Cost           $3,010



Problem
A large software company with 4 different buildings in different states, needs a large supply
of diskettes on a monthly basis in each of those buildings. The company has 3 different offers
from several floppy disk manufacturers. However, Manufacturer 1 is only interested in
contracts of 15,000 diskettes or more. Which offer or combination of offers should the
company accept to minimize cost?


Solution
On the surface this problem seems to be no different from the one in Award1. However, we
have the problem that the number of diskettes bought from Manufacturer 1 should either be 0
or greater than 15000. This is a frequently occurring constraint and Award2 shows us how to
handle this type of condition. The key is to introduce 4 new binary integer variables that tell us
whether a contract is bought from manufacturer 1 or not, for each building.


1) The variables are the contracts to be awarded, and the binary integer variables as discussed
above. In worksheet Award2 these are given the names Contracts and Contract_decisions.
2) First, we still have the constraints used in Award1:
                   Contracts_given >= Contracts_required
                   Total_contracts <= Contracts_available
                   Contracts >= 0 via the Assume Non-Negative option
Second, we have the logical constraints for the binary integer variables:
                   Contract_decisions = binary
The 15000 diskettes constraint is now handled by:
                   Awarded_to_1 <= Maximum_diskettes
                   Awarded_to_1 >= Minimum_diskettes
3) The objective is still to minimize total cost, defined on this worksheet as Total_Cost.




                                                                    Page 3
                                           189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


Remarks
The introduction of binary integer variables often allows us to express the effect of more
complex conditions as seen in this model. It would also be possible to handle other types of
constraints. For example, if Manufacturer 2 only distributes diskettes in multiples of 5000,
we could model this constraint with binary integer variables.




                                                                   Page 4
                                             189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


Inventory Policy 1
What is the best ordering policy for a warehouse to minimize cost, while meeting demands?
The warehouse has a limited storage capacity of 50000 cubic meters (m3).
                                                                              Storage
                                  Storage                     Ordering         space
                     Holding     Space per    Demand per      cost per        available
                      Cost       unit (m3)      month          order            (m3)
Product 1             $25          440           200            $50            50000
Product 2             $20          850           325            $50
Product 3             $30         1260           400            $50
Product 4             $15          950           150            $50
Quantity to order each month
                                   EOQ                          Cost        Space used (m3)
Product 1                   25   28.28427                        $713             5500
Product 2                   25   40.31129                        $900            10625
Product 3                   25   36.51484                      $1,175            15750
Product 4                   25   31.62278                        $488            11875
                                                     Total     $3,275            43750



Problem
A warehouse sells 4 products with a different demand for each product. Each product has a different holding cost
and requires a certain amount of space. What should the ordering policy for the warehouse be, given its limited
storage capacity?


Solution
There is an analytical solution for this problem, which is known as the Economic Order Quantity (EOQ) and is
given by the following formula: q = SQRT(2 k d/h), where q is the quantity to order, k is the cost to place an order,
d is the demand and h is the holding cost of the product. Unfortunately, this formula doesn't always work in the real
world. Demand usually fluctuates, ordering time is variable, and other factors arise to further complicate the
problem. In this model we have one such factor, a limited storage space.


1) The variables are the amounts to order each month for each product. These are defined as Quantities in this
worksheet. By changing these variables we change the total cost.
2) The constraints are very simple. We have a logical constraint and the storage capacity constraint. This gives
                 Quantities >= 0 via the Assume Non-Negative option
                 Space_used <= Available_space
If the latter constraint wasn’t present, the solution to the problem could be calculated by the formula given above.
3) The objective is to minimize the total cost, which is defined as Total_cost. It is calculated by adding the
individual costs for each product. Those costs are calculated by using the formula:
Cost = h q /2 + k d /q, where h, q, k and d are as above.
This formula is easy to understand if we realize that the average inventory level is q/2 and the average number of
orders is d/q.


Remarks
In this worksheet we have also calculated the EOQ with the formula given above. Check to see that when you
increase the storage capacity and thus relax that constraint, the answers found by the Solver will approach the
analytic solution.


This model is an example of a non-linear problem, as can easily be seen by looking at the cost formula. Whereas in
linear problems it does not matter what are starting values for the variables are, it can be very important to have




                                                                    Page 5
                                            189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


reasonable starting values in non-linear problems. In this model it is not possible to start with a quantity of 0, since
this would cause an error in the calculation of the cost.


Please see for yourself that the Solver will still find the correct answer, even when the starting values are close
(but not equal to) zero.




                                                                     Page 6
                                           189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


Inventory Policy 2
What is the best ordering policy for a warehouse to minimize cost, while meeting demands?
The warehouse has a limited storage capacity of 50000 cubic meters (m3) and a budget of $30,000.
                                        Storage
                          Holding      Space per       Demand per Ordering cost           Price per
                           Cost        unit (m3)         month      per order                unit
           Product 1       $25             440            200            $50                $200
           Product 2       $20             850            325            $50                $300
           Product 3       $30            1260            400            $50                $275
           Product 4       $15             950            150            $50                $400
Storage Capacity             50000            Budget    $30,000
Quantity to order each month                                         Cost of holding        Space
                                          EOQ                         and ordering        used (m3)
           Product 1           25       28.28427                            $713               5500
           Product 2           25       40.31129                            $900              10625
           Product 3           25       36.51484                          $1,175              15750
           Product 4           25       31.62278                            $488              11875
Cost of products         $29,375                             Total        $3,275              43750



Problem
This model continues to build on the first inventory policy model. We expand the model by giving the warehouse a
budget for buying new products. In other words: A warehouse sells 4 products with a different demand for each
product. Each product has a different holding cost and requires a certain amount of space. What should the ordering
policy for the warehouse be, given its limited storage capacity and limited budget?


Solution
The variables are exactly the same as in the first model. So is the objective, and the way it is calculated. The
difference is that we have an extra constraint which keeps us within the budget. This new constraint is expressed as:
                       Cost_of_products <= Available_money and we also have
                       Space_used <= Available_space as before
We still have Quantities >= 0 via the Assume Non-Negative option. This time, we also require integer quantities:
                       Quantities = integer


Remarks
Once again, we have calculated the EOQs as discussed in the first inventory policy model. If we would give a
unlimited budget and unlimited storage space, the Solver would find exactly those values.


There is one more change we made in this model compared to the one on worksheet Invent1. This time we
required the variables to be integers. Whether this is a valid assumption would depend completely on the type of
product that is dealt with. If a model is trying to determine how many cars, airplanes or other such articles to buy, it
could be very important to use integer variables. If the model, on the other hand, is giving an indication how much
sugar to buy, for example, it would not be appropriate to use integer variables.




                                                                     Page 7
                                           189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


Media Buying
A company wants its advertisements to reach at least 1.5 million people through different media.
There is a maximum number of ad impressions considered effective in each medium. How should
the company advertise to minimize total cost while satisfying the limits on reach and frequency?

Media Requirements
                               TV          Radio           Mail        Newspaper
Audience Size             50,000         25,000          20,000          15,000
Cost / Impression         $500           $200            $250            $125
Max Impressions             20             15              10              15
Investments
                               TV          Radio           Mail        Newspaper             Total
Amount                         $0           $0             $0              $0                $0
Impressions                     0            0              0               0
Audience                        0            0              0               0                 0



Problem
A company wants its advertisements to reach at least 1.5 million people. It is considering advertising
through TV, radio, direct mail, and newspapers. Each medium has a certain cost per run of an ad,
a certain audience that will see the ad, and a maximum number of ad impressions before response
to the ad falls off too much. How should the company advertise in order to reach its target audience
at the lowest possible cost?


Solution
1) The variables are the amounts of money to spend on each medium. In worksheet Media these
are given the name Investments.
2) The constraints are very simple.
                       Investments >= 0 via the Assume Non-Negative option
                       Impressions <= Max_Impressions for each medium
                       Total_Audience >= 1500000
3) The objective is to minimize total cost. In worksheet Media this is defined as Total_investment.


Remarks
Often, there are discounts for placing ads with greater frequency in different media. This could be
expressed in a model with a 'piecewise-linear' constraint, using binary integer variables.




                                                                   Page 8
                                          189d6c8e-9950-49ed-ab8b-f57e9742264b.xls


Purchasing/Transportation Model
A cake mix manufacturer has 4 different plants that all require a certain amount of sugar. There are
5 different companies where the sugar can be bought. Where should the company buy the
sugar and how much should it buy, to minimize cost of sugar and shipping?

                             Company 1      Company 2      Company 3      Company 4       Company 5
Sugar prices (per ton)           $40            $49             $47            $45            $44

Cost of shipping from companies to plants (per ton)
                             Company 1      Company 2      Company 3      Company 4       Company 5
Plant 1                           $8             $4             $5              $4             $3
Plant 2                           $7             $6             $3              $2             $4
Plant 3                           $7             $3             $7              $5             $2
Plant 4                           $8             $2             $5              $6             $7
Amounts of sugar to buy (tons)
                             Company 1      Company 2      Company 3      Company 4       Company 5         Total     Demand
Plant 1                            0              0              0              0               0             0        420
Plant 2                            0              0              0              0               0             0        360
Plant 3                            0              0              0              0               0             0        400
Plant 4                            0              0              0              0               0             0        375
Total                              0              0              0              0               0
Available supply                  350            250            200            300             500
Cost of sugar                     $0             $0             $0              $0             $0            $0
Cost of shipping                  $0             $0             $0              $0             $0            $0
                                                                                          Total cost         $0



Problem
A cake-mix manufacturer has 4 different plants throughout the country. It can buy sugar from 5 different companies.
The cost of the sugar and the transportation costs from each company to each plant are known. Where should the
company buy sugar and how much should it buy, to meet the demand and minimize cost?


Solution
1) The variables are the amounts of sugar to be bought from each company for each plant. On worksheet Purchase
these are given the name Amounts_to_buy.
2) The constraints are simple and straightforward:
                             Amounts_to_buy >= 0 via the Assume Non-Negative option
                             Total_amounts_to_buy >= Demand
                             Total_sold <= Supply
3) The objective is to minimize cost. This is defined as Total_cost on the worksheet.


Remarks
Even though this model is very simple, it is one of the most used models in the industry. It routinely saves many
companies thousands or even millions of dollars a year.




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