A local tavern has contracted with 4 companies to distribute three of their brew

Question

A local tavern has contracted with 4 companies to distribute three of their brews, a Milk Stout, IPA, and Wheat. Each company has requested the maximum amount of the three beers (in cases) that they wish to receive. The table below shows this MAXIMUM, which is the most that the brewery will provide to them – they can provide less in some/all cases.

The cost per case is shown in the table below.

The brewery has only 800 cases of Milk Stout, 500 cases of IPA, and 625 cases of Wheat that it can allocate to these new distributors. Therefore, it cannot meet the requested demand by the distributors.

The two constraints for the case are that the requested amount of beer by each distributor and the available beer that the brewery has available to allocate to the distributors. These are maximum limit constraints.

Determine the best way to allocate the beer to distributors based upon the following prioritized order of goals.

Goal 1: Provide at least 75% of the total requested cases (not by product type) of each distributor. Round UP to the nearest whole number for your targets.

Goal 2: Fremont is a preferred distributor. Try to meet its requested allotment of beer by product.

Goal 3: Each distributor must get at least 115 cases of IPA.

Goal 4: Spend no more than $225,000 total with the distributors (combined).

Goal 5: Use up all the milk stout

Fremont Rosebud Flint Hills Panhandle Milk Stout 200 175 300 180 IPA 175 150 175 240 Wheat 200 200 250 130 575 525 725 550

Explanation / Answer

Define decision variables -

Let xij be the number of cases sent of the i-th brew to the j-th distributors. i=1,2,3; j=1,2,3,4

Also, let 'ok' and 'uk' be the overachievement and underachievement of the k-th goal.

Objective Function

Minimum Z = 5(u11+u12+u13+u14) + 4(u21+u22+u23) + 3(u31+u32+u33+u34) + 2o4 + 1u5

Subject to,

Goal 1:

x11 + x21 + x31 + u11 - o11 = 431 (Fremont)
x12 + x22 + x32 + u12 - o12 = 394 (Rosebud)
x13 + x23 + x33 + u13 - o13 = 544 (Flint Hills)
x14 + x24 + x34 + u14 - o14 = 413 (Panhandle)

Goal 2:

x11 + u21 - o21 = 200 (Milk Stout)
x21 + u22 - o22 = 175 (IPA)
x31 + u23 - o23 = 200 (Wheat)

Goal 3:

x21 + u31 - o31 = 115 (Fremont)
x22 + u32 - o32 = 115 (Rosebud)
x23 + u33 - o33 = 115 (Flint Hills)
x24 + u34 - o34 = 115 (Panhandle)

Goal 4:

125x11+115x12+140x13+132x14+98x21+100x22+112x23+110x24+145x31+160x32+172x33+155x34 + u4 - o4 = 225000

Goal 5:

x11 + x12 + x13 + x14 + u5 - o5 = 800

Non-goal constraints:

x11 <= 200; x12 <= 175;......; x34 <= 130
x11 + x12 + x13 + x14 <= 800
x21 + x22 + x23 + x24 <= 500
x31 + x32 + x33 + x34 <= 625

Nonnegativity: xij >= 0; ok >= 0; uk >= 0

--------------------------------------------

Global optimal solution found.
Objective value: 774.3430
Infeasibilities: 0.000000
Total solver iterations: 12
Elapsed runtime seconds: 0.18

Model Class: LP

Total variables: 38
Nonlinear variables: 0
Integer variables: 0

Total constraints: 29
Nonlinear constraints: 0

Total nonzeros: 98
Nonlinear nonzeros: 0

Variable Value Reduced Cost
U11 0.000000 4.784884
U12 0.000000 0.3488372
U13 39.66860 0.000000
U14 0.000000 0.4941860
U21 55.00000 0.000000
U22 60.00000 0.000000
U23 29.00000 0.000000
U31 0.000000 2.622093
U32 0.000000 3.000000
U33 0.000000 3.000000
U34 0.000000 2.563953
O4 0.000000 1.970930
U5 0.000000 0.000000
X11 145.0000 0.000000
X21 115.0000 0.000000
X31 171.0000 0.000000
O11 0.000000 0.2151163
X12 175.0000 0.000000
X22 150.0000 0.000000
X32 69.00000 0.000000
O12 0.000000 4.651163
X13 300.0000 0.000000
X23 120.0000 0.000000
X33 84.33140 0.000000
O13 0.000000 5.000000
X14 180.0000 0.000000
X24 115.0000 0.000000
X34 118.0000 0.000000
O14 0.000000 4.505814
O21 0.000000 4.000000
O22 0.000000 4.000000
O23 0.000000 4.000000
O31 0.000000 0.3779070
O32 35.00000 0.000000
O33 5.000000 0.000000
O34 0.000000 0.4360465
U4 0.000000 0.2906977E-01
O5 0.000000 1.000000

Optimal allocation

Fremont Rosebud Flint Hills Panhandle Milk Stout x11 x12 x13 x14 IPA x21 x22 x23 x24 Wheat x31 x32 x33 x34

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