UNIVERSITY OF SOUTHAMPTON

MathematicalProgrammingCoursework

MATH6002

1/2/2014

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Table of Contents1. Introduction...................................................12. Case by case and the corresponding solution....................1Case1: Consider a distribution network with two factories, five depots and ten customers. The objective is to minimize the total cost incurred....................................................1i. Define the decision variables...............................1ii. Assumption.................................................1iii. The objective function....................................2iv. The constraints............................................2v. Sensitivity analysis........................................3vi. Diagram of results.........................................3vii. Comment on the results....................................4

Case 2.1: Consider two new depots in Winchester and Salisbury and expanding the existing depot in Oxford while keeping the five initial depots...................................................5Case 2.2: Consider two new depots in Winchester and Salisbury, closing depots in Reading or Bath (or both) and expanding the existing depot in Oxford. The maximum number of depots is 5......5i. Define the decision variables...............................5ii. Assumption.................................................5iii. The objective function....................................5iv. The constraints............................................6v. Table of results............................................7vi. Comment on the results.....................................9

Case 3: Consider operating a distribution network with the participation of many factories, many depots and many customers. 10

3. Conclusion....................................................10APPENDIX

1. IntroductionThis report is conducted with the aim of presenting a designof supply chain distribution network to the companyALPHATRANS. This distribution network is created to apply tothe current state of the company but it is also expected towork in the case the company enlarge its business. To beginwith, ALPHATRANS has two main factories which are located inSouthampton and Basingstoke and five depots, namely Oxford,Reading, Portsmouth, Bournemouth and Bath. The customers ofthe company are domestic customers. They are labelled as R1,R2 …., R10. Normally, the company can distribute theirproducts to customers by one of the two following ways:

+ The products are distributed directly to customers from thefactory.

+ The products are delivered from the factories to the depotsfirst and then sold to the customers.

The requirement for this distribution network is to minimizethe total cost within the business while it also needs to meetthe demand of the customers and the constraints on thecapacity of the factories and depots. Thus, only limitednumber of routes of (factory, depot), (depot, customer) and(factory, customer) are chosen for the distribution networkalthough a factory or a depot can serve many differentcustomers.

In order to solve this problem, a linear program isconstructed via Xpress IVE software to find the optimalsolution. The following part will lay out the possiblesituations in the business activities of ALPHATRANS andprovides the corresponding solution.

2. Case by case and the corresponding solution

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Case1: Consider a distribution network with two factories, five depots and ten customers. The objective is to minimize the total cost incurred.

i. Define the decision variablesLet

ftd_u denote the number of products from the factoryto the depot

dtc_u denote the number of products from the depot tothe customer

ftc_u denote the number of products directly from thefactory to customer

ii. Assumption

The distribution cost per unit of each route of (factory,depot), (depot, customer) or (factory, customer) is fixed. Itmeans that the cost for distributing each unit from factory inSouthampton to depot in Bournemouth, for example, does notdepend on the total number of units distributed via thisroute. Specifically, if the distribution cost from Southamptonto Bournemouth is 0.2 £ per ton, then it costs the companyftd_u*0.2 £ for delivering ftd_u units.(*In practice, the distribution cost per unit might lower when the number of units distributed is highand vice versa).

Another assumption is that the integrality requirement on thevariables is relaxed and LP problem is assumed to produceinteger solution.

iii. The objective function

Minimize the total distribution cost

totalcost≔ftddist+dtcdist+ftcdist

Equation 1

Where:

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ftd_dist: total distribution cost from all factories todepots

dtc_dist: total distribution cost from all depots tocustomers

ftc_dist: total distribution cost from all factories tocustomers

iv. The constraints

*From factory The number of units delivered from factory to depots plus thenumber of units delivered from factory to customers each monthshould not exceed the monthly capacity of each factory. Thisrequirement can be applied to two factories via “FOR ALL”commands

¿(f):=ftmd¿(f)+ftmc¿(f)≤factorysupply(f)

Equation 2

*From depotThe total number of units distributed from the factories to each depot or the total number of units distributed from each depot to the customers does not exceed monthly throughput of each depot. This procedure is repeated for all five depots.

dtmc¿ (d):=∑ (c∈customer)dtcu(c,d)≤depotcap(d)

mftd¿ (d):=∑ (f∈factory)ftdu(f,d)≤depotcap (d)

Equation 3, 4

It is obvious that the total number of units delivered from adepot to the customers cannot be higher than the total numberof units delivered from the factories to that depot.

mftd¿ (d)≥dtmc¿(d)

Equation 5

*From customer

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The number of units delivered from factories to each customerplus the number of units delivered from depots to eachcustomer each month must meet the demand of each customer.This requirement is applied to all ten customers.

mftc¿ (c)+mdtc¿ (c)≥customerdemand(c)

Equation 6

*Non-negativity constraintsThe number of units delivered via any of the routes must behigher than or equal to zero

∀(f∈factory,d∈depot)ftdu(f,d)≥0

∀(d∈depot,c∈customer)dtcu(c,d)≥0

∀(f∈factory,c∈customer)ftcu(f,c)≥0

Equation 7, 8, 9

v. Sensitivity analysisApply the sensitivity analysis on the optimal strategy whenchanging the capacity of the Basingstoke factory by adding twolines of command

Writeln ("Lower possible capacity: ",getrange(XPRS_LOACT,sup(2)))

Writeln ("Higher possible activity: ",getrange(XPRS_UPACT,sup(2)))*sup (2) is the capacity of the Basingstoke factory

vi. Diagram of results

Total cost: 414500

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3.2.1.

1. 2.Basingstok

Figure 1

Figure 2

(Source: Google map)*Black colour circles represent two factories*Blue colour circles represent seven depots

vii. Comment on the results

Among two factories, Basingstoke factory is used to deliverthe products to the depots only while Southampton candistribute the products to both depots and customers. Inregard to the depots, depot in Oxford is not in use and thereason might be the considerably higher distribution cost perunit of this depot in comparison to the others depots. Incontrast, two depots in Bournemouth and Portsmouth are the

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4. 5.

R10R6 R7 R8 R9R1 R2 R3 R4 R5

most frequently used ones. Generally, with the exception ofdepot in Oxford, all the other depots are used with themaximum throughput while the efficiency percentages of twofactories (Southampton and Basingstoke) are 76% and 90%,respectively.

Turning to the sensitivity analysis, the obtained result is asfollow

+Lower possible capacity: 150000

+Higher possible activity: 220000

These extreme values mean that the capacity of Basingstoke canstay within this range while still keeping the same optimalsolution to the original problem. It can also be seen fromthese results that the gap between lower border and the realcapacity of Basingstoke factory (50000) is wider than the gapbetween the higher border and the real capacity of thisfactory (20000).

Case 2.1: Consider opening two new depots inWinchester and Salisbury and expanding the existingdepot in Oxford while keeping the five initialdepots.

Case 2.2: Consider opening two new depots inWinchester and Salisbury, closing depots in Readingor Bath (or both) and expanding the existing depot inOxford. The maximum number of depots is 5.

In these two cases, it is necessary to add some variables andchange some commands in the “The objective function” and “Theconstraints” parts.

i. Define the decision variables

Add two binary variables oc and x oc denote the possibility of opening of each depot oc = 0 the depot is closed

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oc = 1 the depot is opened x denote the possibility of expanding of each

depot x = 0 the depot is not expanded x = 1 the depot is expanded

ii. Assumption

From the case 2.1, the obtained results show that Oxford doesnot contribute to the distribution network but this depot isstill assumed to be kept opening in these two cases (2.1 and2.2).

Another assumption is that it costs ALPHATRANS only when thecompany opens new depots or expand the existing ones. Thecompany does not have to pay for keeping the existing depotsopen. In contrast, the company will save some cost if itcloses one or many depots among the existing ones.

iii. The objective function

Minimize the total cost:

buildingtc:=∑ (d∈depot)oc (d)∗buildingcost(d)

expansiontc:=∑ (d∈depot )x(d)∗expansioncost(d)

saving¿:=∑ (d∈depot )(oc(d)∗saving¿(d)−saving¿(d))

totalcost:=ftddist+dtcdist+ftcdist+buildingtc+expansiontc+saving¿

Equation 10, 11, 12, 13

The two binary variables work as the switches and help thecompany to make the optimal decision (minimizing thetotal_cost). There are three main decisions related tototal_cost:

The decision of opening a new depot:

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If opening a new depot d is the optimal solution, oc (d) willget the value of 1. From the equation 10, it can be seen thatthe total building cost (building_tc) and total_cost will beincreased by an amount of building_cost (d) for each type ofcost. However, this decision is optimal so it is understoodthat the benefits this new depot brings to company exceed itscorresponding added cost. From the equation 12, it is clearthat the contribution of this new depot to the saving_close_tcis zero. This result is logical because the consideringdecision here is about the opening a new depot.

The decision of closing an existing depot:

If closing an existing depot is the optimal solution, oc (d)will get the value of 0. From the equation 12, it can be seenthat this decision helps the company to save some cost.Specifically, the total_cost will be reduced by an amount ofsaving_via_close (d). The saving_via_close (d) cost isbelieved to be higher than the benefits this depot cancontribute to the company if it is still opened. From theequation 10, the contribution of this decision to building_tcis zero. This result is reasonable because the decision ofclosing an existing depot is examining.

The decision of expanding an existing depot

If expanding an existing depot d is the optimal solution, x(d) will get the value of 1. From the equation 11, it is clearthat the total expansion cost (expansion_tc) and total_costwill be increased by an amount of saving_via_close (d) foreach type of cost. However, this decision is optimal so it isexpected that the benefits of the expansion of this depot willexceed its corresponding added cost.

iv. The constraints

Added constraints for both cases∀(d∈depot∨d>1)DOx(d)=0END−DO

Equation 14∀(d∈depot)DO

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oc (d )≥x (d )END−DO

Equation 15

dtmc¿ (d)≤depotcap(d)∗oc(d)+expansioncap(d)∗x(d)

mftd¿ (d)≤depotcap(d)∗oc(d)+expansioncap(d)∗x(d)

Equation 16, 17

Case 2.1 and 2.2 consider whether Oxford depot is expanded soit is reasonable to assume that the value of x (d) of all theother depots are zero (equation 14). Besides, if Oxford depotis opened but it is not expanded, oc (d) will get the value ofone and x (d) will get the value of zero. It can be also seenfrom equation 15 that the case when oc (d) value of a depot iszero and its x (d) get the value of one cannot exist.

From “The objective function” part, oc (d) and x (d) work asthe switches and help the company to adjust the building_tc,expansion_tc and saving_close_tc. In “The constraints” part,these binary variables will help the company to adjust thedistribution cost of products.

Specifically, if closing a depot is the optimal solution, oc(d) and x (d) will get the value of zero. As the result, theright hand sides of equation 16, 17 also get the value ofzero. It means that the routes via this depot do not work sothis depot contributes nothing to the distribution cost andtotal cost.

Added constraint for Case 2.1

oc(1)+oc (2)+oc(3)+oc(4)+oc (5)=5

Equation 18

In this case, ALPHATRANS considers opening two new depots in Winchester and Salisbury but this company intends to keep all the five initial depots open so oc (d) value of these depots should get the value of one.

Added constraint for Case 2.2

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totaldepot:=∑ (d∈depot)oc(d)≤nbdepotsoc(1)+oc (3)+oc(4)=3

Equation 19, 20

In this case, ALPHATRANS can open at most 5 depots (Equation19) so if this company wants to open two new depots inWinchester and Salisbury, it will have to close the existingdepots. Among the first five depots, only Reading or Bath (orboth) can be closed while the other three depots (Oxford,Portsmouth and Bournemouth) must be opened (equation 20).

v. Table of results

*Number of units distributed from factories to depots

ftd_u

Oxford Reading PortsmouthBournemo

uth BathCase2.1

Case2.2

Case2.1

Case2.2

Case2.1

Case2.2

Case

2.1

Case

2.2

Case

2.1

Case2.2

Southampton - - - - - -

20000

40000 - -

Basingstoke - -

50000 -

100000

100000

20000 -

30000

30000

Table 1

ftd_uWinchester Salisbury

Case2.1

Case2.2

Case2.1 Case 2.2

Southampton - - 60000 60000Basingstoke - - - -

Table 2

*Number of units delivered from factories to customers

ftc_u

R1 R2 R3 R4 R5Case2.1

Case2.2

Case2.1

Case2.2

Case

2.1

Case2.2

Case

2.1

Case

2.2

Case

2.1Case2.2

Southampt 50000

50000

- - 40000

40000

- - - -

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onBasingsto

ke - - - - - - - - - -Table 3

ftc_u

R6 R7 R8 R9 R10Case2.1

Case2.2

Case2.1

Case2.2

Case2.1

Case2.2

Case2.1

Case

2.2

Case2.1

Case2.2

Southampton -

20000 -

10000 - - - - -

20000

Basingstoke - - - - - - - - - -

Table 4

*Number of units distributed from the depots to customers

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10OxfordCase 2.1 - - - - - - - - - -Case 2.2 - - - - - - - - - -Reading

Case 2.1 - - - - - - -20000

30000 -

Case 2.2 - - - - - - - - - -Portsmout

h

Case 2.1 - 5000 -35000 - -

40000 - -

20000

Case 2.2 - - -35000 - - - 5000

30000 -

Bournemouth

Case 2.1 - 5000 - - -20000 -

15000 - -

Case 2.2 -10000 - - - -

30000

30000 - -

Bath

Case 2.1 - - - - - - - -30000 -

Case 2.2 - - - - - - - -30000 -

Winchester

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Case 2.1 - - - - - - - - - -Case 2.2 - - - - - - - - - -Salisbury

Case 2.1 - - - -60000 - - - - -

Case 2.2 - - - -60000 - - - - -

Table 5

OX RG PO BH BA SO2 SPTotalcost

Case 1 O O O O O - - 414500Case2.1 O/NE O O O O - O 374500Case2.2 O/NE C O O O - O 410500Case2.3* C O O O O - O 374500

Table 6*OX: Oxford, RG: Reading, PO: Portsmouth, BH: Bournemouth, BA: Bath, SO2: Winchester, SP:Salisbury *O: Open, C: Close, NE: Not expand*Case 2.3 is recommended

vi. Comment on the results

Similar to the Case 1, in both Case 2.1 and 2.2, Basingstokefactory is used mainly for distribution of products to thedepots while Southampton factory can deliver the products toboth depots and customers. Besides, Oxford depot is again notin use whereas Bournemouth and Portsmouth are the mostfrequently used depots. These two depots can be used todistribute the products to many different customers (at leastthree customers for each). Another thing to note is that incase 2.1 and 2.2, the new depot in Salisbury is used to servethe customer R5 only while depot in Bath only delivers theproducts to customer R9.

In both case 2.1 and 2.2, with the exception of Oxford, allthe depots are used with the maximum throughput. Turning tothe factories, the efficiency percentages of two factories inSouthampton and Basingstoke in two cases show contrast

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numbers. In case 2.1, efficiency percentages of two factoriesin Southampton and Basingstoke are 68% and 100%, respectivelywhereas in case 2.2, these numbers are 96% and 65%.

It seems that there will be main factory which contributesmore to the distribution network for each specific case. Thedifference in role of these two factories in two cases can beseen from red highlight cells in these tables above. In Case2.2, the depot in Southampton works more frequently and it candeliver the products to 5 customers in comparison to just 2customers in Case 2.1.

In Case 2.1 when the company can open more than five depots,the optimal solution is to open one more depot in Salisbury.It shows improvement on the result of the total cost when thetotal cost is now 374500 (improvement of 9.6% in comparison tothe Case 1). In Case 2.2, when the number of depot is limitedto five depots and only Reading or Bath (or both) can beclosed, the optimal solution is to close the depot in Readingand open a new depot in Salisbury. However, this solution doesnot show a reasonable improvement in the value of total cost(just a small improvement of 0.96% in comparison to the Case1).

Case 2.3 is a recommended case when opening depot in Oxford isnot compulsory. In this case, depot in Oxford will be closedand five depots in Reading, Portsmouth, Bournemouth, Bath andSalisbury are opened. The total cost for this case is equal tothe that number in Case 2.1 (374500).

Case 3: Consider operating a distribution networkwith the participation of many factories, many depotsand many customers. Specifically, the company intends to set up an optimal networkwith the maximum number of factories, depots and customersbeing 3, 5 and 10 respectively. In regard to the depots,ALPHATRANS can choose 5 depots from a portfolio of 10 depots.The model used for the Case 2.1 and 2.2 can be employed forthis case after a small adjustment of the number of factories

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(nf --> 3) and the number of depots (nd --> 10). The obtainedresults are as follow:

Total cost is 208310

*Number of units delivered from Factory to Depot

Depot 1 Depot 2 Depot 3 Depot 4 Depot 8Factory

1 69862 88056 - - -Factory

2 - - 52533 87761 41234Table 6

*Number of units delivered from Depot to CustomerCustomer1

Customer 2

Customer3

Customer4

Customer 5

Depot 1 25155 8442 - 36265 -Depot 2 - - 39620 - -Depot 3 - - - - -Depot 4 - 28473 - - -Depot 8 - - 15892 - 25342

Customer6

Customer 7

Customer8

Customer9

Customer10

Depot 1 - - - - -Depot 2 - - 48436 - -Depot 3 11723 - - 12976 27834Depot 4 - 45767 - - 13521Depot 8 - - - - -

Table 7 It can be seen these two tables above that there should beonly two factories for the distribution network, namelyfactory 1 and 2. The efficiency percentages of two factoriesare 79.9 % and 98.01 % respectively. Five chosen depots aredepot 1, 2, 3, 4 and 8 respectively. Among five depots, depot1, 2, 4 are used with maximum throughput while the efficiencypercentages of two depots (3 and 8) are 90.24% and 44.62%respectively.

3. ConclusionThe model used for Case 2.1, 2.2 and 3 is a general model sothe company can continue to use it for different problem.

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However, it should bear in mind that the information in thedata file must match with declaration of parameters andvariables in the model to avoid any mistakes. For furtherinvestigation, these assumptions about fixed distribution costper unit or zero keeping cost of existing depots should be re-examined because they somehow do not reflect the real lifesituation exactly.

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