LP8 ASSIGNMENT DISTRIBUTION SYSTEM DESIGN By: Jeffrey L. Blake Course: MT4210 Quantitative Analysis Instructor: Paul Larson Distribution System Design 1. If the company does not change its current distribution strategy, what will its distribution costs be for the follow quarter? Original shipping plan model 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 DISTRIBUTION CONSTRAINTS 1. x1+x2+x3? 30,000 2. x4+x5? 20,000 3. –x1+x6+x7+x8+x9=0 4. –x2-x4+x10+x11+x12=0 5. –x3-x5+x13+x14=0 6. x6=3600 7. x7=4880 8. x8=2130 . x9=1210 10. x10=6120 11. x11=4830 12. x12=2750 13. x13=8580 14. x14=4460 I used this distribution model because the question states that the company in not changing its current distribution strategy. Constraints 1-2 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 number of units demanded at each customer zone and are in place to guarantee the demand is satisfied.
Now to show how to calculate the totals we must set up the model to show no limitations as the problem is saying. To set that up I am going to list the new formulas without 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. 0x15+5. 2×16+5. 4×17+4. 5×18+6. 0x19+3. 3×20+2. 7×21+ 5. 4×22+3. 3×23+2. 4×24 S. T. (subject to): 1. x1+x2+x3? 30,000 2. x4+x5? 20,000 3. –x1+x6+x7+x8+x9+x15=0 4. –x2-x4+x10+x11+x12+x16+x17+x18+x19+x20+x21=0 5. –x3-x5+x13+x14+x22+x23+x24=0 6. x6+x16=3600 7. x7+x17=4880 8. x8+x18=2130 9. x9+x19=1210 0. x10+x15+x22=6120 11. x11+x23=4830 12. x12+x24=2750 13. x13+x20=8580 14. x14+x21=4460 If Darby keeps the current distribution plan, the total cost will be $620,770 for the following quarter. This is computed by taking the information from above and putting it into the objective function which shows the minimized shipping cost to be $194,060. To find the manufacturing cost, we take the values for x1 through x6 and multiple by their respective costs per unit. For example, the values x1, x2 and x3 are the number of items shipped from the El Paso plant, where the manufacturing cost is $10. 0 per unit. These values are 14,520 units to ship to the Ft. Worth distribution center, 13,700 units to ship to the Santa Fe plant and 0 to ship to the Las Vegas distribution plant. In order to ship the meters, they must produce them, therefore the manufacturing cost at the El Paso plant is $152,460 (14520×10. 50) + $148,850 (13,700×10. 50) = $296,310. The San Bernardino plant only has a value for shipping to the Las Vegas distribution center and this value is 13,040 units. Since the cost of manufacturing at that plant is only $10 per unit, their cost is $130,400.
When the cost of manufacturing at each plant is added to the cost of shipping throughout the system the total is $296,310 + $130,400 + $194,060 = $620,770. If the company does not change its current distribution strategy the following quarter the distribution cost for the following quarter will be $620,060. 2. Suppose that the company is willing to consider dropping the distribution center limitations: that is, customers could be served by any of the distribution centers for which costs are available. Can costs be reduced? By how much? Min 3. x1 + 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. 0x15 + 5. 2×16 + 5. 4×17 + 4. 5×18 + 6. 0x19 + 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. : 1. x1 + x2 + x3 + x27 ? 30,000 2. x4 + x5 + x25 + x26 ? 20,000 3. -x1 + x6 + x7 + x8 + x9 + x15 = 0 4. -x2 – x4 + x10 + x11 + x12 + x16 + x17 + x18 + x19 + x20 + x21 = 0 5. –x3 – x5 + x13 + x14 + x22 + x23 + x24 = 0 6. x6 + x16 =3600 7. x7 + x17 + x27 = 4880 8. x8 + x18 = 2130 9. 9 + x19 = 1210 10. x10 + x15 + x22 = 6120 11. x11 + x23 = 4830 12. x12 + x24 = 2750 13. x13 + x20 + x25 =8580 14. x14 + x21 + x26 = 4460 If Darby Company drops the shipping limitations and allows any distribution center to supply to any customer zone for which they have shipping information, their total cost would be reduced to $600,942, which is a savings of $19,828 or 3. 2%. By putting the above information into the objective function, the distribution plan without restrictions reduces the cost of shipping to $177,712 – a savings of $16,888.
In addition, this new plan allows the company produce more of its meters at its more cost efficient San Bernardino plant – in fact, under this plan it produces at capacity, which is 20,000 units, meaning that its cost of manufacturing is $200,000. These units are then shipped to Las Vegas. The El Paso plant produces 14520 units for the Ft. Worth center and only 6740 for the Santa Fe center. The cost of manufacturing these units are $152460 and $70,770 respectively, making the cost at the El Paso plant $223,230.
Adding all these costs together gives us the total cost as such: $177,712 + $200,000 + $223,230 = $600,942. This plan allows more than one center to supply to a specific zone and the solution shows that this is the case for the San Diego customer zone. The Santa Fe center supplies 620 units to them and the Las Vegas center supplies the other 3,840, satisfying their 4460 demanded units. In addition, this plan allows a different center to service customers if it is cheaper. Yes, the costs can be reduced if there were no limitations.
They would be able to reduce it by $19,828 or 3. 2% compared to the model of limitation being used. 3. The company wants to explore the possibility of satisfying some of the customer demand directly from the production plant. In particular, the shipping cost is $0. 30 per unit from San Bernardino to Los Angeles and $0. 70 from San Bernardino to San Diego. The cost for direct shipments from El Paso to San Antonio is $3. 50 per unit. Can distribution cost be further reduced by considering these direct plant to customer shipments?
In order to reduce costs even further, Darby Company could not only drop it’s shipping limitations, but also ship directly to its Los Angeles, San Diego and San Antonio customer zones. By doing this, Darby Company’s total cost will be $553,534 – which is a savings of $47,408, or 7. 9% from the limitation-free distribution system and a savings of $67,236, or 10. 8% from the original distribution system. Darby should definitely adopt this plan of allowing customer zones to be supplied by numerous distribution centers as well as shipping directly the aforementioned customer zones.
Under this system, the San Bernardino plant again operates at full capacity, and ships 6,960 units to the Las Vegas distribution center and as also satisfies the entire demand at the Las Angeles and San Diego, which is 8,580 and 4,460 respectively. Satisfying these demands makes the manufacturing cost at the San Bernardino plant $200,000. Meanwhile, the El Paso plant produces 21,260 units in total, shipping 9,640 units to the Ft. Worth distribution center, 6,740 to the Santa Fe and 4,880 (all of the demand) to San Antonio, making the manufacturing cost $223,230.
These manufacturing costs are the same as those in question #2, but the cost of shipping in this system that reduces the total cost. The shipping cost is only $130,304. This is because it is a great deal cheaper to supply directly to the customers in question, which is usually the case for all direct shipping and especially in this case as San Bernardino is much close to LA and San Diego than Las Vegas or Santa Fe, as well as San Antonio from El Paso rather than Ft.
Worth. 4. Over the next five years, Darby is anticipating moderate growth (5,000 meters) to the north and west. Would you recommend that they consider plant expansion at this time? I would not recommend expanding production capacity for the moderate growth of 5,000 units. First of all, the El Paso plant is producing 8,740 units below their capacity, which could be used to satisfy an increased demand. This increased capacity can then possibly be shipped to the Santa Fe istribution center, which can service any customer zone. The plant capacity should only be expanded if the majority of the demand comes from the west, where it is cheaper to produce and ship from San Bernardino plant. If this demand is focused in the west, the current projects of shipping costs are only valid up to 20,620 units. This means production capacity would have to be expanded. The cost of expanding a plant should of course be weighed against the cost of shipping to determine whether it is worth it.