Darby Case Study
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I. Problem Description
The Darby Company is re-evaluating its current production and distribution system in order to determine whether it is cost-effective or if a different approach should be considered. The company produces meters that measure the consumption of electrical power. Currently, they produce these meters are two locations — El Paso, Texas and San Bernardino, California. The San Bernardino plant is newer, and therefore the technology is more effective, meaning that their cost per unit is $10.00, while the El Paso plant produces at $10.50. However, the El Paso plant has a higher capacity at 30,000 to San Bernardino’s 20,000. Once manufactured, the meters are sent to one of three distribution centers — Ft. Worth, Texas, Santa Fe, New Mexico and Las Vegas. Due to the proximity of El Paso to Ft. Worth, they are only plant to ship to Ft. Worth. The costs associated with each shipment are described in detail in Appendix 2.2A. From these distribution centers, meters are shipped to one of nine customer zones. The Ft. Worth center services Dallas, San Antonio, Wichita and Kansas City, the Santa Fe center services Denver, Salt Lake City, and Phoenix, and the Las Vegas center ships to Los Angeles and San Diego.
The purpose of this report to compare the current distribution system to a distribution system without the aforementioned limitations on which distribution centers are allowed to service a specific area. In order to determine which system would be better, Darby Company has gathered additional information about the costs of shipping to other areas. For example, the Ft. Worth center could also service Denver, in addition to its current zones, Santa Fe could ship to any customer and Las Vegas could ship to Denver, Salt Lake City, and Phoenix, as well as LA and San Diego. In Appendix 2.2B, specific costs of shipping from distribution centers to customers are detailed.
Darby Company is also considering supplying a number of customers directly. From the San Bernardino plant, they would ship direct Los Angeles and San Diego, and from the El Paso plant they would supply direct to San Antonio (exact costs in Appendix 2.2C). In order to determine what will be the most cost effective way to distribution the meters, I will use network mapping and linear programming to minimize costs.
II. Model Description
In order to minimize the total shipping costs in the Darby distribution system, linear programming can be used. In the case of the original shipping plan the model would be as such:
Min 3.2×1 + 2.2×2 + 4.2×3 + 3.9×4 + 1.2×5 + 0.3×6 + 2.1×7 + 3.1×8 + 4.4×9 + 2.7×10 + 4.7×11 + 3.4×12 + 2.1×13 + 2.5×14
In this objective function, each coefficient is equal the cost per unit of shipping on that route. For example, it costs $3.20 per meter to ship on route 1 — which is from the El Paso plant to the Ft. Worth distribution center. For a full description of each route see Appendix 2.1. In addition, the illustration of this distribution strategy can be viewed in Appendix 1.5.
This distribution is subject to the following constraints:
x1 + x2 + x3 ≤ 30,000
x4 + x5 ≤ 20,000
– x1 + x6 + x7 + x8 + x9 = 0
-x2 – x4 + x10 + x11 + x12 = 0
—x3 — x5 + x13 + x14 = 0
x6 =3600
x7 = 4880
x8 = 2130
x9 = 1210
x10 = 6120
x11 = 4830
x12 = 2750
x13 =8580
x14 = 4460
Constraints 1 and 2 are the number of units that the two plants are able to produce. Constraints 3 — 5 are transshipment constraints; they guarantee that the number of units shipped into the distribution center is equal to the number shipped out. Constraints 6 — 14 are the numbers of units demanded at each customer zone and are in place to guarantee that demand is satisfied.
In order to determine the minimized cost of the two other distribution plans, the objection function and constraints must be modified. The additional cost information must be placed in the objective function and the constraints must be modified to reflect the additional distribution centers supplying each area. The new formulas are as follows.
Without distribution limitations:
Min 3.2×1 + 2.2×2 + 4.2×3 + 3.9×4 + 1.2×5 + 0.3×6 + 2.1×7 + 3.1×8 + 4.4×9 + 2.7×10 + 4.7×11 + 3.4×12 + 2.1×13 + 2.5×14 + 6.0×15 + 5.2×16 + 5.4×17 + 4.5×18 + 6.0×19 + 3.3×20 + 2.7×21 + 5.4×22 + 3.3×23 + 2.4×24
S.T. (subject to):
x1 + x2 + x3 ≤ 30,000
x4 + x5 ≤ 20,000
-x1 + x6 + x7 + x8 + x9 + x15 = 0
-x2 – x4 + x10 + x11 + x12 + x16 + x17 + x18 + x19 + x20 + x21 = 0
—x3 — x5 + x13 + x14 + x22 + x23 + x24 = 0
x6 + x16 =3600
x7 + x17 = 4880
x8 + x18 = 2130
x9 + x19 = 1210
x10 + x15 + x22 = 6120
x11 + x23 = 4830
x12 + x24 = 2750
x13 + x20 =8580
x14 + x21 = 4460
Without distribution limitations and supplying direct to Los Angeles, San Diego and San Antonio:
Min 3.2×1 + 2.2×2 + 4.2×3 + 3.9×4 + 1.2×5 + 0.3×6 + 2.1×7 + 3.1×8 + 4.4×9 + 2.7×10 + 4.7×11 + 3.4×12 + 2.1×13 + 2.5×14 + 6.0×15 + 5.2×16 + 5.4×17 + 4.5×18 + 6.0×19 + 3.3×20 + 2.7×21 + 5.4×22 + 3.3×23 + 2.4×24 + 0.3×25 +0.7×26 + 3.5×27
s.t.:
x1 +