Capacity Constraint OPRE 6377 Lecture Notes by Metin Çakanyıldırım Compiled at 15:30 on Tuesday 22 nd August, 2017

Transcription

1 apacity onstraint OPRE 6377 Lecture Notes by Metin Çakanyılırım ompile at 5:30 on Tuesay 22 n August, 207 Solve Exercises. [Marginal Opportunity ost of apacity for Deman with onstant Elasticity] We suppose that p) = p for. Let the profit uner capacity b be Πb) = max p {p c)p) : p) b}. We want to obtain the marginal opportunity cost of the capacity, i.e., b Πb). ANSWER From our basic price optimization iscussion, we know that the unconstraine optimal price is p 0 = c. This price generates a eman of p 0 ) = c/ )). If the capacity is more than this price, its marginal opportunity costs is zero: ) Πb) = 0 for b c. b We now suppose that b < c/ )) which implies /b > c/ )). On the other han, the inverse eman function is foun from setting p) = p = b an solving p in terms of b: b) = p = ) /. b Then b Πb) = ) b b b) c = b b b ) / c = = ) ) / c + b b ) / ) c b b ) ) / This marginal opportunity cost is nonnegative because /b > c/ )). It ecreases in the capacity b. It starts with infinity when b = 0 an becomes zero when b = c/ )). We can check the last assertion: b Πb) = c/ )) ) / ) c = 0. b=c/ ))

2 2. onsier selling tickets for UTDallas an UTAustin laies volleyball game to be hel at a staium with seating capacity. Demans for general public an stuents are given by These lea to the following pricing problem g p g ) = a g b g p g an s p s ) = a s b s p s. max p g a g b g p g ) + p s a s b s p s ) a g b g p g + a s b s p s Fin the optimal prices from the optimization problem above. p g a g /b g p s a s /b s. ANSWER The maximization objective above is equivalent to the following objectives max p g a g b g p g ) + p s a s b s p s ) min b g p 2 g a g p g + b s p 2 s a s p s min b g p g a g /2b g )) 2 + b s p s a s /2b s )) 2 Thus, the equivalent optimization problem is min b g p g a g /2b g )) 2 + b s p s a s /2b s )) 2 b g p g + b s p s a g + a s p g a g /b g p s a s /b s. The unconstraine optimal price is a g /2b g ), a s /2b s )) which satisfies both constraints p g a g /b g an p s a s /b s. Then there are two possibilities: Either a g + a s 2 or a g + a s > 2. These cases can be foun from the problem ata wgich specify a g, a s,. If a g + a s 2, we have b g a g /2b g ) + b s a s /2b s ) = a g /2 + a s /2 a g + a s, so the capacity constraint is satisfie at the unconstraine price. In summary, the unconstraine price satifies all theree constraints an it is the constraine optimal as well. If a g + a s > 2, we have b g a g /2b g ) + b s a s /2b s ) = a g /2 + a s /2 < a g + a s, so the capacity constraint is not satisfie at the unconstraine price. The optimal constraine price will use up all the available capacity: b g p g + b s p s = a g + a s. In aition to the last equality, we obtain another equality by consiering the graient of the objective Πp g, p s ) = b g p g a g /2b g )) 2 + b s p s a s /2b s )) 2 : [ ] Π Π Π =, = [2b g p g a g /2b g )), 2b s p s a s /2b s ))]. p g p s 2

3 At the optimal price, the garient must be perpenicular to the slope of the capacity constraint which can be represente by either [b s ; b g ] or [ b s ; b g ] in coorinate system where x-axis is p g an y axis is p s. Note that while computing the graient, we have treate p g as the first variable of the function Π. The consistency of the first an secon variables is important. The perpenicular vectors must give zero when they are multiplie: [2b g p g a g /2b g )), 2b s p s a s /2b s ))][b s ; b g ] = 2b g b s p g a g /2b g )) 2b s b g p s a s /2b s )) This implies the optimality equation = [2b g b s ][p g a g /2b g ) p s + a s /2b s )] = 0. p g p s = a g /2b g ) a s /2b s ). When we intersect this equation with the constraint b g p g + b s p s = a g + a s, we obtain the optimal prices p g = {a g + b ) s + a } s b g + b s 2b g 2 p s = {a s + b ) g + a } g b g + b s 2b s 2 We summarize the results: a g + a s 2 = a g + a s > 2 = { pg = a g /2b g ) p s = a s /2b s ) }, { ) } p g = b g +b s a g + b s 2b g + a s 2 { ) } p s = b g +b s a s + b g 2b s + a g 2. 2 Homework Questions. The cost of an orinary haircut is c = $4. On the other han, the eman can vary a lot with the price. The eman function of regular customers is r p) = 50 5p for 0 p 30. If the price rops below $0, opportunistic customers also consier having a haircut an their eman function is o p) = p for 0 p 0. a) What is the total eman function p) for 0 p 30? b) Fin the optimal price for the haircut. c) ompute the eman corresponing to the optimal price an check that it is above the capacity of b = 60. Fin a new price that maximizes the profit an yiels a eman that can be ealt with the capacity of b = 60. ) How oes the price change if we expan the capacity to b = 80. e) How oes the price change if we expan the capacity to b = For the haircut eman function p) efine in the last exercise, obtain the marginal opportunity cost of capacity b for 0 b 400. Draw this cost as function of capacity b for 0 b onsier selling tickets for a concert at UTD Theater which has a seating capacity of 400. Demans for general public an stuents are given by g p g ) = 400 0p g an s p s ) = p s. What shoul be the optimal ticket prices for general public an stuents? 3

4 4. onsier selling tickets for a concert at UTD Theater which has a seating capacity of 200. Demans for general public an stuents are given by g p g ) = 400 0p g an s p s ) = p s. What shoul be the optimal ticket prices for general public an stuents? 5. onsier the 3-perio variable pricing formulation for a system constraine by 0 an for the parameter s 0: st. x i D i m i p i + sp j p i ) + sp k p i ) for i = j = k an i, j, k {, 2, 3}, x i for i {, 2, 3}, a) Besies the formulation above, we are presente with another one: st. x D m p sp p 2 ) sp p 3 ), x 2 D 2 m 2 p 2 + sp p 2 ) sp 2 p 3 ), x 3 D 3 m 3 p 3 + sp p 3 ) + sp 2 p 3 ), Determine if this formulation is written uner the conition p 3 p 2 p or p 2 p p 3 or p p 2 p 3 or no conition on prices, explain. b) For paramters s, r, t 0, the formulation in a) is moifie as: st. x D m p sp p 2 ) rp p 3 ), x 2 D 2 m 2 p 2 + sp p 2 ) tp 2 p 3 ), x 3 D 3 m 3 p 3 + rp p 3 ) + tp 2 p 3 ), Express parameters s an r in English. c) For paramters s, s 2 0, the formulation in a) is moifie as: st. x D m p s p p 2 ) s 2 p p 3 ), x 2 D 2 m 2 p 2 + s p p 2 ) s p 2 p 3 ), x 3 D 3 m 3 p 3 + s 2 p p 3 ) + s p 2 p 3 ), Express parameters s an s 2 in English. 6. The art museum in your town is close on Monays an charges three ifferent entrance fees. The fee p on Tuesays an Wenesays is the lowest. The fee on p 3 on weekens is the highest. The fee on p 2 charge on Thursays an Friays satisfies p < p 2 < p 3. The museum has 3 two-ay perios every week: ={Tue, We}, 2={Thu, Fri}, 3={Sat, Sun} an we use inices j, k {, 2, 3} to enote perios. Recall that s in general represents the number of customers switching between perios which have a price ifferential of one monetary unit. The same switching behavior from origin perio j to estination perio k is expecte This question eals with elicitation/parameterization of is)utility of visiting a museum on alternative ays. Note that a isutility can be compensate by an appropriate incentive. 4

5 from museum patrons an can be moelle with Origin-estination epenent, non-symmetric: s jk that epens on the origin perio j an estination perio k an s jk = s kj ; Origin-estination epenent, symmetric: s jk that epens on the origin perio j an estination perio k an s jk = s kj ; Only origin epenent: s j that epens on the origin perio j but oes not epen on estination perio k; Only estination epenent: s k that epens on the estination perio k but oes not epen on origin perio j; Only switching length epenent: s m that epens on the number m of perios between the origin an estination perios; onstant: s that is a constant inepenent of perios. Decie an iscuss which one of the six switching incentive parameterizations is more appropriate for patrons. You shoul focus on your own scheule. For example, if you plan to go to the museum on a weeken perio 3) but offere two incentives to go in perio an perio 2, shoul the incentives be the same or ifferent? If you say the same, we have s 3 = s 32 ; otherwise s 3 = s 32. If, in aition to s 3 = s 32, you claim s 2 = s 3 an s 2 = s 23, you require only origin epenent incentives. For another example, if you plan to go to the museum in perio 2 or 3 but offere two incentives to go in perio, shoul the incentives be the same or ifferent? If you say the same, we have s 2 = s 3 ; otherwise s 2 = s 3. If, in aition to s 2 = s 3, you claim s 2 = s 32 an s 3 = s 23, you require only estination epenent incentives. If you require the same incentive to switch from perio 3 to perio or vice versa from perio to perio 3), you have s 3 = s 3. If, in aition to s 3 = s 3, you claim s 2 = s 2 an s 23 = s 32, you require symmetric incentives. If, in aition to symmetric incentives, you claim s 2 = s 2 = s 23 = s 32 for switching one perio, you can set only switching length epenent incentives s an s 2 as s = s 2 = s 2 = s 23 = s 32 an s 2 = s 3 = s 3. The four parameterizations liste above are intermeiate with respect to the two extreme cases: originestination epenent, non-symmetric incentives an constant incentives. In the former case, there is no equality between s jk pairs; In the latter case, all s jk are equal to each other an hence they can be enote by a single number s. 7. A permanent collection remains in the museum for many weeks unlike a temporary one a collection that tours museums) that may be there for just one week. Woul your answer about incentives change when consiering a temporary collection as oppose to a permanent collection in a museum? 8. Suppose that the switching behavior from perio j to perio k is moelle with s jk that epens on the perio j an perio k an s jk = s kj. Moifying the 3-perio variable pricing formulation, we obtain. st. x D m p + s 2 p 2 p ) + + s 3 p 3 p ) + s 2 p p 2 ) + s 3 p p 3 ) +, x 2 D 2 m 2 p 2 s 2 p 2 p ) + + s 2 p p 2 ) + + s 32 p 3 p 2 ) + s 23 p 2 p 3 ) +, x 3 D 3 m 3 p 3 +???, a) Is the constraint for the secon perio correct? b) omplete the right-han sie of the constraint for the thir perio. 9. If you are a PhD stuent, you may attempt to prove that the optimal solutions with an without constraint x i 0 are the same. In other wors, the optimal solution cannot have negative x i even without x i 0 constraint. 5