Question

(10 points) Develop the algebraic model. Define the decision variables, identify the objective function, and identify all of the constraints.

The exercise is going to be 4 weeks long in the mountains of North Carolina. You have been asked to set up a staging base that the executing units can draw their supplies from. As the sizes of the units participating vary for week to week so do their requirements. The nature of the exercise requires that once each unit departs the staging base they will not return until the end of their maneuvers at the end of the week, as such all of their supplies must be on hand at the start of the week. The two closes bases for you to draw support from are Ft Bragg and Ft Jackson. As each of these bases have their own missions, the capacity they can provide you is limited Atlanta Savannah Supply limit week 1: 700 tons 600 tons Supply limit week 2: 600 tons 800 tons Supply limit week 3: 500 tons 400 tons Supply limit week 4: 600 tons 500 tons You have been assigned a limited number of trucks to transport your supplies. You have been allocated ten trucks with a 34 ton capacity, fifteen trucks with an 18 ton capacity, and twelve trucks with an 11 ton capacity. Due to maintenance requirements each of these trucks can make a maximum of 2 trips a week.

Answer 1

Algebraic model

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Decision variables:

Let Xijk be the number of truck trips of capacity i, supplied from base j in week k,

Vk be the quantity carried over at the end of week k

where i={1,2,3} for truck capacity 34,18,11 tons

j={1,2} for Atlanta,Savannah

k={1,2,3,4} for weeks 1,2,3,4

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Objective function:

Min 360*(X111+X112+X113+X114)+600*(X121+X122+X123+X124)+240*(X211+X212+X213+X214)+390*(X221+X222+X223+X224)+195*(X311+X312+x313+X314)+330*(X321+X322+X323+X324)

(Total cost of delivering supplies over 4 weeks)

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Constraints:

34X111+34X121+18X211+18X221+11X311+11X321-V1 = 1000 (total tonnage required in week 1)

34X112+34X122+18X212+18X222+11X312+11X322-V2+V1 = 1200 (total tonnage required in week 2)

34X113+34X123+18X213+18X223+11X313+11X323-V3+V2 = 1200 (total tonnage required in week 3)

34X114+34X124+18X214+18X224+11X314+11X324-V4+V3 = 1000 (total tonnage required in week 3)

34X111+18X211+11X311 <= 700     (maximum supply from Atlanta in week 1)

34X121+18X221+11X321 <= 600     (maximum supply from Savannah in week 1)

34X112+18X212+11X312 <= 600     (maximum supply from Atlanta in week 2)

34X122+18X222+11X322 <= 800     (maximum supply from Savannah in week 2)

34X113+18X213+11X313 <= 500     (maximum supply from Atlanta in week 3)

34X123+18X223+11X323 <= 400     (maximum supply from Savannah in week 3)

34X114+18X214+11X314 <= 600     (maximum supply from Atlanta in week 4)

34X124+18X224+11X324 <= 500     (maximum supply from Savannah in week 4)

X111+X121 <= 20   (maximum number of trips by 34-ton truck in week 1=10*2)

X211+X221 <= 30   (maximum number of trips by 18-ton truck in week 1=15*2)

X311+X321 <= 24   (maximum number of trips by 11-ton truck in week 1=12*2)

X112+X122 <= 20   (maximum number of trips by 34-ton truck in week 2=10*2)

X212+X222 <= 30   (maximum number of trips by 18-ton truck in week 2=15*2)

X312+X322 <= 24   (maximum number of trips by 11-ton truck in week 2=12*2)

X113+X123 <= 20   (maximum number of trips by 34-ton truck in week 3=10*2)

X213+X223 <= 30   (maximum number of trips by 18-ton truck in week 3=15*2)

X313+X323 <= 24   (maximum number of trips by 11-ton truck in week 3=12*2)

X114+X124 <= 20   (maximum number of trips by 34-ton truck in week 4=10*2)

X214+X224 <= 30   (maximum number of trips by 18-ton truck in week 4=15*2)

X314+X324 <= 24   (maximum number of trips by 11-ton truck in week 4=12*2)

V1, V2, V3, V4 <= 300   (inventory carried over at the end of each week)

All variables >= 0